Recent zbMATH articles in MSC 20https://zbmath.org/atom/cc/202021-01-08T12:24:00+00:00WerkzeugCompactness in singular cardinals revisited.https://zbmath.org/1449.030152021-01-08T12:24:00+00:00"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: This is the second combinatorial proof of the compactness theorem for singular from 1977. In fact it gives a somewhat stronger theorem.Involutory Latin quandles of order \(pq\).https://zbmath.org/1449.200662021-01-08T12:24:00+00:00"Jedlička, Přemysl"https://zbmath.org/authors/?q=ai:jedlicka.premyslSummary: We present a construction of a family of involutory latin quandles, a family that contains all non-Alexander involutory latin quandles of order \(pq\), for \(p < q\) odd primes. Such quandles exist if and only if \(p\) divides \(q^2-1\).Commuting conjugacy class graph of finite CA-groups.https://zbmath.org/1449.200062021-01-08T12:24:00+00:00"Salahshour, Mohammad Ali"https://zbmath.org/authors/?q=ai:salahshour.mohammad-ali"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: Let \(G\) be a finite nonabelian group. The commuting conjugacy class graph \(\Gamma(G)\) is a simple graph with the noncentral conjugacy classes of \(G\) as its vertex set and two distinct vertices \(X\) and \(Y\) in \(\Gamma(G)\) are adjacent if and only if there are \(x \in X\) and \(y \in Y\) with this property that \(xy = yx\). The aim of this paper is to obtain the structure of the commuting conjugacy class graph of finite CA-groups. It is proved that this graph is a union of some complete graphs. The commuting conjugacy class graph of certain groups are also computed.(\(U,V\))-interval-valued fuzzy bi-ideals in semigroups.https://zbmath.org/1449.200782021-01-08T12:24:00+00:00"Wang, Fengxiao"https://zbmath.org/authors/?q=ai:wang.fengxiaoSummary: The concept of (\(U, V\))-interval-valued fuzzy bi-ideals was introduced in semigroups, and its basic properties were discussed. The author gave the relations among (\(U, V\))-interval-valued fuzzy bi-ideals, fuzzy bi-ideals and interval-valued fuzzy bi-ideals of semigroups, proved that the intersection of (\(U, V\))-interval-valued fuzzy bi-ideals was still (\(U, V\))-interval-valued fuzzy bi-ideals of semigroups, and discussed the correlation properties of direct product of (\(U, V\))-interval-valued fuzzy bi-ideals of semigroups.Rank and relative rank of the semigroup \(\mathcal{OPD} (n,r)\).https://zbmath.org/1449.200482021-01-08T12:24:00+00:00"Li, Xiaomin"https://zbmath.org/authors/?q=ai:li.xiaomin"Luo, Yonggui"https://zbmath.org/authors/?q=ai:luo.yonggui"Zhao, Ping"https://zbmath.org/authors/?q=ai:zhao.pingSummary: Let \(\mathcal{OPD}_n\) be the semigroup of all order-preserving and distance-preserving partial one-to-one singular transformations on a finite-chain \([n]\) (\(n\geq 3\)), and let \(\mathcal{OPD} (n, r) = \{\alpha \in \mathcal{OPD}_n:|{\mathrm{Im}} (\alpha)| \le r\}\) be the two-sided star ideal of the semigroup \(\mathcal{OPD}_n\) for an arbitrary integer \(r\) such that \({0 \le r \le n-1}\). By analyzing the elements of rank \(r\) and star Green's relations, the minimal generating set and rank of the semigroup \(\mathcal{OPD} (n, r)\) are obtained, respectively. Furthermore, the relative rank of the semigroup \(\mathcal{OPD} (n, r)\) with respect to its star ideal \(\mathcal{OPD} (n, l)\) is determined for \(0 \le l \le r\).The rank of semigroup of similar complete partial order-preserving transformations \(\mathcal{H} ({\mathcal{SPO}_n},r)\).https://zbmath.org/1449.200532021-01-08T12:24:00+00:00"Yan, Huaying"https://zbmath.org/authors/?q=ai:yan.huaying"You, Taijie"https://zbmath.org/authors/?q=ai:you.taijie"Liu, Bing"https://zbmath.org/authors/?q=ai:liu.bing|liu.bing.1Summary: Let \(n \ge 5\), \({X_n} =\{1,2,\ldots, n\}\), we give the order of magnitude of the natural numbers. Let \(\mathcal{H} ({\mathcal{SPO}_n}, r) = {\mathcal{SPO}_n}\cup\mathcal{L} (n,r) (5 \le n, 2 \le r \le n - 3)\), \(\mathcal{H} ({\mathcal{SPO}_n}, r)\) is called the semigroup of complete similar partial order-preserving transformations on \({X_n}\). In this paper, the rank of the semigroup \(\mathcal{H} ({\mathcal{SPO}_n}, r)\) is proved to be \(\begin{pmatrix}n-1\\ r-1\end{pmatrix}+2n-2\).Local properties of \(p\)-subgroups and \(p\)-nilpotency of finite groups.https://zbmath.org/1449.200162021-01-08T12:24:00+00:00"Han, Lingling"https://zbmath.org/authors/?q=ai:han.lingling"Guo, Xiuyun"https://zbmath.org/authors/?q=ai:guo.xiuyunSummary: Let \(G\) be a finite group and \(P\) be a Sylow \(p\)-subgroup of \(G\). Several sufficient conditions for a finite group \(G\) to be \(p\)-nilpotent group are given by means of some kinds of commutativity of a special \(p\)-subgroup \(\Omega (P \cap {O^p} (G))\) in \(G\).Finite groups with \(X\)-\(ss\)-semipermutable subgroups of order \({P^2}\).https://zbmath.org/1449.200192021-01-08T12:24:00+00:00"Xie, Fengyan"https://zbmath.org/authors/?q=ai:xie.fengyanSummary: Let \(G\) be a finite group and \(X\) be a nonempty subset of \(G\). A subgroup \(H\) of \(G\) is said to be \(X\)-\(ss\)-semipermutable in \(G\) if \(H\) has a supplement \(T\) in \(G\), such that \(H\) is \(X\)-permutable with every Sylow \(p\)-subgroups of \(T\) with \( (p, |H|) = 1\). By investigating \(X\)-\(ss\)-semipermutability of some subgroups of order \({P^2}\), some criteria of \(p\)-nilpotency of finite groups are obtained.The influence of some embedded subgroups on the structure of finite groups.https://zbmath.org/1449.200202021-01-08T12:24:00+00:00"Bao, Hongwei"https://zbmath.org/authors/?q=ai:bao.hongwei"Zhang, Jia"https://zbmath.org/authors/?q=ai:zhang.jia"Li, Decai"https://zbmath.org/authors/?q=ai:li.decaiSummary: Using \(\mathcal{M}\)-supplemented subgroups property in normalizer of the given order subgroup of Sylow subgroups, we study the structure of finite groups with nearly \(m\)-embedded subgroups in \(\mathcal{H} (P)\). Some sufficient conditions for \(p\)-nilpotent groups and supersolvable groups are obtained, and the structure of generalized hypercentre is further discussed.Number of homomorphisms between groups with Abel direct factors and finite groups.https://zbmath.org/1449.200392021-01-08T12:24:00+00:00"Li, Ningying"https://zbmath.org/authors/?q=ai:li.ningying"Guo, Jidong"https://zbmath.org/authors/?q=ai:guo.jidong"Hai, Jinke"https://zbmath.org/authors/?q=ai:hai.jinkeSummary: Let \(H\), \(G\) be finite group. If the subgroup \(A\) of \(H\) is an Abel direct factor of \(H\), then the number of homomorphism from \(H\) to \(G\) is a multiple of the largest common factor of \(|A|\) and \(|G|\). The famous T. Yoshida's theorem is generalized.Maximal subsemigroup of the semigroup \(PC{S_n}\).https://zbmath.org/1449.200512021-01-08T12:24:00+00:00"Li, Yalei"https://zbmath.org/authors/?q=ai:li.yalei"Xu, Bo"https://zbmath.org/authors/?q=ai:xu.bo"Dai, Xiansheng"https://zbmath.org/authors/?q=ai:dai.xianshengSummary: Let \({C_n}\) be the cyclic group on \({X_n}\), \(S{P_n} = {P_n}\backslash {S_n}\) be the partial singular transformation semigroup on \({X_n}\). By analyzing the elements of the transformation semigroup \(PC{S_n} = {C_n} \cup S{P_n}\), the classification of the maximal subsemigroup of the transformation semigroup \(PC{S_n}\) is obtained.The number of conjugacy classes of non-abelian subgroups in finite groups.https://zbmath.org/1449.200292021-01-08T12:24:00+00:00"Yang, Guifang"https://zbmath.org/authors/?q=ai:yang.guifang"Meng, Wei"https://zbmath.org/authors/?q=ai:meng.wei"Lu, Jiakuan"https://zbmath.org/authors/?q=ai:lu.jiakuanSummary: Let \(G\) be a finite group and \(\tau (G)\) denote the number of conjugacy classes of all non-abelian subgroups of \(G\). The symbol \(|\pi (G)|\) denotes the set of the prime divisors of \(|G|\). This paper investigates the properties of finite groups with \(\tau (G) = |\pi (G)| + 1\). It is proved that \(\tau (G) = |\pi (G)| + 1\) implies \(|\pi (G)|\le 3\).Interval value \(Q\)-fuzzy subsemigroup of semigroups.https://zbmath.org/1449.200792021-01-08T12:24:00+00:00"Wang, Fengxiao"https://zbmath.org/authors/?q=ai:wang.fengxiaoSummary: Fuzzy subsemigroups are an important research field of semigroup fuzzy theory. In this paper, the concept of interval value \(Q\)-fuzzy subsemigroups of semigroups is introduced, and the related properties of interval value \(Q\)-fuzzy subsemigroups of semigroups are discussed. The relations among interval value \(Q\)-fuzzy subsemigroups and fuzzy subsemigroups and interval value fuzzy subsemigroups are given. It is proved that the intersections and homomorphisms of interval value \(Q\)-fuzzy subsemigroups of semigroups are still interval value \(Q\)-fuzzy subsemigroups.On strongly \(H_v\)-groups.https://zbmath.org/1449.200672021-01-08T12:24:00+00:00"Arabpour, Fatemeh"https://zbmath.org/authors/?q=ai:arabpour.fatemeh"Jafarpour, Morteza"https://zbmath.org/authors/?q=ai:jafarpour.mortezaSummary: The largest class of hyperstructures is the one which satisfies the weak properties; these are called \(H_v\)-structures. In this paper, we introduce a special product of elements in \(H_v\)-group \(H\) and define a new class of \(H_v\)-groups called strongly \(H_v\)-groups. Then, we show that in strongly \(H_v\)-groups \(\beta=\beta^{\ast}\). Also, we express \(\theta\)-hyperoperation and investigate some of its properties in connection with strongly \(H_v\)-groups.On the residual of a finite group with semi-subnormal subgroups.https://zbmath.org/1449.200152021-01-08T12:24:00+00:00"Trofimuk, Alexander"https://zbmath.org/authors/?q=ai:trofimuk.alexander-aA subgroup \(A\) of a group \(G\) is called \textit{seminormal} if there exists a subgroup \(B\) of \(G\) such that \(G=AB\) and \(AX=XA\) for all subgroups \(X\) of \(B\). Moreover, the subgroup \(A\) is said to be \textit{subseminormal} if it is either subnormal or seminormal.
In the paper under review, the author proves the following interesting result. Let \(\mathcal{F}\) be a saturated formation containing all finite supersoluble groups, and let the finite group \(G=AB\) be the product of two semisubnormal subgroups \(A\) and \(B\). If \(A\) and \(B\) belong to \(\mathcal{F}\), then the \(\mathcal{F}\)-residual of \(G\) is contained in the residual of \(G'\) with respect to the formation of finite nilpotent groups.
Reviewer: Francesco de Giovanni (Napoli)Simplicial volume with \(\mathbb{F}_p\)-coefficients.https://zbmath.org/1449.570082021-01-08T12:24:00+00:00"Löh, Clara"https://zbmath.org/authors/?q=ai:loh.claraThe classical simplicial volume \(||\cdot||_\mathbb{R}\) of an oriented closed connected manifold, introduced by \textit{M. Gromov} [Publ. Math., Inst. Hautes Étud. Sci. 56, 5--99 (1982; Zbl 0516.53046)], is defined in terms of the \(l^1\)-norm on the singular chain complex with real coefficients. Ignoring weights the author studies a simplicial volume \(|| \cdot ||_{(R)}\) for any subring \(R \subseteq \mathbb{R}\). This leads to an \(\mathbb{F}_p\)-version \(||\cdot||_{(\mathbb{F}_p)}\). The author discusses relations between such volumes (varying coefficients) and Betti numbers. The following questions are addressed:
What is the topological/geometric meaning of these invariants?
How do these invariants relate to other topological invariants?
(How) does the choice of the prime \(p\) effect the simplicial volume?
Do these invariants relate to the weightless \(\mathbb{Q}\)-simplicial volume?
What happens if \(\mathbb{F}_p\)-simplicial volumes are stabilized along a tower of finite coverings?
Reviewer: Karl Heinz Dovermann (Honolulu)On varietal fuzzy subgroups.https://zbmath.org/1449.200742021-01-08T12:24:00+00:00"Javadi, A."https://zbmath.org/authors/?q=ai:javadi.ardalan|javadi.akbar-a|javadi.ali"Gholami, A."https://zbmath.org/authors/?q=ai:gholami.atena|gholami.asghar|gholami.a-h|gholami.azam|gholami.amir|gholami.ahmad|gholami.amin|gholami.ali|gholami.aref|gholami.amirhoseinSummary: Varieties of groups and fuzzy subgroups are two important concepts in mathematics. In this paper, after presenting concepts of verbal and marginal fuzzy subgroups and discussing some of their most important characteristics of these concepts, we define variety of fuzzy subgroups and study the complete structure of this. At the end, we devote isologism of fuzzy subgroups.The influence of weakly \(M\)-supplemented subgroups on composition factors of finite groups.https://zbmath.org/1449.200222021-01-08T12:24:00+00:00"Gao, Baijun"https://zbmath.org/authors/?q=ai:gao.baijun"Zhang, Jia"https://zbmath.org/authors/?q=ai:zhang.jia"Zhu, Zhenyang"https://zbmath.org/authors/?q=ai:zhu.zhenyangSummary: Let \(G\) be a finite group and \(P\) be a Sylow \(p\)-subgroup of \(G\), where \(P\) has a subgroup \(D\) with \(1 < D \le P\). In this paper, considering prime divisors 5 and 7 of \(|G|\), we investigate the structure of the composition factors of \(G\) by using the weakly \(M\)-supplemented property of every subgroups \(H\) of \(P\) with \(|H| = |D|\).Characterization of pomonoids by weakly pullback flat \(S\)-posets.https://zbmath.org/1449.060232021-01-08T12:24:00+00:00"Liang, Xingliang"https://zbmath.org/authors/?q=ai:liang.xingliang"Long, Bin"https://zbmath.org/authors/?q=ai:long.bin"Xu, Panpan"https://zbmath.org/authors/?q=ai:xu.panpanSummary: Let \(S\) be a pomonoid. By using the theory of modules over rings and the theory of \(S\)-acts over semigroups, the weakly pullback flat \(S\)-posets in the category of \(S\)-posets are investigated. Pomonoids over weakly pullback flatness of \(S\)-posets are preserved under direct products, and pomonoids over weakly pullback flatness coincides with other flatness for \(S\)-posets are characterized. Moreover, some conditions under which a cyclic \(S\)-poset has a weakly pullback flat cover are discussed, and some important results on weakly pullback flat right \(S\)-acts are extended.Complex fuzzy and generalized complex fuzzy subpolygroups of a polygroup.https://zbmath.org/1449.200712021-01-08T12:24:00+00:00"Al-Tahan, M."https://zbmath.org/authors/?q=ai:tahan.m-al|al-tahan.madline-ali"Davvaz, B."https://zbmath.org/authors/?q=ai:davvaz.bijn|davvaz.bijanSummary: The objective of this paper is to combine the innovative concept of complex fuzzy sets and polygroups. We introduce the concepts of complex fuzzy subpolyrgroups and generalized complex fuzzy subpolyrgroups of a polygroup. We provide some examples and properties of them.On certain \(\mathfrak{F}\)-perfect groups.https://zbmath.org/1449.200332021-01-08T12:24:00+00:00"Arikan, Aynur"https://zbmath.org/authors/?q=ai:arikan.aynurIf \(\mathfrak{X}\) is a class of groups, a group \(G\) is called ``minimal non-\(\mathfrak{X}\)'' (or a ``\(\mathrm{MN}\mathfrak{X}\)-group'') if it is not an \(\mathfrak{X}\)-group but all its proper subgroups belong to \(\mathfrak{X}\). The structure of \(\mathrm{MN}\mathfrak{X}\)-groups has been investigated for many relevant choices of the group class \(\mathfrak{X}\). The paper under review deals with \(\mathrm{MN}\mathfrak{S}\)-groups, where \(\mathfrak{S}\) is the class of soluble groups. Although it is still unknown if primary locally finite \(\mathrm{MN}\mathfrak{S}\)-groups exist, in recent years several restrictions on their structure have been obtained. Here, the author provides new information on perfect primary locally finite \(\mathrm{MN}\mathfrak{S}\)-groups.
Reviewer: Francesco de Giovanni (Napoli)A note on tetravalent \(s\)-arc-regular Cayley graphs of finite simple groups.https://zbmath.org/1449.200012021-01-08T12:24:00+00:00"Ling, Bo"https://zbmath.org/authors/?q=ai:ling.boSummary: A Cayley graph \(\Gamma=\mathrm{Cay}(G,S)\) is said to be normal if \(G\) is normal in \(\Aut\Gamma\), otherwise, \(\Gamma\) is called non-normal. By earlier work of \textit{J. J. Li} et al. [J. Algebra Appl. 16, No. 10, Article ID 1750195, 17 p. (2017; Zbl 1404.20001)], the only possibilities for non-normal connected tetravalent \(s\)-arc-regular Cayley graphs of finite simple groups must arise from one of two graphs of the alternating group \(A_{35}\). In this paper, we prove that the full automorphism groups of these two graphs are isomorphic to \(A_{36}\) and show that these two graphs are not isomorphic, and so this paper completes the classification of nonnormal connected tetravalent \(s\)-arc-regular Cayley graphs of finite simple groups.Hyperelliptic curves and arithmetic functions.https://zbmath.org/1449.110742021-01-08T12:24:00+00:00"Davis, Simon"https://zbmath.org/authors/?q=ai:davis.simon-brianSummary: The precise form of the correspondence between Dirichlet series, modular forms and Riemann surfaces is given. An upper bound for the prime \(p\) following from maximum value of the number of points of an unramified algebraic curve of finite genus over \(\mathbb{F}_q\) is derived. The rational characters of absolute invariants of arithmetic subgroups of the modular group are verified and their role in rational conformal field theories is elucidated. The transcendental limit at infinite genus is examined.The adjacency-Jacobsthal sequence in finite groups.https://zbmath.org/1449.110792021-01-08T12:24:00+00:00"Karaduman, Erdal"https://zbmath.org/authors/?q=ai:karaduman.erdal"Aküzüm, Yeşim"https://zbmath.org/authors/?q=ai:akuzum.yesim"Deveci, Ömür"https://zbmath.org/authors/?q=ai:deveci.omurSummary: The adjacency-Jacobsthal sequence and the adjacency-Jacobsthal matrix were defined by \textit{Ö. Deveci} and \textit{G. Artun} (see [Commun. Algebra 47, No. 11, 4520--4532 (2019; Zbl 07098140)]). In this work, we consider the cyclic groups which are generated by the multiplicative orders of the adjacency-Jacobsthal matrix when read modulo \(\alpha\) (\(\alpha > 1\)). Also, we study the adjacency-Jacobsthal sequence modulo \(\alpha\) and then we obtain the relationship among the periods of the adjacency-Jacobsthal sequence modulo \(\alpha\) and the orders of the cyclic groups obtained. Furthermore, we redefine the adjacency-Jacobsthal sequence by means of the elements of 2-generator groups which is called the adjacency-Jacobsthal orbit. Then we examine the adjacency-Jacobsthal orbit of the finite groups in detail. Finally, we obtain the periods of the adjacency-Jacobsthal orbit of the dihedral group \(D_{10}\) as applications of the results obtained.t-norms over \(Q\)-fuzzy subgroups of a group.https://zbmath.org/1449.200762021-01-08T12:24:00+00:00"Rasuli, R."https://zbmath.org/authors/?q=ai:rasuli.rasul"Naraghi, H."https://zbmath.org/authors/?q=ai:naraghi.hassan|naraghi.hosseinSummary: In this paper, \(Q\)-fuzzy subgroups and normal \(Q\)-fuzzy subgroups of group \(G\) with respect to t-norm \(T\) are defined and investigated some of their properties and structured characteristics. Next the properties of them under homomorphisms and anti-homomorphisms are discussed.Three dimensional semigroup algebras.https://zbmath.org/1449.200552021-01-08T12:24:00+00:00"Ji, Yingdan"https://zbmath.org/authors/?q=ai:ji.yingdan"Luo, Yanfeng"https://zbmath.org/authors/?q=ai:luo.yan-fengSummary: In this paper, the authors provide the idempotent sets and Jacobson radicals of all three dimensional semigroup algebras, and then classify these algebras up to isomorphism, where the results depend on the characteristic of the ground algebraically closed field. Then the representation type of these semigroup algebras are investigated by computing certain quivers. The authors also prove that a three (resp., two) dimensional semigroup algebra is cellular if and only if it is commutative. As a by-product, it is shown that the semigroup algebra of a left zero band is cellular if and only if it is a semilattice.On \(ss\)-quassinormal subgroups and \(c\)-normal subgroups.https://zbmath.org/1449.200242021-01-08T12:24:00+00:00"Cheng, Dan"https://zbmath.org/authors/?q=ai:cheng.dan"Xu, Yingwu"https://zbmath.org/authors/?q=ai:xu.yingwuSummary: Let \(G\) be a finite group, \(H\) is a subgroup of \(G\). A subgroup \(H\) is said to be an \(ss\)-quasinormal subgroup of \(G\), if there is a supplement \(B\) of \(H\) to \(G\) such that \(H\) permutes with every Sylow subgroup of \(B\). However, A subgroup \(H\) of \(G\) is called \(c\)-normal in \(G\) provided that there exists a normal subgroup \(K\) such as \(G = HK\) and \(H \cap K \le {H_G}\), where \({H_G}\) is the normal core of \(H\) in \(G\). The two concepts are considered in a group at the same time, and we apply the ``or'' method of group theory to the research. It is concluded as follows: suppose that \(p\) is a prime dividing \(|G|\) with \( (|G|, p - 1) = 1\) and let \(P\) be a Sylow \(p\)-subgroup of a group \(G\), and if every maximal subgroup of \(P\) is either \(ss\)-quasinormal or \(c\)-normal, then \(G\) is \(p\)-nilpotent.Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals.https://zbmath.org/1449.430042021-01-08T12:24:00+00:00"Askari-Sayah, M."https://zbmath.org/authors/?q=ai:askari-sayah.m"Pourabbas, A."https://zbmath.org/authors/?q=ai:pourabbas.abdolrasoul"Sahami, A."https://zbmath.org/authors/?q=ai:sahami.amirSummary: In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup \(S\), we show that \(\ell^1(S)\) is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra \(A\), we show that \(A\) has no non-zero complemented closed nilpotent ideal.On localized \(\mathcal{M}_p\)-embedded subgroups.https://zbmath.org/1449.200212021-01-08T12:24:00+00:00"Liu, Liping"https://zbmath.org/authors/?q=ai:liu.liping"Miao, Long"https://zbmath.org/authors/?q=ai:miao.long"Tang, Juping"https://zbmath.org/authors/?q=ai:tang.jupingSummary: A subgroup \(H\) of \(G\) is called \(\mathcal{M}_p\)-embedded in \(G\) if there exists a \(p\)-nilpotent subgroup \(B\) of \(G\) such that \({H_p} \in {\mathrm{Syl}}_p(B)\) and \(B\) is \(\mathcal{M}_p\)-supplemented in \(G\), where \({H_p} \in {\mathrm{Syl}}_p(H)\). Using the \(\mathcal{M}_p\)-embedded properties of some subgroups of \({N_G} (P)\), where \(P\) is a Sylow subgroup of \(G\), and the nearly \(m\)-embedded properties of \(\mathcal{H}\)-subgroups, some sufficient conditions for the \(p\)-nilpotency and \(p\)-supersolvability are obtained.Gröbner-Shirshov basis of non-degenerate affine Hecke algebras of type \({A_n}\).https://zbmath.org/1449.161042021-01-08T12:24:00+00:00"Munayim, Dilxat"https://zbmath.org/authors/?q=ai:munayim.dilxat"Abdukadir, Obul"https://zbmath.org/authors/?q=ai:abdukadir.obulSummary: In this paper, by computing the compositions, we give a Gröbner-Shirshov basis of non-degenerate affine Hecke algebra of type \({A_n}\). By using this Gröbner-Shirshov basis and the composition-diamond lemma of associative algebras, we give a linear basis of the non-degenerate affine Hecke algebra of type \({A_n}\).The properties of lattice on a bicyclic semigroup.https://zbmath.org/1449.200612021-01-08T12:24:00+00:00"Tian, Zhenji"https://zbmath.org/authors/?q=ai:tian.zhenji"Wang, Yaning"https://zbmath.org/authors/?q=ai:wang.yaningSummary: It is proved that an arbitrary normal subsemigroup \(N\) in a bicyclic semigroup can be expressed by \({B_d}\). Meanwhile, it is also proved that the set \(B\) comprised of normal subsemigroups \({B_d}\) is a distributive lattice.Highly-arc-transitive and descendant-homogeneous digraphs with finite out-valency.https://zbmath.org/1449.051192021-01-08T12:24:00+00:00"Amato, Daniela A."https://zbmath.org/authors/?q=ai:amato.daniela-aA digraph \(D\) is said to have property \(Z\) if there is a digraph homomorphism from \(D\) onto the digraph \(Z\). The descendant set of a vertex in a digraph \(D\) is the induced subdigraph on the set of vertices reachable from the given vertex by an outward-directed path. A digraph \(D\) is descendant-homogeneous if it is vertex-transitive and any isomorphism between finitely generated subdigraphs of \(D\) extends to an automorphism of \(D\). The present paper investigates highly-arc-transitive digraphs with a homomorphism onto \(Z\) which are, additionally, descendant-homogeneous. It is shown that if \(D\) is a highly-arc-transitive descendant-homogeneous digraph with property \(Z\) and \(F\) is the subdigraph spanned by the descendant sets of a line in \(D\), then \(F\) is a locally finite 2-ended digraph with property \(Z\). In addition if \(D\) has prime out-valency, then there is only one possibility for the subdigraph \(F\), which is then used to classify the highly-arc-transitive descendant-homogeneous digraphs of prime out-valency with property \(Z\).
Reviewer: Wai-Kai Chen (Fremont)Act of fractions and flatness properties.https://zbmath.org/1449.200562021-01-08T12:24:00+00:00"Irannezhad, Setareh"https://zbmath.org/authors/?q=ai:irannezhad.setareh"Madanshekaf, Ali"https://zbmath.org/authors/?q=ai:madanshekaf.aliSummary: In this paper, we study the fractional \(S\)-acts and the localization functor \({C^{-1}}\), corresponding to a multiplicative subset \(C\) of a commutative monoid \(S\). We investigate preservation and reflection of certain properties by the functor \({C^{-1}}\). We show that the functor \({C^{-1}}\) is a reflector and also it is a left exact functor.Exterior isoclinism on finite groups.https://zbmath.org/1449.200302021-01-08T12:24:00+00:00"Hakima, H."https://zbmath.org/authors/?q=ai:hakima.h"Hadi, Jafari S."https://zbmath.org/authors/?q=ai:hadi.jafari-sSummary: In 1940, the notion of isoclinism was introduced as a structurally motivated classification of finite groups. In this paper we define the exterior isoclinism of groups which yields a new classification on the class of all groups. Among the other results, we obtain some group theoretical properties which are invariant of the families of finite exterior isoclinic groups. Moreover, we compute the exterior degrees of some abelian groups.On the nonperiodic groups, whose subgroups of infinite special rank are transitively normal.https://zbmath.org/1449.200312021-01-08T12:24:00+00:00"Kurdachenko, L. A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Subbotin, I.Ya."https://zbmath.org/authors/?q=ai:subbotin.igor-ya"Velychko, T. V."https://zbmath.org/authors/?q=ai:velychko.t-vSummary: This paper devoted to the nonperiodic locally generalized radical groups, whose subgroups of infinite special rank are transitively normal. We proved that if such a group \(G\) includes an ascendant locally nilpotent subgroup of infinite special rank, then \(G\) is abelian.Abian's relation on semirings.https://zbmath.org/1449.160962021-01-08T12:24:00+00:00"Khatun, Sarifa"https://zbmath.org/authors/?q=ai:khatun.sarifa"Sircar, Jayasri"https://zbmath.org/authors/?q=ai:sircar.jayasri"Abu Nayeem, Sk. Md."https://zbmath.org/authors/?q=ai:abu-nayeem.sk-mdSummary: In this paper, we introduce a relation called Abian's relation on a semiring. A semiring endowed with Abian's order is called an Abian's semiring. Here we obtain a sufficient condition for an Abian's semiring to be a partially ordered semiring with respect to Abian's relation. Also we study different properties of the comparability graph of an Abian's semiring.A generalization of Kramer's theory.https://zbmath.org/1449.200122021-01-08T12:24:00+00:00"Chi, Z."https://zbmath.org/authors/?q=ai:chi.ziying|chi.zhizhen|chi.zhongxian|chi.zongtao|chi.zuohe|chi.zhiyi|chi.zimeng|chi.ziqiang|chi.zheru|chi.zhedong|chi.zhang"Skiba, A. N."https://zbmath.org/authors/?q=ai:skiba.aleksandr-nikolaevich|skiba.alexander-nLet \(\sigma\) be a partition of the set of all prime numbers, and for each positive integer \(n\) let \(\sigma(n)\) denote the set of all elements of \(\sigma\) that contain some divisor of \(n\). Then the positive integers \(m\) and \(n\) are called \textit{\(\sigma\)-coprime} if \(\sigma(m)\cap\sigma(n)=\emptyset\). A group class \(\mathcal{F}\) (consisting only of finite groups) is said to be \textit{\(\Gamma_t^\sigma\)-closed} for some integer \(t>1\) if \(\mathcal{F}\) contains every finite group \(G\) admitting \(t\) subgroups \(A_1,\ldots,A_t\) such that \(G=A_iA_j\) whenever \(i\neq j\) and the indices \(|G:N_G(A_1)|,\ldots,|G:N_G(A_t)|\) are pairwise \(\sigma\)-coprime.
In the paper under review, the authors study the above concepts, and in particular they prove that certain formations of generalized metanilpotent groups (related to the partition \(\sigma\)) are \(\Gamma_4^\sigma\)-closed.
Reviewer: Francesco de Giovanni (Napoli)On anti-regular semigroups of \(N (2,2,0)\) algebra.https://zbmath.org/1449.200442021-01-08T12:24:00+00:00"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.luSummary: In the paper, the concept of anti-regular element of \(N (2, 2, 0)\) algebra was introduced. Some examples about regular element, inverse element and anti-regular element were provided, and some properties related to anti-regular semigroup of \(N (2, 2, 0)\) algebra were investigated. Finally, a quotient algebra was constructed by anti-regular element of \(N (2, 2, 0)\) algebra.The influence of nearly \(\mathcal{M}\)-supplemented subgroups on the structure of two classes of quasi-\(\mathcal{F}\)-groups.https://zbmath.org/1449.200132021-01-08T12:24:00+00:00"Gao, Baijun"https://zbmath.org/authors/?q=ai:gao.baijun"Wang, Keke"https://zbmath.org/authors/?q=ai:wang.kekeSummary: Let \(G\) be a finite group and \(E\) a normal subgroup of \(G\). Some sufficient conditions about \(p\)-quasisupersoluble groups and quasisupersoluble groups were obtained by using the nearly \(\mathcal{M}\)-supplementation of the maximal subgroup of \(P\), in which \(P\) was a Sylow \(p\)-subgroup of \(E\), \(p\) was the smallest prime divisor of \(|E|\) or the non-cyclic Sylow \(p\)-subgroup of \(E\) or \({F^*} (E)\), respectively.Weak commutativity of lattice-valued finite state machines.https://zbmath.org/1449.680472021-01-08T12:24:00+00:00"Huang, Feidan"https://zbmath.org/authors/?q=ai:huang.feidan"Li, Xuejia"https://zbmath.org/authors/?q=ai:li.xuejia"Wu, Lingling"https://zbmath.org/authors/?q=ai:wu.linglingSummary: In this paper, the definition of weak commutativity of lattice-valued finite state machine and the definition of weak commutativity of lattice-valued transformation semigroup are given. Weak commutativity of lattice-valued finite state machine is characterized by using matrix and semigroup, and some equivalent conditions of weak commutativity of lattice-valued finite state machine are obtained. The relation between weak commutativity of lattice-valued finite state machine and weak commutativity of second class lattice-valued transformation semigroup is also found out. Moreover, weak commutativity of full direct product, restricted direct product, cascade product, wreath product and sum of lattice-valued finite state machines are studied. Weak commutativity of full direct product and restricted direct product of second class lattice-valued transformation semigroups associated with lattice-valued finite state machines are also studied, and a sufficient condition is obtained.Rank of the semigroup \(PCS_n\).https://zbmath.org/1449.200502021-01-08T12:24:00+00:00"Li, Yalei"https://zbmath.org/authors/?q=ai:li.yalei"Luo, Yonggui"https://zbmath.org/authors/?q=ai:luo.yonggui"Xu, Bo"https://zbmath.org/authors/?q=ai:xu.boSummary: Let \({C_n}\) be the cyclic group on \(X_n\), \(SP_n = P_n\backslash S_n\) be the partial singular transformation semigroup on \(X_n\). By analyzing the generating set of the transformation semigroup \(PCS_n = C_n \bigcup SP_n\), we obtain the rank of the transformation semigroup \(PCS_n\), which is \([\frac{n}{2}] + 2\).The translational hull of right \(e\)-wlpp semigroups.https://zbmath.org/1449.200432021-01-08T12:24:00+00:00"Wang, Chunru"https://zbmath.org/authors/?q=ai:wang.chunruSummary: The definition of adequate right \(e\)-wlpp semigroup is given according to the right \(e\)-wlpp semigroup. The mapping \({\lambda^+}\) and \({\rho^+}\) from \(S\) to itself is introduced using the relation between adequate right \(e\)-wlpp semigroup and idempotent set \(E (S)\). It is proved that the translation hull of right \(e\)-wlpp semigroup is still right \(e\)-wlpp semigroup. The translation hull of an adequate right \(e\)-wlpp semigroup is still an adequate right \(e\)-wlpp semigroup.On some new criteria of \(p\)-supersolubility and \(p\)-nilpotency of finite groups.https://zbmath.org/1449.200102021-01-08T12:24:00+00:00"Chen, Deping"https://zbmath.org/authors/?q=ai:chen.deping"You, Ze"https://zbmath.org/authors/?q=ai:you.ze"Li, Baojun"https://zbmath.org/authors/?q=ai:li.baojun|li.baojun.1Summary: A subgroup \(H\) of a group \(G\) is called weakly \(s\)-permutable in \(G\), if there is a subnormal subgroup \(T\) of \(G\) such that \(HT = G\) and \(T \cap H \le {H_{sG}}\). By discussing the influence of weakly \(s\)-permutable subgroups on the structure of finite groups, some new criteria of \(p\)-supersolubility and \(p\)-nilpotency of finite groups are obtained.Nearly \(CAP^*\)-subgroups and \(p\)-supersolvability of finite groups.https://zbmath.org/1449.200232021-01-08T12:24:00+00:00"Zhong, Xianggui"https://zbmath.org/authors/?q=ai:zhong.xianggui"Sun, Yue"https://zbmath.org/authors/?q=ai:sun.yue"Wu, Xianghua"https://zbmath.org/authors/?q=ai:wu.xianghuaSummary: A subgroup \(H\) of a finite group \(G\) is called a \(CAP^*\)-subgroup of \(G\), if \(H\) either covers or avoids every non-Frattini chief factor of \(G\). A subgroup \(H\) of \(G\) is said to be a nearly \(CAP^*\)-subgroup of \(G\) if there exists a subnormal subgroup \(K\) of \(G\) such that \(HK = G\) and \(H\bigcap K\) is a \(CAP^*\)-subgroup of \(G\). This paper investigates the structure of \(G\) under the assumption that certain subgroups of prime power orders are nearly \(CAP^*\)-subgroups of \(G\), and a series of known results are generalized.A note on the Erdős-Faber-Lovász conjecture: quasigroups and complete digraphs.https://zbmath.org/1449.050862021-01-08T12:24:00+00:00"Araujo-Pardo, G."https://zbmath.org/authors/?q=ai:araujo-pardo.gabriela"Rubio-Montiel, C."https://zbmath.org/authors/?q=ai:rubio-montiel.christian"Vázquez-Ávila, A."https://zbmath.org/authors/?q=ai:vazquez-avila.adrianSummary: A decomposition of a simple graph \(G\) is a pair \((G,P)\) where \(P\) is a set of subgraphs of \(G\), which partitions the edges of \(G\) in the sense that every edge of \(G\) belongs to exactly one subgraph in \(P\). If the elements of \(P\) are induced subgraphs then the decomposition is denoted by \([G,P]\).
A \(k\)-\(P\)-coloring of a decomposition \((G,P)\) is a surjective function that assigns to the edges of \(G\) a color from a \(k\)-set of colors, such that all edges of \(H\in P\) have the same color, and, if \(H_1, H_2\in P\) with \(V(H_1)\cap V(H_2)\neq\emptyset\) then \(E(H_1)\) and \(E(H_2)\) have different colors. The chromatic index \(\chi^\prime((G,P))\) of a decomposition \((G,P)\) is the smallest number \(k\) for which there exists a \(k\)-\(P\)-coloring of \((G,P)\).
The well-known Erdős-Faber-Lovász conjecture states that any decomposition \([K_n,P]\) satisfies \(\chi^\prime([K_n,P])\leq n\). We use quasigroups and complete digraphs to give a new family of decompositions that satisfy the conjecture.On the rank and \( (0,1)\)-square idempotent rank of the semigroup \(G (n, r)\).https://zbmath.org/1449.200492021-01-08T12:24:00+00:00"Li, Xiaomin"https://zbmath.org/authors/?q=ai:li.xiaomin"Luo, Yonggui"https://zbmath.org/authors/?q=ai:luo.yonggui"Zhao, Ping"https://zbmath.org/authors/?q=ai:zhao.pingSummary: We introduce one-to-one singular transformation semigroup of ascending-preserving finite parts. By analyzing the \( (0, 1)\)-square idempotent elements and star Green's relations, the unique minimal \( (0, 1)\)-square idempotent generating set, rank and \( (0, 1)\)-square idempotent rank of the semigroup \(G (n, r)\) are obtained, respectively. Furthermore, the relative rank of the semigroup \(G (n, r)\) with respect to its each star ideal \(G (n, l)\) is determined if \(0 \le l \le r\).Isomorphism of commutative group algebras of finite abelian \(p\)-groups.https://zbmath.org/1449.200032021-01-08T12:24:00+00:00"Mollov, Todor Zh."https://zbmath.org/authors/?q=ai:mollov.todor-zh"Nachev, Nako A."https://zbmath.org/authors/?q=ai:nachev.nako-aSummary: Let \(R\) be a direct product of commutative indecomposable rings with identities and let \(G\) be a finite abelian \(p\)-group. In the present paper we give a complete system of invariants of the group algebra \(RG\) of \(G\) over \(R\) when \(p\) is an invertible element in \(R\). These investigations extend some classical results of \textit{S. D. Berman} [Dokl. Akad. Nauk SSSR, n. Ser. 91, 7--9 (1953; Zbl 0050.25504) and Mat. Sb., Nov. Ser. 44(86), 409--456 (1958; Zbl 0080.02102)], \textit{S. K. Sehgal} [J. Number Theory 2, 500--508 (1970; Zbl 0209.05804)] and \textit{G. Karpilovsky} [Period. Math. Hung. 15, 259--265 (1984; Zbl 0526.20004)] as well as a result of the first author [PLISKA, Stud. Math. Bulg. 8, 54--64 (1986; Zbl 0655.16004)].A natural partial order on the semigroups \(S_n^-\) of order-decreasing transformations.https://zbmath.org/1449.200522021-01-08T12:24:00+00:00"Sun, Lei"https://zbmath.org/authors/?q=ai:sun.lei"Li, Jiayang"https://zbmath.org/authors/?q=ai:li.jiayangSummary: Let \({T_X}\) be the full transformation semigroup on a total order set \(X = \{1 < 2 < \cdots < n\}\). Then \(S_n^- = \{f \in {T_X}:\forall x \in X, f (x) \le x\}\) is a subsemigroup of \({T_X}\). We endow the order-decreasing transformation semigroup \(S_n^-\) with the natural partial order. With respect to this partial order, we investigate when two elements of \(S_n^-\) are related, then find elements of \(S_n^-\) which are compatible with the order. Also, we characterize the minimal elements and the maximal elements of \(S_n^-\).Block pentagons cannot occur as character degree graphs of finite solvable groups.https://zbmath.org/1449.200052021-01-08T12:24:00+00:00"Meng, Qingyun"https://zbmath.org/authors/?q=ai:meng.qingyun"Du, Ni"https://zbmath.org/authors/?q=ai:du.niSummary: In a literature, it has been proved that pentagons cannot occur as character degree graphs of any finite solvable groups. In this short paper, we generalize the definition of a pentagon graph to graphs that are block pentagons. We investigate character degree graphs of finite solvable groups and show that block pentagons cannot occur as character degree graphs of any solvable groups, either. This discovery generalized the result of the literature.The Fitting length of a product of mutually permutable finite groups.https://zbmath.org/1449.200252021-01-08T12:24:00+00:00"Jabara, E."https://zbmath.org/authors/?q=ai:jabara.enricoA group \(G\) is said to be a mutually permutable product of its subgroups \(A\) and \(B\) if \(G=AB\), \(AY=YA\) and \(XB=BX\) for all subgroups \(X\) of \(A\) and \(Y\) of \(B\). It has recently been proved by \textit{J. Cossey} and \textit{Y. Li} [Arch. Math. 110, No. 6, 533--537 (2018; Zbl 06897419)] that if a finite group \(G=AB\) is a mutually permutable product of two \(p\)-soluble subgroups \(A\) and \(B\), then \(G\) is \(p\)-soluble and \(l_p(G)\leq \max\{ l_p(A),l_p(B)\}+1\). In the paper under review, the author extends this result to the case of more factors.
Reviewer: Francesco de Giovanni (Napoli)A new characterization of symmetric groups \({S_n}\) (\(n \le 13\)).https://zbmath.org/1449.200022021-01-08T12:24:00+00:00"Xia, Xueqin"https://zbmath.org/authors/?q=ai:xia.xueqin"He, Liguan"https://zbmath.org/authors/?q=ai:he.liguanSummary: Let \(G\) be a finite group, \({o_1} (G)\) denote the largest element order of \(G\) and \({n_1} (G)\) denote the number of the elements of order \({o_1} (G)\). Assume that \(G\) totally has \(r\) elements of order \({o_1} (G)\), the centralizers of it are of different orders, and \({c_i} (G)\) denote the order of the \(i\)-th centralizer of order \({o_1} (G)\). Define \({\mathrm{ONC}}_1 (G): = \{{o_1} (G); {n_1} (G); {c_1} (G), {c_2} (G), \dots, {c_r} (G)\}\). We call \({\mathrm{ONC}}_1 (G)\) is the 1st ONC-degree of \(G\). In this paper, we characterize symmetric groups \({S_n}\) (\(n \le 13\)) by their 1st ONC-degree \({\mathrm{ONC}}_1 (G)\).Some notes of fuzzy congruences on weakly type B semigroups.https://zbmath.org/1449.200752021-01-08T12:24:00+00:00"Li, Chunhua"https://zbmath.org/authors/?q=ai:li.chunhua"Pei, Zhi"https://zbmath.org/authors/?q=ai:pei.zhi"Xu, Baogen"https://zbmath.org/authors/?q=ai:xu.baogen"Huang, Huawei"https://zbmath.org/authors/?q=ai:huang.huaweiSummary: The definition of a fuzzy strong congruence on semi-abundant semigroups is introduced by using the notion of a fuzzy relation, and some properties and characterizations of fuzzy strong congruences on such semigroups are obtained. On this base, some properties of fuzzy strong congruences and fuzzy unipotent congruences on weakly type B semigroups are given, and sufficient and necessary conditions for a fuzzy strong congruence on a weakly type B semigroup to be fuzzy unipotent are proved.Coleman automorphisms of extensions of the finite groups by some groups.https://zbmath.org/1449.200262021-01-08T12:24:00+00:00"Zhao, Lele"https://zbmath.org/authors/?q=ai:zhao.lele"Hai, Jinke"https://zbmath.org/authors/?q=ai:hai.jinkeSummary: Let \(G\) be an extension of a finite characteristically simple group by a finite abelian group or a finite non-abelian simple group. It is shown that under some conditions every Coleman automorphism of \(G\) is an inner automorphism.Axiomatic system defining an order-embedding between infinite \(\sigma\)-algebras.https://zbmath.org/1449.280012021-01-08T12:24:00+00:00"Agbeko, Nutefe Kwami"https://zbmath.org/authors/?q=ai:agbeko.nutefe-kwamiSummary: The purpose of the present paper is twofold: on the one hand to set up an axiomatic system defining an order-embedding between infinite \(\sigma\)-algebras to generalize the powering mapping and investigate some additional necessary and sufficient condition for the postulate of powering to hold in the system, and on the other hand to provide some theoretical applications.\(E\)-solid locally inverse semigroups.https://zbmath.org/1449.200412021-01-08T12:24:00+00:00"Dékány, Tamás"https://zbmath.org/authors/?q=ai:dekany.tamas"Szendrei, Mária B."https://zbmath.org/authors/?q=ai:szendrei.maria-b"Szittyai, István"https://zbmath.org/authors/?q=ai:szittyai.istvanA regular semigroup \(S\) is \textit{locally inverse} if for each idempotent \(e\) of \(S\), \(eSe\) is inverse; it is \textit{\(E\)-solid} if its idempotents generate a completely regular semigroup. In general terms, these are the two types of regular semigroups for which research has progressed most deeply, especially in universal algebraic terms. The locally inverse, \(E\)-solid semigroups are common generalizations of generalized inverse semigroups and normal bands of groups. The main result is that if \(S\) is such a semigroup and \(\rho\) is an inverse semigroup congruence on \(S\) whose idempotent classes are completely simple, then \(S\) is embeddable into a \(\lambda\)-semidirect product of a completely simple semigroup by \(S/\rho\). This generalizes a result of \textit{B. Billhardt} and \textit{I. Szittyai} [Semigroup Forum 61, No. 1, 26--31 (2000; Zbl 0957.20043)] on inverse semigroups themselves. It follows from the main result that the locally inverse, \(E\)-solid semigroups are, up to isomorphism, the regular subsemigroups of the \(\lambda\)-semidirect products of completely simple semigroups by inverse semigroups. An intermediate specialization is then to the case of generalized inverse semigroups. The \(\lambda\)-semidirect product of a semigroup \(K\) by an inverse semigroup \(T\) was introduced by \textit{B. Billhardt} [Semigroup Forum 45, No. 1, 45--54 (1992; Zbl 0769.20027)]: if \(T\) acts on \(K\) on the left by \((a,t) \mapsto \,^t \!a\), then it consists of \(\{(a,t) \in K \times T : \,^{tt^{-1}}\!a = a\}\), with \( (a,t)(b,u) = (^{(tu)(tu)^{-1}}\!a \cdot \,^t \! b, tu)\).
Reviewer: Peter R. Jones (Milwaukee)Topological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical.https://zbmath.org/1449.220032021-01-08T12:24:00+00:00"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Ficzere, Kornélia"https://zbmath.org/authors/?q=ai:ficzere.kornelia"Al-Abayechi, Ameer"https://zbmath.org/authors/?q=ai:al-abayechi.ameerSummary: Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.Algebraic properties of \(\lambda\)-fuzzy subgroups.https://zbmath.org/1449.200772021-01-08T12:24:00+00:00"Shuaib, Umer"https://zbmath.org/authors/?q=ai:shuaib.umer"Asghar, Wassem"https://zbmath.org/authors/?q=ai:asghar.wassemSummary: In this paper, we initiate the study of \(\lambda\)-fuzzy sets. We define the notion of \(\lambda\)-fuzzy subgroup and prove that every fuzzy subgroup is \(\lambda\)-fuzzy subgroup. We introduce the notion of \(\lambda\)-fuzzy cosets and establish their algebraic properties. We also initiate the study of \(\lambda\)-fuzzy normal subgroups and quotient group with respect to \(\lambda\)-fuzzy normal subgroup and prove some of their various group theoretic properties. We also investigate effect on the image and inverse image of \(\lambda\)-fuzzy subgroup (normal subgroup) under group homomorphism and establish an isomorphism between the quotient group with respect to \(\lambda\)-fuzzy normal subgroup and quotient group with respect to the normal subgroup \(G_{\rho^\lambda}\).Asymptotic metric behavior of random Cayley graphs of finite abelian groups.https://zbmath.org/1449.051362021-01-08T12:24:00+00:00"Shapira, Uri"https://zbmath.org/authors/?q=ai:shapira.uri"Zuck, Reut"https://zbmath.org/authors/?q=ai:zuck.reutSummary: Using methods of \textit{J. Marklof} and \textit{A. Strömbergsson} [ibid. 33, No. 4, 429--466 (2013; Zbl 1340.05063)] we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen set of generators of a fixed size. Doing so we settle a conjecture of \textit{G. Amir} and \textit{O. Gurel-Gurevich} [Groups Complex. Cryptol. 2, No. 1, 59--65 (2010; Zbl 1194.05054)].\(n\)-tilting torsion classes and \(n\)-cotilting torsion-free classes.https://zbmath.org/1449.200372021-01-08T12:24:00+00:00"He, Donglin"https://zbmath.org/authors/?q=ai:he.donglin"Li, Yuyan"https://zbmath.org/authors/?q=ai:li.yuyanSummary: In this paper, we consider some generalizations of tilting torsion classes and cotilting torsion-free classes, give the definition and characterizations of \(n\)-tilting torsion classes and \(n\)-cotilting torsion-free classes, and study \(n\)-tilting preenvelopes and \(n\)-cotilting precovers.A characterization of regular, intra-regular, left quasi-regular and semisimple hypersemigroups in terms of fuzzy sets.https://zbmath.org/1449.200692021-01-08T12:24:00+00:00"Kehayopulu, Niovi"https://zbmath.org/authors/?q=ai:kehayopulu.nioviSummary: We prove that an hypersemigroup \(H\) is regular if and only, for any fuzzy subset \(f\) of \(H\), we have \(f\preceq f \circ 1\circ f\) and it is intra-regular if and only if, for any fuzzy subset \(f\) of \(H\), we have \(f \preceq 1 \circ f \circ f \circ 1\). An hypersemigroup \(H\) is left (resp. right) quasi-regular if and only if, for any fuzzy subset \(f\) of \(H\) we have \(f \preceq 1 \circ f \circ 1 \circ f\) (resp. \(f \preceq f \circ 1 \circ f \circ 1\)) and it is semisimple if and only if, for any fuzzy subset \(f\) of \(H\) we have \(f \preceq 1 \circ f \circ 1 \circ f \circ 1\). The characterization of regular and intra-regular hypersemigroups in terms of fuzzy subsets are very useful for applications.The inclusion graph of \(S\)-acts.https://zbmath.org/1449.051382021-01-08T12:24:00+00:00"Sun, Shuang"https://zbmath.org/authors/?q=ai:sun.shuang"Liu, Hongxing"https://zbmath.org/authors/?q=ai:liu.hongxing.1|liu.hongxingSummary: Let \(S\) be a semigroup and \(M\) be an \(S\)-act. The inclusion graph of \(M\), denoted by \(G (M)\), is the undirected simple graph whose vertices are all non-trivial subact of \(M\) and defining two distinct vertices \(I\) and \(J\) to be adjacent if and only if \(I \subseteq J\) or \(J \subseteq I\). Some results on completeness, connectivity, diameter, girth, the clique number and the chromatic number of \(G (M)\) are presented.Automorphism groups of some graphs for the ring of Gaussian integers modulo \({p^s}\).https://zbmath.org/1449.051412021-01-08T12:24:00+00:00"Zhang, Hengbin"https://zbmath.org/authors/?q=ai:zhang.hengbin"Nan, Jizhu"https://zbmath.org/authors/?q=ai:nan.ji-zhuSummary: In this paper, the automorphism group is completely determined. The unitary Cayley graph, the unit graph and the total graph are defined to be simple graphs over the ring of Gaussian integers modulo \({p^s}\).Some notes on infix chains -- several studies on combinatorial semigroups. II.https://zbmath.org/1449.200622021-01-08T12:24:00+00:00"Leng, Jing"https://zbmath.org/authors/?q=ai:leng.jing"Guo, Yuqi"https://zbmath.org/authors/?q=ai:guo.yuqiSummary: Some notes on infix chains and a characterization of completely dense languages are given in this paper.
For Part I see [the authors, ibid. 54, No. 6, 2--7 (2019; Zbl 1449.20064)].Gelfand pairs over hypergroup joins.https://zbmath.org/1449.200702021-01-08T12:24:00+00:00"Vati, K."https://zbmath.org/authors/?q=ai:vati.kedumetseGelfand pairs over a locally compact hypergroup are characterized by the commutativity of double coset hypergroups.
Reviewer: Wiesław A. Dudek (Wrocław)The number of homomorphisms from dihedral groups to a class of metacyclic groups.https://zbmath.org/1449.200282021-01-08T12:24:00+00:00"Li, Hongxia"https://zbmath.org/authors/?q=ai:li.hongxia"Guo, Jidong"https://zbmath.org/authors/?q=ai:guo.jidong"Hai, Jinke"https://zbmath.org/authors/?q=ai:hai.jinkeSummary: Based on group theory, the structure of metacyclic groups and the characteristics of group elements, and by using the basic method of algebra and number theory, we calculate the number of homomorphism between dihedral groups and a class of metacyclic groups.Characterization of pomonoids by inverse \(S\)-posets.https://zbmath.org/1449.060242021-01-08T12:24:00+00:00"Qiao, Husheng"https://zbmath.org/authors/?q=ai:qiao.husheng"Feng, Leting"https://zbmath.org/authors/?q=ai:feng.letingSummary: Let \(S\) be a pomonoid. The inverse \(S\)-acts are extended, the properties and homological classification problem of inverse \(S\)-posets are investigated.Characterization of monoids by strong faithful \(S\)-acts.https://zbmath.org/1449.200582021-01-08T12:24:00+00:00"Qiao, Husheng"https://zbmath.org/authors/?q=ai:qiao.husheng"Chen, Qian"https://zbmath.org/authors/?q=ai:chen.qianSummary: Let \(S\) be a monoid. The new properties of strong faithful \(S\)-acts are investigated, and the homological classification problem of strong faithful property and flatness properties are discussed.The concatenation of thin languages and \(r\)-disjunctive languages -- several studies on combinatorial semigroups. I.https://zbmath.org/1449.200642021-01-08T12:24:00+00:00"Liu, Zuhua"https://zbmath.org/authors/?q=ai:liu.zuhua"Guo, Yuqi"https://zbmath.org/authors/?q=ai:guo.yuqiSummary: The background of this paper is from two papers in literature. We merge some results in above two papers as Proposition 1: \({L_1}L \in \mathrm{bf}{D}_f (\mathrm{bf}{D}_t, \mathrm{bf}{D}_r)\) implies \(L \in \mathrm{bf}{D}_f (\mathrm{bf}{D}_t, \mathrm{bf}{D}_r)\), where \({L_1}, L\) are languages over alphabet \(A\) and \({L_1}\) is finite. In this paper, a new and simple proof of Proposition 1 is given for \(\mathrm{bf}{D}_r\). It is proved that Proposition 1 is also true for \(\mathrm{bf}{D}\) and \(\mathrm{bf}{D}_i\). Replacing ``finite'' with ``thin'', Proposition 1 is true for \(\mathrm{bf}{D}\), \(\mathrm{bf}{D}_f\) and \(\mathrm{bf}{D}_t\). Finally, some examples are given which show that Proposition 1 is not true for \(\mathrm{bf}{D}_i\) and \(\mathrm{bf}{D}_r\).Some properties of weakly projective acts.https://zbmath.org/1449.200602021-01-08T12:24:00+00:00"Qiao, Husheng"https://zbmath.org/authors/?q=ai:qiao.husheng"Zhang, Xiaoqin"https://zbmath.org/authors/?q=ai:zhang.xiaoqinSummary: Weakly projective property of acts is investigated, the conditions that preserve weakly projective property under coproducts and products are discussed, and transitivity of weakly projective property on Rees short exact sequences is also considered.A note on 1-\( (k,m)\)-comma-codes -- several studies on combinatorial semigroups. III.https://zbmath.org/1449.200632021-01-08T12:24:00+00:00"Liu, Haiyan"https://zbmath.org/authors/?q=ai:liu.haiyan"Guo, Yuqi"https://zbmath.org/authors/?q=ai:guo.yuqiSummary: Let \(L\) be a nonempty language over \(A\), \(k \in \mathbb{N}^0\) and \(m \in \mathbb{N}\). If \(L\) satisfies \((LA^k)^m L \cap A^ + (LA^k)^{m - 1}LA^+ = \emptyset \), then it is a code called \( (k,m)\)-comma code. If every singleton of \(L\) is a \( (k,m)\)-comma code, then it is called a 1-\( (k,m)\)-comma code. It is known that the class of 1-\( (k,m)\)-comma code is \(2^{Xk}\backslash \{ \emptyset \}\), where \(X_k = \{ u \in A^+ \mid (\forall w \in {A^k} )uwu \cap A^+ uA^+ = \emptyset\}\). Some previous researchers showed that \(X_0\) is the set of primitive words, and gave a characterization of \(X_1\) in terms of bordered words, unbordered words, and primitive words. In this paper, we discuss \(X_k\) for any \(k \ge 2\).
For Part I and II see [the authors, ibid. 54, No. 6, 2--7 (2019; Zbl 1449.20064); ibid. 54, No. 6, 8--10 (2019; Zbl 1449.20062)].Stabilizer limits of character triples.https://zbmath.org/1449.200042021-01-08T12:24:00+00:00"Chang, Xuewu"https://zbmath.org/authors/?q=ai:chang.xuewu"Wen, Mengli"https://zbmath.org/authors/?q=ai:wen.mengli"Jin, Ping"https://zbmath.org/authors/?q=ai:jin.pingSummary: The stabilizer limits and linear limits of character triples are the basic contents in the representation theory of finite groups. It is proved that a nilpotent linear limit of a given character triple is a stabilizer limit, and a quasiprimitive subnormal inductor of the triple is also a stabilizer limit. The results generalize some relevant theorems in literatures.On \(n\)-\(\sigma\)-embedded subgroups of finite groups.https://zbmath.org/1449.200092021-01-08T12:24:00+00:00"Cao, C."https://zbmath.org/authors/?q=ai:cao.conghui|cao.chunhua|cao.cuixia|cao.chuansheng|cao.chao|cao.chongsheng|cao.cuiwen|cao.chongying|cao.chun.1|cao.chun|cao.changyong|cao.chengxuan|cao.charles|cao.chunhong|cao.chunxiang|cao.changguang|cao.chongyan|cao.cuizhen|cao.chuqing|cao.chengdong|cao.changwei|cao.chunjun|cao.cong|cao.chen|cao.cen|cao.chenlei|cao.congzhe|cao.chuanshu|cao.cangjian|cao.chong|cao.chunzheng|cao.chang|cao.chunyun|cao.chung|cao.chenjie|cao.chunfang|cao.caixia|cao.cungen|cao.chunjie|cao.cewen|cao.changqi|cao.chongguang|cao.chengyu|cao.chunlei|cao.chenchen|cao.changxiu|cao.chuan|cao.chenxi|cao.chunling|cao.cheng|cao.clewen|cao.chengming|cao.chuqi|cao.chengtao|cao.chu|cao.chunguang"Hussain, M. T."https://zbmath.org/authors/?q=ai:hussain.muhammad-tanveer|hussain.mushtaq-taleb"Zhang, L."https://zbmath.org/authors/?q=ai:zhang.lei.25|zhang.linyan|zhang.lingqian|zhang.lianying|zhang.lianjun|zhang.liang|zhang.linlin|zhang.lianhua|zhang.leiming|zhang.liangying|zhang.leike|zhang.laicheng|zhang.liantang|zhang.lifei|zhang.lingyuan|zhang.lilun|zhang.lingyu|zhang.lili|zhang.liumei|zhang.ligang|zhang.lige|zhang.lvyuan|zhang.liqiang|zhang.lunkai|zhang.luyi|zhang.lupeng|zhang.liangyun|zhang.linjie|zhang.liwei|zhang.lina|zhang.lizhu|zhang.lingling|zhang.longsheng|zhang.lingyue|zhang.lianyi|zhang.lanlan|zhang.lingyan|zhang.lefeng|zhang.luping|zhang.lailiang|zhang.lumei|zhang.luying|zhang.lan|zhang.liyuan|zhang.lizhi|zhang.lingzhong|zhang.liangxin|zhang.lei.18|zhang.lirong|zhang.lianfang|zhang.linda|zhang.limin|zhang.lixu|zhang.leilei|zhang.lianhe|zhang.linghua|zhang.leying|zhang.linjun|zhang.lequn|zhang.lei.7|zhang.lansheng|zhang.liyan|zhang.lefei|zhang.liangcheng|zhang.lukun|zhang.lijiang|zhang.lichao|zhang.lida|zhang.lianzhu|zhang.libing|zhang.liwei.1|zhang.lingwei|zhang.li.8|zhang.linghong|zhang.lezhong|zhang.liangwei|zhang.lei.5|zhang.liren|zhang.liangchi|zhang.lizhao|zhang.liquing|zhang.lei.22|zhang.lingsong|zhang.lingyun|zhang.liuliu|zhang.lingfu|zhang.liangyue|zhang.linan|zhang.lei.2|zhang.laobing|zhang.linsen|zhang.laihui|zhang.li.1|zhang.liying|zhang.longfei|zhang.lei.15|zhang.li.5|zhang.luyu|zhang.luyang|zhang.linna|zhang.libo|zhang.lilin|zhang.lihong|zhang.lixiang|zhang.lulu|zhang.lilian|zhang.libao|zhang.lanhua|zhang.lianwen|zhang.lin.2|zhang.lingxia|zhang.liwen|zhang.ling|zhang.lingjun|zhang.linghai|zhang.lisa|zhang.lifa|zhang.lingjie|zhang.lei.24|zhang.le|zhang.longbin|zhang.liu|zhang.li.6|zhang.lingli|zhang.lang|zhang.lianmin|zhang.liabiao|zhang.lunchuan|zhang.lianmeng|zhang.lipan|zhang.liangrui|zhang.liangjin|zhang.langwen|zhang.luyin|zhang.lejie|zhang.lun|zhang.lei.13|zhang.lian|zhang.leyan|zhang.lei.20|zhang.lisheng|zhang.liya|zhang.lihui|zhang.li.7|zhang.lixian|zhang.lanxia|zhang.li.10|zhang.lei.17|zhang.linwen|zhang.longyao|zhang.lei-hong|zhang.lingxin|zhang.li.11|zhang.lijing|zhang.lei|zhang.liqiong|zhang.lingchuan|zhang.lanzhu|zhang.lichuan|zhang.lanyong|zhang.lianhai|zhang.libang|zhang.liangqi|zhang.liangxiu|zhang.lingfeng|zhang.liqing|zhang.lifan|zhang.liangquan|zhang.lixun|zhang.liancheng|zhang.lei.14|zhang.leyou|zhang.lianzhen|zhang.linqing|zhang.lei.11|zhang.li.12|zhang.luojia|zhang.linmiao|zhang.longge|zhang.li.9|zhang.lihai|zhang.lingrui|zhang.lanyu|zhang.lingbo|zhang.liuhua|zhang.luming|zhang.linxi|zhang.lijia|zhang.lipeng|zhang.liang.2|zhang.libin|zhang.lejun|zhang.lingfan|zhang.likai|zhang.linke|zhang.linqiao|zhang.lai|zhang.lianxing|zhang.lanju|zhang.lingping|zhang.lingming|zhang.longxiang|zhang.lei.1|zhang.luona|zhang.liangyin|zhang.lequan|zhang.liang.3|zhang.linchuang|zhang.lixuan|zhang.linrang|zhang.linru|zhang.lizhen|zhang.lihao|zhang.lianzeng|zhang.liuwei|zhang.lintao|zhang.liqi|zhang.lizao|zhang.lianyang|zhang.lei.9|zhang.ledi|zhang.lixin|zhang.lianping|zhang.liangzhe|zhang.leping|zhang.letian|zhang.liang.1|zhang.leishi|zhang.liyi|zhang.linyuan|zhang.lijuan|zhang.liangzhong|zhang.longting|zhang.lianshui|zhang.lin.3|zhang.lixing|zhang.lu|zhang.licen|zhang.long|zhang.lizou|zhang.lele|zhang.liping|zhang.longbing|zhang.linli|zhang.lianzheng|zhang.longjie|zhang.lingmin|zhang.li|zhang.longteng|zhang.litao|zhang.liehui|zhang.lichen|zhang.luo|zhang.lidong|zhang.liquan|zhang.li.2|zhang.liyu|zhang.letao|zhang.lingmi|zhang.liao|zhang.lipu|zhang.liming|zhang.lei.10|zhang.lei.16|zhang.liyun|zhang.liaojun|zhang.liguo|zhang.lingyi|zhang.liuqing|zhang.leon|zhang.liuping|zhang.lifang|zhang.louzin|zhang.linnan|zhang.lianyong|zhang.lisha|zhang.lijie|zhang.lishi|zhang.linhua|zhang.luchan|zhang.lihe|zhang.li.3|zhang.liufeng|zhang.lei.4|zhang.lianhong|zhang.li.4|zhang.liangyong|zhang.lei.21|zhang.lingqin|zhang.lingying|zhang.lanling|zhang.lemei|zhang.lihua|zhang.lidan|zhang.lin.1|zhang.lixia|zhang.landing|zhang.laping|zhang.lanfang|zhang.liangliang|zhang.liruo|zhang.laiwu|zhang.liangdi|zhang.lei.23|zhang.lijian|zhang.lingqi|zhang.lianzhong|zhang.liansheng|zhang.lilong|zhang.luchao|zhang.lixiu|zhang.lifeng|zhang.lianming|zhang.luning|zhang.lipai|zhang.lufei|zhang.lifu|zhang.liangbin|zhang.leigang|zhang.lyuyuan|zhang.lingmei|zhang.lanhui|zhang.liqin|zhang.longbo|zhang.lingxiang|zhang.lixi|zhang.longchuan|zhang.linlan|zhang.lanhong|zhang.linfeng|zhang.lingxian|zhang.lichun|zhang.longjun|zhang.luzou|zhang.lingchen|zhang.liyou|zhang.lin|zhang.linxia|zhang.lide|zhang.laixi|zhang.liangjun|zhang.lijun|zhang.limei|zhang.linbo|zhang.lie|zhang.lianfeng|zhang.luoping|zhang.louxin|zhang.lianmei|zhang.lingjuan|zhang.liting|zhang.linyun|zhang.liyang|zhang.liandi|zhang.liqian|zhang.lieping|zhang.luwan|zhang.linzi|zhang.luyao|zhang.liqun|zhang.licheng|zhang.lufang|zhang.longhui|zhang.liling|zhang.likun|zhang.liuyue|zhang.libiao|zhang.liyong|zhang.liangcai|zhang.lining|zhang.laiping|zhang.liangpei|zhang.li-xinSummary: Let \(\sigma=\{\sigma_i\mid i\in I\}\) be some partition of the set of all primes \(\mathbb{P}\), \(G\) be a finite group and \(\sigma(G)=\{\sigma_i\mid \sigma_i\cap \pi (G)\ne \emptyset\}\). A set \(\mathcal{H}\) of subgroups of \(G\) is said to be a complete Hall \(\sigma\)-set of \(G\) if every non-identity member of \(\mathcal{H}\) is a Hall \(\sigma_i\)-subgroup of \(G\) and \(\mathcal{H}\) contains exactly one Hall \(\sigma_i\)-subgroup of \(G\) for every \(\sigma_i\in\sigma(G)\). A subgroup \(H\) of \(G\) is \(\sigma\)-permutable in \(G\) if \(G\) possesses a complete Hall \(\sigma\)-set \(\mathcal{H}\) such that \(HA^x= A^xH\) for all \(A\in\mathcal{H}\) and all \(x\in G\). We say that a subgroup \(H\) of \(G\) is \(n\)-\(\sigma\)-embedded in \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(HT\) is \(\sigma\)-permutable in \(G\) and \(H\cap T\le H_{\sigma G}\), where \(H_{\sigma G}\) is the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(\sigma\)-permutable in \(G\).
In this paper, we study the properties of the \(n\)-\(\sigma\)-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.A nilpotency criterion for finite groups.https://zbmath.org/1449.200172021-01-08T12:24:00+00:00"Tǎrnǎuceanu, M."https://zbmath.org/authors/?q=ai:tarnauceanu.mariusSummary: Let \(G\) be a finite group. We give a criterion of nilpotency of \(G\) based on the existence of elements of certain order in each section of \(G\).Some properties of \(n\)-autocamina pairs of groups.https://zbmath.org/1449.200322021-01-08T12:24:00+00:00"Mahmudi, F."https://zbmath.org/authors/?q=ai:mahmudi.fatemeh"Gholami, A."https://zbmath.org/authors/?q=ai:gholami.ahmadSummary: The concept of auto con-cos groups and autocamina groups was introduced in 2015. In this paper, we define \(n\)-auto con-cos groups and obtain some properties of them. In final section, we introduce the concept of \(n\)-autocamina groups and investigate some equivalent conditions of them.The automorphism groups of the involution \(G\)-graph and Cayley graph.https://zbmath.org/1449.051302021-01-08T12:24:00+00:00"Ashrafi, A. R."https://zbmath.org/authors/?q=ai:ashrafi.ali-reza"Bretto, A."https://zbmath.org/authors/?q=ai:bretto.alain"Gholaminezhad, F."https://zbmath.org/authors/?q=ai:gholaminezhad.farzanehSummary: Let \(G\) be a finite group and \(\Phi (G, S)\) be the \(G\)-graph of a group \(G\) with respect to a non-empty subset \(S\). The aim of this paper is to study the structure and the automorphism group of a simple form of \(G\)-graphs for some finite groups like alternating groups, dihedral, semi-dihedral, dicyclic, and \({\mathbb{Z}_m}\rtimes_\delta{\mathbb{Z}_{2n}}\) groups, where \(\delta\) is an inverse mapping and \({V_{8n}} = \{a,b|{a^{2n}} = {b^4} = 1, aba = b^{-1}, ab^{-1}a = b\}\). Then we compare it with the automorphism group of the corresponding Cayley graph. Also we study the structure of involution \(G\)-graphs when \(S = Inv\) is the set of all involutions of \(G\).On the \(S\)-embedded and \({\mathfrak{F}_s}\)-quasinormal subgroups of finite groups.https://zbmath.org/1449.200142021-01-08T12:24:00+00:00"Liu, Yufeng"https://zbmath.org/authors/?q=ai:liu.yufeng"Mao, Yuemei"https://zbmath.org/authors/?q=ai:mao.yuemeiSummary: A subgroup \(H\) of \(G\) is said to be \(S\)-embedded in \(G\) if \(G\) has a normal subgroup \(T\) such that \(HT\) is \(s\)-permutable in \(G\) and \(H \cap T \le {H_{sG}}\), where \({H_{sG}}\) is the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(s\)-permutable in \(G\). Let \(\mathfrak{F}\) be a formation of finite groups. \(A\) subgroup \(H\) of \(G\) is said to be \({\mathfrak{F}_s}\)-quasinormal in \(G\) if \(G\) has a normal subgroup \(T\) such that \(HT\) is \(s\)-permutable in \(G\) and \( (H \cap T){H_G}/{H_G} \le Z_\infty^\mathfrak{F} (G/{H_G})\), where \(Z_\infty^\mathfrak{F} (G/{H_G})\) is the \(\mathfrak{F}\)-hypercenter of \(G/{H_G}\). In this paper, we investigate the influence of \(S\)-embedded subgroups and \({\mathfrak{F}_s}\)-quasinormal subgroups on the structure of finite groups.The classification of finite simple groups with some conditions.https://zbmath.org/1449.200072021-01-08T12:24:00+00:00"Wang, Lingli"https://zbmath.org/authors/?q=ai:wang.lingli"Zhang, Liangcai"https://zbmath.org/authors/?q=ai:zhang.liangcaiSummary: The prime graph of the finite nonabelain simple group is further investigated and the influence of vertex degrees on the structure was considered. Using number theory, the classification of complete prime graph and the adjacency criterion for the prime graph of the finite nonabelain simple groups, the classification of the finite nonabelain simple group \(G\) with complete prime graph components satisfying \(|G| < {10^{10}}\), \(7 \in \pi (G)\) and \({d_G} (7) = 1\) is obtained. The classification problem further completed the information of prime graph. And using this method, the classification of the finite nonabelain simple group with the other prime can also be solved.Automorphism groups of Suzuki-Ree groups can be characterized by their order components.https://zbmath.org/1449.200082021-01-08T12:24:00+00:00"Chen, Yanheng"https://zbmath.org/authors/?q=ai:chen.yanheng"Jia, Songfang"https://zbmath.org/authors/?q=ai:jia.songfangSummary: In a previous paper, it was proved that \(\Aut({^2{F_4}(q)})\), \(q = 2^f\) and \(\Aut({^2{G_2}(q)})\), \(q = 3^f\) can be characterized by their order components, where \(f = 3^s\), \(s\) is a positive integer. In this paper, we proved that \(\Aut({^2{B_2}(q )})\), \(q = 2^f\) and \(\Aut({^2{G_2}(q)})\), \(q = 3^f\) also can be characterized by their order components, where \(f\) is an odd prime. Combining both, we can obtain that the automorphism groups of the Suzuki-Ree simple groups whose prime graphs are not connected can be characterized by their order components.The final Moufang variety: FRUTE loops.https://zbmath.org/1449.200652021-01-08T12:24:00+00:00"Drápal, Aleš"https://zbmath.org/authors/?q=ai:drapal.ales"Phillips, J. D."https://zbmath.org/authors/?q=ai:phillips.jason-d|phillips.james-d|phillips.jon-dFRUTE loops are loops that satisfy the identity \((x\cdot xy)z=(y\cdot zx)x\). It is proved that locally finite FRUTE loops can be considered as the products \(O\times H\) of a commutative Moufang loop in which all elements are of odd order a \(2\)-group \(H\) such that the derived subloop \(H\) is of exponent two and \(H\leq Z(H)\).
Reviewer: Wiesław A. Dudek (Wrocław)On the unit groups of commutative group algebras.https://zbmath.org/1449.160482021-01-08T12:24:00+00:00"Kuneva, Velika"https://zbmath.org/authors/?q=ai:kuneva.velika-n"Mollov, Todor"https://zbmath.org/authors/?q=ai:mollov.todor-zh"Nachev, Nako"https://zbmath.org/authors/?q=ai:nachev.nako-aSummary: Let \(RG\) be the group algebra of an abelian group \(G\) over a commutative indecomposable ring \(R\) with identity of prime characteristic p and \(U(RG)\) be the unit group of \(RG\). \textit{R. B. Warfield} jun. [Bull. Am. Math. Soc. 78, 88--92 (1972; Zbl 0231.13004)] introduced the concept \(KT\)-module \(M\) over a discrete valuation ring and invariants \(W_{\alpha,q}(M)\), denoted by \(h(\alpha,M)\), for an arbitrary limit ordinal \(\alpha\) and prime \(q\). \textit{P. V. Danchev} [J. Algebra Appl. 8, No. 6, 829--836 (2009; Zbl 1183.16031)] calculated the values \(W_{\alpha,q}(U(RG))\), when the quotient group \(G_t/G_p\) is finite (Proposition 10), where \(G_t\) is the torsion subgroup of \(G\) and \(G_p\) is the \(p\)-component of \(G\). ln the present paper we establish that Proposition 10 is not valid and does not have a sense, since for an arbitrary prime \(r\ne p\) and \(G=A\times B\), where \(A\) is a \(p\)-group and \(B\) is a cyclic group of order \(r^n\), \(n\in\mathbb{N}\), we obtain the contradiction that \(W_{\alpha,q}(U(RG))\) is a fraction when \(\zeta_{p^n}\) is a primitive \(p^n\)-th root of identity over \(R\) such that \(\zeta_{r^n}\) is not a root of a polynomial over \(R\) of degree les than \(r^n\).Good congruences on abundant semigroups with quasi-ideal adequate transversals.https://zbmath.org/1449.200422021-01-08T12:24:00+00:00"Wang, Aifa"https://zbmath.org/authors/?q=ai:wang.aifaA semigroup \(S\) is \textit{abundant} if all classes of the generalized Green relations \(\mathcal{L}^*\) and \(\mathcal{R}^*\) contain idempotents. Such a semigroup is, further, \textit{adequate} if its idempotents form a semilattice, in which case the relations induce unary operations \(a \mapsto a^*\) and \(a \mapsto a^\dagger\) (more usually, now, \(a^+\)), respectively. An \textit{adequate transversal} of an abundant semigroup \(S\) is an adequate subsemigroup \(S^o\) such that (i) the generalized Green relations restrict appropriately and (ii) for any \(x \in S\), there exist a unique \(x^o \in S^o\) and idempotents \(e,f\) such that \(x = e x^o f\), where \(e \mathcal{L}^* {x^o}^\dagger\) and \(f \mathcal{R} {x^o}^*\).
\textit{J. Chen} [Semigroup Forum 60, No. 1, 57--79 (2000; Zbl 0944.20044)] introduced and studied these semigroups, especially those with quasi-ideal or multiplicative transversals, as generalizations of the regular semigroups with inverse transversals studied by various authors to that point in time. The \textit{good} congruences on an abundant semigroup are those that respect the generalized Green relations. The author describes the good congruences on an abundant semigroup with either a quasi-ideal or a multiplicative adequate tranversal in terms of equivalence triples, defined relative to the \((E^o, I, \Lambda)\)-systems used by Chen to coordinatize such semigroups.
Reviewer: Peter R. Jones (Milwaukee)Monomorphisms in categories of firm acts.https://zbmath.org/1449.200572021-01-08T12:24:00+00:00"Laan, Valdis"https://zbmath.org/authors/?q=ai:laan.valdis"Reimaa, Ülo"https://zbmath.org/authors/?q=ai:reimaa.uloA right \(S\)-act \({A_S}\) of a semigroup \({\mathcal{S}}\) is firm if the mapping \(A \otimes S \to A\) is bijective; \(S\) is firm if it is firm as a right (left) \(S\)-act; \({A_S}\) is unitary if \(AS = A\). In the category of firm acts, monomorphisms are not necessarily injective and here an example of a non-injective monomorphism is presented. It is also shown that in the category of firm acts over a firm semigroup monomorphisms coincide with regular monomorphisms and extremal monomorphisms. For the category of unitary right \(S\)-acts and firm right \(S\)-acts, necessary and sufficient conditions for all monomorphisms to be injective are provided.
Reviewer: Jaak Henno (Tallinn)On the constructions of two semigroups defined by presentations and their tests by GAP.https://zbmath.org/1449.200472021-01-08T12:24:00+00:00"Liu, Jingguo"https://zbmath.org/authors/?q=ai:liu.jingguoSummary: In this paper, the author reinvestigates two semigroups defined by presentations, which were used in a paper of the author to answer an open problem in this field. The Cayly tables of these two semigroups are given, and then the two semigroups are realized by means of transformations by GAP, aiming to show that elements in each semigroup are all distinct. The attributes of the two semigroups are also verified by GAP.The structure of totally ordered E-unitary inverse residuated lattice-ordered monoids.https://zbmath.org/1449.060222021-01-08T12:24:00+00:00"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.1|chen.wei|chen.wei.2|chen.wei.3|chen.wei.4"Ruan, Xiaojun"https://zbmath.org/authors/?q=ai:ruan.xiaojunSummary: This paper studies totally ordered E-unitary inverse residuated lattice-ordered monoids. The structure theorem for such residuated lattice-ordered monoids is established, which generalizes the result of a literature.Subgroup growth of virtually cyclic right-angled Coxeter groups and their free products.https://zbmath.org/1449.200362021-01-08T12:24:00+00:00"Baik, Hyungryul"https://zbmath.org/authors/?q=ai:baik.hyungryul"Petri, Bram"https://zbmath.org/authors/?q=ai:petri.bram"Raimbault, Jean"https://zbmath.org/authors/?q=ai:raimbault.jeanLet \(\Gamma\) be a finitely generated group, and \(n\) a positive integer, so that the number \(s_{n}(\Gamma)\) of subgroups of \(\Gamma\) of index \(n\) is finite. In [J. Lond. Math. Soc., II. Ser. 101, No. 2, 556--588 (2020; Zbl 07216611)], the authors studied the factorial growth rate of \(s_{n}(\Gamma)\) for right-angled Artin and Coxeter groups \(\Gamma\), that is, the limits \[\lim_{n \to \infty} \frac{\log(s_{n}(\Gamma))}{n \log(n)}.\] The goal of the paper under review is to treat in detail the subclass of virtually cyclic Coxeter groups and their free products. It turns out that one can obtain much finer estimates than it was possible in the general case, where the limits were determined explicitly for Artin groups, but not for all Coxeter groups.
Recall that, given a graph \(\mathcal{G}\) with vertex set \(V\) and edge set \(E\), the right-angled Coxeter group associated to it is \[\Gamma^{\mathrm{Cox}}({\mathcal{G}})=\langle\sigma_{v} : v \in V,\sigma_{v}^{2} = e, \; {\text{for}}\; v \in V,[\sigma_{v}, \sigma_{w}] = e, \; {\text{for}}\; \{ v, w \} \in E \rangle. \] In the proofs, an asymptotic formula for the number \(n_{n}(\Gamma)\) of permutation representations of degree \(n\) of a free product \(\Gamma\) of certain right-angled Coxeter groups plays a crucial role; this generalises the asymptotic formula implicit in \textit{S. Chowla} et al. [Can. J. Math. 3, 328--334 (1951; Zbl 0043.25904)]. This is based on the fact that the exponential generating function for the sequence of the \(h_{n}(\Gamma)\) converges.
Reviewer: Andrea Caranti (Trento)Quasi-orthodox quasi completely regular semirings.https://zbmath.org/1449.160982021-01-08T12:24:00+00:00"Maity, S. K."https://zbmath.org/authors/?q=ai:maity.sunil-kumar"Ghosh, R."https://zbmath.org/authors/?q=ai:ghosh.rituparnaSummary: The aim of this paper is to characterize a class of additively quasi regular semirings which are subdirect products of an idempotent semiring and a \(b\)-lattice of quasi skew-rings.Regular bisimple \({\omega^2}\)-semigroups.https://zbmath.org/1449.200452021-01-08T12:24:00+00:00"Wang, Limin"https://zbmath.org/authors/?q=ai:wang.limin"Shang, Yu"https://zbmath.org/authors/?q=ai:shang.yu"Feng, Yingying"https://zbmath.org/authors/?q=ai:feng.yingyingSummary: In this paper, we study the \({\omega^2}\)-chain of idempotent elements and generalized Bruck-Reilly expansion. By using the expansion method, the structure theorem of regular bisimple \({\omega^2}\)-semigroups is obtained.The inverse monoid associated to a group and the semidirect product of groups.https://zbmath.org/1449.200462021-01-08T12:24:00+00:00"Ghadbane, N."https://zbmath.org/authors/?q=ai:ghadbane.nacerSummary: In this paper, we construct an inverse monoid \(M(G)\) associated a given group \(G\) by using the notion of the join of subgroups and then, by applying the left action of monoid \(M\) on a semigroup \(S\), we form a semigroup \(S\omega M\) on the set \(S\times M\). The finally result is to build the semi direct product of groups associated to the group action on an another group.Difference bases in finite abelian groups.https://zbmath.org/1449.050342021-01-08T12:24:00+00:00"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Gavrylkiv, Volodymyr"https://zbmath.org/authors/?q=ai:gavrylkiv.volodymyr-mSummary: A subset \(B\) of a group \(G\) is called a difference basis of \(G\) if each element \(g\in G\) can be written as the difference \(g =ab^{-1}\) of some elements \(a,b\in B\). The smallest cardinality \(|B|\) of a difference basis \(B\subset G\) is called the difference size of \(G\) and is denoted by \(\Delta [G]\). The fraction \(\partial [G]:=\Delta [G]/\sqrt{|G|}\) is called the difference characteristic of \(G\). Using properties of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite abelian groups. In particular, we prove that for a prime number \(p\geq 11\), any finite abelian \(p\)-group \(G\) has difference characteristic \(\partial [G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot \sup \partial [C_{p^k}]<\sqrt{2}\cdot \frac{\sqrt{p}-1}{\sqrt{p}-3}\). Also we calculate the difference sizes of all abelian groups of cardinality less than 96.Rough fuzzy ternary subsemigroups based on fuzzy ideals with three-dimensional congruence relation.https://zbmath.org/1449.200732021-01-08T12:24:00+00:00"Bashir, Shahida"https://zbmath.org/authors/?q=ai:bashir.shahida"Abbas, Hasnain"https://zbmath.org/authors/?q=ai:abbas.hasnain"Mazhar, Rabia"https://zbmath.org/authors/?q=ai:mazhar.rabia"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammadSummary: The main objective of the proposed work in this paper is to introduce a generalized form of rough fuzzy subsemigroups, which is rough fuzzy ternary subsemigroups (RFTSs) combining the notions of fuzziness and roughness in ternary semigroups. In RFTSs, we deal with vague and incomplete information in decision-making problems. RFTSs are characterized by lower and upper approximations using fuzzy ideals. In this research, we propose the three-dimensional k-level relation and proved that this relation is a congruence relation on a ternary semigroup. Furthermore, comparing it with the previous literature, we conclude that our proposed technique is better and effective because it deals with vague problems and there are many structures which are not handled using binary multiplication such as all the sets of negative numbers. In addition, we have proved by counterexamples that converses of many parts of many results do not hold which have negated the results proved in \textit{Q. Wang} and \textit{J. Zhan}'s paper [Open Math. 14, 1114--1121 (2016; Zbl 1372.20060)].On covers of \(S\)-act \(A (I)\).https://zbmath.org/1449.200592021-01-08T12:24:00+00:00"Qiao, Husheng"https://zbmath.org/authors/?q=ai:qiao.husheng"Chen, Qian"https://zbmath.org/authors/?q=ai:chen.qianSummary: Let \(S\) be a monoid and \(I\) a proper right ideal of \(S\), and let \(A (I)\) be the amalgam of two copies of \(S\) with the core \(I\). As an important tool, \(A (I)\) is used to investigate homological classification problems of \(S\) acts, but its flatness covers have not been considered yet. We investigated the flatness covers of \(A (I)\) and extended some known results.Some classes of semihypergroups of type \(U\) on the right.https://zbmath.org/1449.200682021-01-08T12:24:00+00:00"Heidari, D."https://zbmath.org/authors/?q=ai:heidari.dariushSummary: In this paper, we prove various inclusion relationships among different classes of algebraic structures and hyperstructures of type \(U\) on the right of finite size. In particular, we consider in detail the inclusion properties \(\mathcal{G}_n\subseteq \mathcal{PUR}_n\subseteq \mathcal{HUR}_n\subseteq \mathcal{SUR}_n\) between the classes of groups \(\mathcal{G}_n\), polygroups \(\mathcal{PUR}_n\), hypergroups \(\mathcal{HUR}_n\) and semihypergroups \(\mathcal{SUR}_n\) of type \(U\) on the right of size \(n\) and provide conditions such that the equality holds. As a particular result, we prove that every polygroup of type \(U\) on the right is a group.A note on the structure of the augmentation quotient group for some non-abelian \(p\)-groups.https://zbmath.org/1449.200182021-01-08T12:24:00+00:00"Wang, Xiulan"https://zbmath.org/authors/?q=ai:wang.xiulan"Zhou, Qingxia"https://zbmath.org/authors/?q=ai:zhou.qingxiaSummary: Let \(G\) be a finite non-abelian \(p\)-group of order \({p^k}\) with a cyclic subgroup of index \(p\), where \(p \ne 2\), \(k \ge 3\). The authors had given a basis of \(n\)-th augmentation ideal and the structure of \(n\)-th augmentation quotient group for the group \(G\), but it is not very exact. In our research, following their method in literature, we construct an exact basis of \(n\)-th augmentation ideal and the structure of \(n\)-th augmentation quotient group for the group \(G\).The collection formula of a nilpotent group of class 3.https://zbmath.org/1449.200342021-01-08T12:24:00+00:00"Fu, Yuan"https://zbmath.org/authors/?q=ai:fu.yuan"Yi, Xiaolu"https://zbmath.org/authors/?q=ai:yi.xiaolu"Gao, Rui"https://zbmath.org/authors/?q=ai:gao.rui"Che, Chenxi"https://zbmath.org/authors/?q=ai:che.chenxiSummary: Based on the basic commutator calculus formula, for the nilpotent group of class 3, by calculating the collection formula of 3, 4 elements respectively, we get the collection formula of finite elements by induction.The unit triangular group over \(Q_{\pi_{ij}}\).https://zbmath.org/1449.200352021-01-08T12:24:00+00:00"Gao, Rui"https://zbmath.org/authors/?q=ai:gao.rui"Fu, Yuan"https://zbmath.org/authors/?q=ai:fu.yuan"Yi, Xiaolu"https://zbmath.org/authors/?q=ai:yi.xiaoluSummary: Let \[G = \left\{ \left(\begin{array}{*{20}{c}}1 & k_{12}\alpha_{12} & \cdots & k_{1n}\alpha_{1n}\\ 0 & 1 & \cdots & k_{2n}\alpha_{2n}\\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \vdots & 1\end{array}\right)\mid \alpha_{ij}\in Q_{\pi_{ij}}\right\},\] where if \(k_{ij} = p_1^{e_1}p_2^{e_2} \cdots p_n^{e_n}\), then \(p_i\notin \pi_{ij}\). Then \(G\) is a group if and only if the element in position \( (ij)\) of matrix satisfies the conditions \(\pi_{ij} \supseteq \mathop \bigcup \limits_{1 \le i < l < j \le n} (\pi_{il} \cup \pi_{lj})\), and \(k_{ij}\) divides \(k_{il}k_{lj}\) (\(1 \le i < l < j \le n\)). When \(G\) is a group, both the upper central series and the lower central series of \(G\) coincide if and only if \(\pi_{ij} \supseteq \mathop \bigcup \limits_{1 \le i < l < j \le n} (\pi_{il} \cup \pi_{lj})\) and \(k_{ij} = d_{ij}^{ (m)}\), where \(d_{ij}^{ (m)}\) denotes the greatest common divisor of all \(k_{ij_1}k_{l_1l_2} \cdots k_{l_{m - 1}j}\) (\(1 \le i < {l_1} < {l_2} < \dots < {l_{m - 1}} < j \le n\)).A finite group with three conjugacy classes of non-abelian subgroups.https://zbmath.org/1449.200272021-01-08T12:24:00+00:00"Chen, Wei"https://zbmath.org/authors/?q=ai:chen.wei.3|chen.wei|chen.wei.4|chen.wei.2|chen.wei.1"Yang, Guifang"https://zbmath.org/authors/?q=ai:yang.guifang"Meng, Wei"https://zbmath.org/authors/?q=ai:meng.weiSummary: Because the number of conjugacy classes of non-abelian subgroups has much influence on the structure of finite groups, there have been quite a few meaningful results in this field. Let \(G\) be a finite group and \(\tau (G)\) denote the number of conjugacy classes of all non-abelian subgroups of \(G\). This research investigates the properties of the finite groups with \(\tau (G) = 3\) and gives a complete classification of these finite groups.Equivalent characterizations of cyclic groups and abelian groups.https://zbmath.org/1449.200382021-01-08T12:24:00+00:00"Shi, Jiangtao"https://zbmath.org/authors/?q=ai:shi.jiangtao"Bi, Lingxiao"https://zbmath.org/authors/?q=ai:bi.lingxiao"Li, Na"https://zbmath.org/authors/?q=ai:li.naSummary: Considering some abelian subgroups with special normalizers, we proved the following equivalent characterizations of cyclic groups and abelian groups by elementary methods: let \(G\) be a finite group, then \(G\) is a cyclic group if and only if the normalizer of every minimal subgroup of \(G\) is a cyclic group; \(G\) is an abelian group if and only if the normalizer of every elementary abelian subgroup of \(G\) is an abelian group.Some criteria on quasi-\(\mathcal{F}\)-groups.https://zbmath.org/1449.200112021-01-08T12:24:00+00:00"Chen, Qiaolin"https://zbmath.org/authors/?q=ai:chen.qiaolin"You, Ze"https://zbmath.org/authors/?q=ai:you.ze"Li, Baojun"https://zbmath.org/authors/?q=ai:li.baojun.1|li.baojunSummary: Let \(\mathcal{F}\) be a class of group. A group \(G\) is said to be a quasi-\(\mathcal{F}\)-group, if \(x\) induces an inner automorphism on \(H/K\) for any \(x \in G\) and any \(\mathcal{F}\)-eccentric chief factor \(H/K\). By using \(\prod\)-normality of some primary subgroups, some new criteria for a group to be a quasi-\(\mathcal{F}\)-group are obtained.New types of soft rough sets in groups based on normal soft groups.https://zbmath.org/1449.200722021-01-08T12:24:00+00:00"Ayub, Saba"https://zbmath.org/authors/?q=ai:ayub.saba"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammad"Mahmood, Waqas"https://zbmath.org/authors/?q=ai:mahmood.waqasSummary: Hybridization of soft sets and rough sets is an important way to deal with uncertainties. This paper aims to study the concept of roughness in soft sets over groups. In this regard, a pair of two soft sets, viz. soft lower and soft upper approximation spaces, are introduced by applying the normal soft groups corresponding to each parameter. Some important results related to these soft approximation spaces over groups are studied with examples. Furthermore, this paper presents a relationship between the soft approximation spaces based on the soft image and soft pre-image of a normal soft group via group homomorphisms. This work can be applicable in the field of information technology to connect two information systems.A description of the automorphism groups of the convex regular 4-polytopes.https://zbmath.org/1449.051312021-01-08T12:24:00+00:00"Cai, Qi"https://zbmath.org/authors/?q=ai:cai.qi"Yu, Lu"https://zbmath.org/authors/?q=ai:yu.lu"Zhang, Hua"https://zbmath.org/authors/?q=ai:zhang.huaSummary: In combinatorial mathematics, it is an important and often a difficult problem of determining the automorphism group of a graph or various combinatorial structures. In this paper, we depict the automorphism groups of the convex regular 4-polytopes by using the basic theory of graphs and permutation groups.Ranks of the semigroup \({\mathcal{O}_n} (k,m)\).https://zbmath.org/1449.200542021-01-08T12:24:00+00:00"Zhang, Chuanjun"https://zbmath.org/authors/?q=ai:zhang.chuanjun"Chen, Songliang"https://zbmath.org/authors/?q=ai:chen.songliangSummary: Let \({\mathcal{O}_n}\) be the semigroup of all order-preserving transformations on a finite-chain \([n]\). For an arbitrary integers \({k, m}\) such that \(1 \le k \le n-1\) and \(2 \le m \le n\), the rank and idempotent rank of the semigroup \({\mathcal{O}_n} (k, m) = \{\alpha \in {\mathcal{O}_n}\mid k\alpha \le k, m\alpha \ge m\}\) were studied.Green's D-relation on a semiring CR\( (n,1)\).https://zbmath.org/1449.160972021-01-08T12:24:00+00:00"Lian, Lifeng"https://zbmath.org/authors/?q=ai:lian.lifengSummary: This paper studied the Green's relation of semiring whose additive semigroup should be a band and multiplicative semigroup should be a completely regular semigroup. The characterization of relations of \({\mathrm{\dot D}} \bigcap \overset{+}{\mathrm{D}}\), \({\mathrm{\dot D}} \bigcap {\overset{+}{\mathrm{L}}}\), \({\mathrm{\dot D}} \bigcap \overset{+}{\mathrm{R}}\) is carried out and sufficient and necessary conditions for \({\overset{+}{\mathrm{D}}} \bigcap {\overset{+}{\mathrm{D}}}\) to be a congruence relation is given.Bipolar valued fuzzy bi-ideals with thresholds \( (\lambda,\mu)\) in semigroups.https://zbmath.org/1449.200802021-01-08T12:24:00+00:00"Wang, Fengxiao"https://zbmath.org/authors/?q=ai:wang.fengxiaoSummary: The concept of bipolar valued fuzzy set is applied to semigroups, and the concepts of bipolar valued fuzzy bi-ideals with thresholds \( (\lambda, \mu)\) and bipolar valued fuzzy generalized bi-ideals with thresholds \( (\lambda, \mu)\) of semigroups are introduced. The basic properties of \( (\lambda, \mu)\)-bipolar valued fuzzy (generalized) bi-ideals of semigroups are discussed. The relations between \( (\lambda, \mu)\)-bipolar valued fuzzy bi-ideals of semigroups and fuzzy bi-ideals and bi-ideals of semigroups are given. It is proved that the intersection and direct product of \( (\lambda, \mu)\)-bipolar valued fuzzy (generalized) bi-ideals of semigroups are still \( (\lambda, \mu)\)-bipolar valued fuzzy (generalized) bi-ideals.Green\(^*\)-relations of the half-transitive semigroup \(M (a)\).https://zbmath.org/1449.200402021-01-08T12:24:00+00:00"Zhu, Jianxin"https://zbmath.org/authors/?q=ai:zhu.jianxin"Yang, Xiuliang"https://zbmath.org/authors/?q=ai:yang.xiuliangSummary: Let \({I_n}\) be the symmetric inverse semigroup of \({X_n} = \{1, 2, \cdots, n\}\), \(M (A)\) be the half-transitive subsemigroup of \({I_n}\). This paper investigates the Green-relations and the Green\(^*\)-relations of \(M (A)\), and proves that \(M (A)\) is a type \(A\) semigroup.