Recent zbMATH articles in MSC 06https://zbmath.org/atom/cc/062021-01-08T12:24:00+00:00WerkzeugSymmetric polynomial-like Boolean functions.https://zbmath.org/1449.060212021-01-08T12:24:00+00:00"Gonda, János"https://zbmath.org/authors/?q=ai:gonda.janosSummary: Polynomial-like Boolean functions form a class of the Boolean functions invariant with respect to a special transform of the linear space of the two-valued logical functions. Another special set of the Boolean-functions are the set of the symmetric functions. In this article we introduce the class of the symmetric polynomial-like Boolean functions and investigate some elementary properties of such functions.Injective hulls of \(L\)-ordered algebras.https://zbmath.org/1449.060292021-01-08T12:24:00+00:00"Tian, Tian"https://zbmath.org/authors/?q=ai:tian.tian.2|tian.tian.1|tian.tian"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin.1|zhao.binSummary: We discussed the injectivity of the category of \(L\)-ordered algebras, and proved that \(\mathrm{M}^\le\)-injective objects in the category of \(L\)-ordered algebras were exactly \(L\)-sup-algebras, and gave the concrete form of \(\mathrm{M}^\le\)-injective hulls of a special class of \(L\)-ordered algebras in the category of \(L\)-ordered algebras.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 \(S_n^-\).Different types of cubic ideals in BCI-algebras based on fuzzy points.https://zbmath.org/1449.060302021-01-08T12:24:00+00:00"Jana, Chiranjibe"https://zbmath.org/authors/?q=ai:jana.chiranjibe"Senapati, Tapan"https://zbmath.org/authors/?q=ai:senapati.tapan"Pal, Madhumangal"https://zbmath.org/authors/?q=ai:pal.madhumangal"Borumand Saeid, Arsham"https://zbmath.org/authors/?q=ai:borumand-saeid.arshamSummary: The notions of \((\in ,\in \vee q)\)-cubic \(p\)- (\(a\)- and \(q\)-) ideals of BCI-algebras are introduced and some related properties are investigated. Several characterization for these generalized \((\in ,\in \vee q)\)-cubic ideals are defined and relationship between \((\in ,\in \vee q)\)-cubic \(p\)-ideals, \((\in ,\in \vee q)\)-cubic \(q\)-deals and \((\in ,\in \vee q)\)-cubic \(a\)-ideals of BCI-algebras are discussed.The category of upper bounded bifinite posets.https://zbmath.org/1449.060122021-01-08T12:24:00+00:00"Li, Jibo"https://zbmath.org/authors/?q=ai:li.jibo"Chen, Yanchang"https://zbmath.org/authors/?q=ai:chen.yanchang"Zhang, Haixia"https://zbmath.org/authors/?q=ai:zhang.haixiaSummary: In this paper, some results about \(\mathcal{D}\)-precontinuous or \(\mathcal{D}\)-prealgebraic posets and \({\mathcal{D}^\Delta}\)-continuous functions are summarized and supplemented. The category BFBP, in which objects are upper bounded bifinite posets and arrows are \({\mathcal{D}^\Delta}\)-continuous functions between them, is shown to be cCartesian closed.Cubic intuitionistic structure of KU-algebras.https://zbmath.org/1449.060352021-01-08T12:24:00+00:00"Senapati, Tapan"https://zbmath.org/authors/?q=ai:senapati.tapan"Jun, Young Bae"https://zbmath.org/authors/?q=ai:jun.young-bae"Shum, K. P."https://zbmath.org/authors/?q=ai:shum.kar-pingSummary: In this paper, the notions of cubic intuitionistic KU-subalgebras and KU-ideals of a KU-algebras are introduced, and several properties are investigated. Characterizations of cubic intuitionistic KU-subalgebras and KU-ideals of KU-algebras are considered. Relations between a cubic intuitionistic KU-subalgebra and a cubic intuitionistic KU-ideal are given.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.\(\oplus\)-supplemented lattices.https://zbmath.org/1449.060142021-01-08T12:24:00+00:00"Biçer, Çiğdem"https://zbmath.org/authors/?q=ai:bicer.cigdem"Nebiyev, Celil"https://zbmath.org/authors/?q=ai:nebiyev.celilSummary: In this work, \(\oplus\)-supplemented and strongly \(\oplus\)-supplemented lattices are defined and investigated some properties of these lattices. Let \(L\) be a lattice and \(1=a_{1}\oplus a_{2}\oplus \dots\oplus a_{n}\) with \(a_{1},a_{2},\dots,a_{n}\in L\). If \(a_{i}/0\) is \(\oplus\)-supplemented for each \(i=1,2,\dots,n\), then \(L\) is also \(\oplus\)-supplemented. Let \(L\) be a distributive lattice and \(1=a_{1}\oplus a_{2}\oplus \dots\oplus a_{n}\) with \(a_{1},a_{2},\dots,a_{n}\in L\). If \(a_{i}/0\) is strongly \(\oplus\)-supplemented for each \(i=1,2,\dots,n\), then \(L\) is also strongly \(\oplus\)-supplemented. A lattice \(L\) is said to have property \(( D1) \) if for every element \(a\) of \(L\), there exist \(a_{1},a_{2}\in L\) such that \(1=a_{1}\oplus a_{2}\), \(a_{1}\in a/0\) and \(a_{2}\wedge a\ll a_{2}/0\). It is shown that a lattice \(L\) has property \(( D1) \) if and only if \(L\) is amply supplemented and strongly \(\oplus\)-supplemented.On the positive definite solution of a class of pair of nonlinear matrix equations.https://zbmath.org/1449.150382021-01-08T12:24:00+00:00"Ali, Hasem"https://zbmath.org/authors/?q=ai:ali.hasem"Hossein, Sk M."https://zbmath.org/authors/?q=ai:hossein.sk-monowarSummary: We find some necessary and sufficient conditions for the existence of Hermitian positive definite solution of a pair of nonlinear matrix equations of the form:
\[\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1 \\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}\]
and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.BF-ideals in negative non-involutive residuated lattices.https://zbmath.org/1449.060072021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhui"Jiang, Zhiting"https://zbmath.org/authors/?q=ai:jiang.zhiting"Qin, Xuecheng"https://zbmath.org/authors/?q=ai:qin.xuechengSummary: In order to study the structural characteristics of negative non-involutive residuated lattices, the concept of BF-ideal of negative non-involutive residuated lattice is introduced and its properties are investigated. It is proved that BF-intersection of some BF-ideals, homomorphism image and inverse image of a BF-ideal are also BF-ideals. At the same time, a condition which makes BF-union of some BF-ideals to be a BF-ideal is given.MV-algebra valued metric-based fuzzy rough sets.https://zbmath.org/1449.030402021-01-08T12:24:00+00:00"Xiong, Xingguo"https://zbmath.org/authors/?q=ai:xiong.xingguo"Lu, Lingxia"https://zbmath.org/authors/?q=ai:lu.lingxiaSummary: MV-algebra valued metric-based fuzzy rough set model is defined. The interrelations between \(\oplus\)-hemimetrics and the standard real valued hemimetrics are investigated. It is shown that \(\oplus\)-hemimetrics and \(\otimes\)-similarities are equivalent to each other. The properties of fuzzy rough approximation operators and the related definable sets are studied.Some properties of prime and z-semi-ideals in posets.https://zbmath.org/1449.060112021-01-08T12:24:00+00:00"Porselvi, Kasi"https://zbmath.org/authors/?q=ai:porselvi.kasi"Elavarasan, Balasubramanian"https://zbmath.org/authors/?q=ai:elavarasan.balasubramanianSummary: We define the notion of z-semi-ideals in a poset \(P\) and we show that if a z-semi-ideal \(J\) satisfies \((\ast )\)-property, then every minimal prime semi-ideal containing \(J\) is also a z-semi-ideal of \(P\). We also show that every prime semi-ideal is a z-semi-ideal or the maximal z-semi-ideals contained in it are prime z-semi-ideals. Further, we characterize some properties of union of prime semi-ideals of \(P\) provided the prime semi-ideals are contained in the unique maximal semi-ideal of \(P\).The primary congruence and the primary decomposition of congruence on semirings.https://zbmath.org/1449.161002021-01-08T12:24:00+00:00"Wu, Ya'nan"https://zbmath.org/authors/?q=ai:wu.yanan"Ren, Miaomiao"https://zbmath.org/authors/?q=ai:ren.miaomiaoSummary: We study primary congruences on commutative idempotent semirings, give definition of \(\rho\)-congruence, and obtain some results of their structures. On this basis, a uniqueness theorem is obtained by studying the minimal primary congruence decomposition.Coaxer pseudo-complemented almost distributive lattices.https://zbmath.org/1449.060162021-01-08T12:24:00+00:00"Rafi, N."https://zbmath.org/authors/?q=ai:rafi.noorbhasha"Bandaru, Ravi Kumar"https://zbmath.org/authors/?q=ai:bandaru.ravi-kumar"Rao, G. C."https://zbmath.org/authors/?q=ai:rao.g-chakradhara|rao.g-chakradharSummary: The concepts of coaxer ideals in a pseudo-complemented almost distributive lattice (PCADL) and coaxer pseudo-complemented almost distributive lattices are introduced. We characterized the coaxer PCADL in terms of coaxer ideals and maximal ideals. Finally, we established equivalent conditions for Boolean algebras in terms of coaxer ideals and congruences.Generalized relative annihilators in \(\mathrm{R}_0\)-algebras.https://zbmath.org/1449.030542021-01-08T12:24:00+00:00"Zhang, Lifang"https://zbmath.org/authors/?q=ai:zhang.lifang"Wu, Hongbo"https://zbmath.org/authors/?q=ai:wu.hongboSummary: Firstly, using the method of extended filters, we introduced the concept of the relative annihilator in \(\mathrm{R}_0\)-algebras, and proposed the concept of the generalized relative annihilator combined with the concept of filters. It was proved that the generalized relative annihilators in \(\mathrm{R}_0\)-algebras were still filters. Secondly, we described the prime filters by using the generalized relative annihilator, and discussed the relationship between the relative annihilators and generalized relative annihilators. Finally, based on two given elements in \(\mathrm{R}_0\)-algebras, we gave a complete residuated lattice structure which satisfied join-infinite distributive law in the family of generalized relative annihilators.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 \(0 \le l \le r\).Idempotent and nilpotent matrices of triangular modules on the distributive lattice.https://zbmath.org/1449.150092021-01-08T12:24:00+00:00"Li, Aimei"https://zbmath.org/authors/?q=ai:li.aimei"Wu, Miaoling"https://zbmath.org/authors/?q=ai:wu.miaoling"Wang, Yaxian"https://zbmath.org/authors/?q=ai:wang.yaxianSummary: In this paper, we introduce some vital properties which include irreflexivity, reflexivity and idempotence of \(T\)-idempotent matrix and \(S\)-idempotent matrix over the distributive lattice. In addition, we also give the sufficient conclusions under which irreflexive matrix becomes \(T\)-nilpotent matrix and \(S\)-nilpotent matrix, and prove them with our own method and improved method.Relative annihilators and filters in almost semilattice.https://zbmath.org/1449.060032021-01-08T12:24:00+00:00"Nanaji, Rao G."https://zbmath.org/authors/?q=ai:nanaji.rao-g"Beyene, Terefe Getachew"https://zbmath.org/authors/?q=ai:beyene.terefe-getachewSummary: In this paper, we firstly introduced the concept of relative annihilators in almost semilattice \( (ASL)\) and proved some basic properties of relative annihilators. The concepts of filter and principal filter are introduced in an \(ASL\). We then proved that the set \(\mathcal{F} (L)\) of all filters of an \(ASL\) \(L\) with unimaximal element is a lattice. Also, we proved that the set of all principal filters \(P\mathcal{F} (L)\) of \(L\) is a semilattice and \(P\mathcal{F} (L)\) is isomorphic to the semilattice \(P\mathfrak{I} (L)\) of all principal ideal of \(L\). Next, we gave that a set of equivalent conditions for an arbitrary family of ideals (filters) is again an ideal (filter). We proved that the lattice of all filters of \(L\) is isomorphic to the lattice of all filters of an \(ASL\) \(P\mathfrak{I} (L)\). Finally, we proved that the lattice \(\mathcal{F} (L)\) of all filters of an \(ASL\) \(L\) is isomorphic to the lattice \(\mathcal{F} (P\mathfrak{I} (L))\) \( (\mathcal{F} (P\mathcal{F} (L)))\) of all filters of an \(ASL\) \(P\mathfrak{I} (L)\) \( (P\mathcal{F} (L))\) and hence \(\mathcal{F} (P\mathcal{F} (L))\) is isomorphic to \(\mathcal{F} (P\mathfrak{I} (L))\). Prime ideal conditions on an \(ASLs\) are investigated in connection with the relative annihilators.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.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.On \( (\sigma, \tau)\)-derivations of BCI-algebras.https://zbmath.org/1449.060332021-01-08T12:24:00+00:00"Muhiuddin, G."https://zbmath.org/authors/?q=ai:muhiuddin.ghulam|muhiuddin.gulamSummary: In this paper, we introduce the notion of \( (\sigma, \tau)\)-derivations of a BCI-algebra and investigate their related properties. Moreover, we study \( (\sigma, \tau)\)-derivations in a \(p\)-semisimple BCI-algebra and establish some results.Fuzzyifying ideals of pseudo BCI-algebras based on complete residuated lattice-valued logic.https://zbmath.org/1449.060342021-01-08T12:24:00+00:00"Peng, Jiayin"https://zbmath.org/authors/?q=ai:peng.jiayinSummary: By a unary fuzzy predicate calculus on complete residuated lattice-valued logic, the classic fuzzy ideal, classic fuzzy \(p\)-ideal, classic fuzzy associative ideal, classic fuzzy \(q\)-ideal and classic fuzzy \(a\)-ideal in pseudo BCI-algebra are recharacterized. The concepts of \(l\)-valued fuzzy ideals, \(l\)-valued fuzzy \(p\)-ideals, \(l\)-valued fuzzy associative ideals, \(l\)-valued fuzzy \(q\)-ideals and \(l\)-valued fuzzy \(a\)-ideals in pseudo BCI-algebra are introduced. Using the semantic method of complete residuated lattice-valued logic, the properties and relations of these \(l\)-valued fuzzy ideals are studied. Conditions for an \(l\)-valued fuzzy ideal to be an \(l\)-valued fuzzy \(p\)-ideal (resp. \(l\)-valued fuzzy \(q\)-ideal) are provided. The invariances of these \(l\)-valued fuzzy ideals under the intersection, homomorphic mapping and Descartes product operations are investigated, and the corresponding and existing conclusions in classical fuzzy cases are generalized.On the invertible elements of quantum B-algebras.https://zbmath.org/1449.030482021-01-08T12:24:00+00:00"Gu, Xiaojuan"https://zbmath.org/authors/?q=ai:gu.xiaojuan"Han, Shengwei"https://zbmath.org/authors/?q=ai:han.shengweiSummary: Quantum B-algebras are a class of non-commutative logical algebras, which can be seen as a generalization of quantales. Quantum B-algebras cover a majority of implication algebras such as partially ordered groups which are a special class of quantum B-algebras. In this paper, the invertible elements of quantum B-algebras are investigated. It is shown that the set of all invertible elements of quantum B-algebras is a partially ordered group.Closure filters of decomposable MS-algebras.https://zbmath.org/1449.060172021-01-08T12:24:00+00:00"El-Mohsen, Badawy Abd"https://zbmath.org/authors/?q=ai:el-mohsen.badawy-abd"El-Fawal, R."https://zbmath.org/authors/?q=ai:el-fawal.ragaaSummary: In this paper, we introduce the notion of boosters in MS-algebras and study their properties. We investigate more properties of boosters for decomposable MS-algebras. We obtain a closure operator in the filter lattice of a decomposable MS-algebra and with using this operator we introduce the concept of closure filters and characterize such filters in terms of boosters. An isomorphism is obtained between the lattice of closure filters and the ideal lattice of boosters. Also, we study many properties of closure filters with respect to homomorphisms and direct products of decomposable MS-algebras.Some results about atoms and branches of BCH-algebra.https://zbmath.org/1449.060322021-01-08T12:24:00+00:00"Li, Jinlong"https://zbmath.org/authors/?q=ai:li.jinlongSummary: In a BCH-algebra \(\langle{X; *, 0}\rangle\), it was proved that the generalized associative product of any element in \(X\) was equal to the atom in its branch, a sufficient condition under which the union of two branches was an ideal of \(X\) was given. In a partial ordering BCH-algebra \(\langle{X; *, 0}\rangle\), a sufficient condition under which the product of two nonempty subsets of \(X\) had the minimal element was given, a sufficient and necessary condition under which the union of two branches was a subalgebra of \(X\) was given.Hesitant fuzzy sets applied to BCK/BCI-algebras.https://zbmath.org/1449.060312021-01-08T12:24:00+00:00"Kim, J."https://zbmath.org/authors/?q=ai:kim.junhui"Lim, P. K."https://zbmath.org/authors/?q=ai:lim.pyung-ki"Lee, J. G."https://zbmath.org/authors/?q=ai:lee.jeong-gon"Hur, K."https://zbmath.org/authors/?q=ai:hur.kulSummary: We define a hesitant fuzzy BCK/BCI-algebra and obtain some of its properties. Next, we introduce the concept of hesitant fuzzy ideal and obtain some of its properties and give some examples. Finally, we define a hesitant fuzzy positive implicative ideal, a hesitant fuzzy implicative ideal, a hesitant fuzzy commutative ideal and investigate some of its properties, respectively and their relations. In particular, we give characterizations of hesitant fuzzy positive implicative ideal, a hesitant fuzzy implicative ideal and a hesitant fuzzy commutative ideal, respectively.\(n\)-absorbing \(\delta\)-primary elements in a multiplicative lattice.https://zbmath.org/1449.060092021-01-08T12:24:00+00:00"Nimbhorkar, Shriram K."https://zbmath.org/authors/?q=ai:nimbhorkar.shriram-khanderao"Nehete, Jaya Y."https://zbmath.org/authors/?q=ai:nehete.jaya-ySummary: We study the concepts of an \(n\)-absorbing and a weakly \(n\)-absorbing \(\delta\)-primary element in a multiplicative lattice. Also, we introduce the concept of an almost \(\delta\)-primary element and an \(n\)-almost \(n\)-absorbing \(\delta\)-primary element in a multiplicative lattice. We give a characterization for a weakly \(n\)-absorbing \(\delta\)-primary element.Some studies in the approximation of \((\in_\gamma, \in_\gamma \vee q_\delta)\)-fuzzy substructures in quantales.https://zbmath.org/1449.060272021-01-08T12:24:00+00:00"Qurashi, Saqib Mazher"https://zbmath.org/authors/?q=ai:qurashi.saqib-mazher"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammadSummary: The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, was used for the first time by \textit{C. J. Mulvey} [Suppl. Rend. Circ. Mat. Palermo (2) 12, 99--104 (1986; Zbl 0633.46065)]. In many applied disciplines like theoretical computer science, algebraic theory, rough set theory, topological theory and linear logic, the use of fuzzified algebraic structures specially quantales plays an important role. In the present paper, the concept of generalized roughness for \(( \in_\gamma,\in_\gamma\vee q_\delta)\)-fuzzy \(f\) ilters in quantales, is introduced. The concept is extended to the approximations of \(( \in_\gamma,\in_\gamma\vee q_\delta)\)-fuzzy ideals and \(( \in_\gamma,\in_\gamma\vee q_\delta)\)-fuzzy subquantales. Moreover, this concept is applied to study the approximations of \(( \in_\gamma,\in_\gamma\vee q_\delta)\)-fuzzy prime ideals and \(( \in_\gamma,\in_\gamma\vee q_\delta)\)-fuzzy semi-prime ideals.The Boole atom of \({R_0}\) algebra and its application.https://zbmath.org/1449.030512021-01-08T12:24:00+00:00"Fan, Fengli"https://zbmath.org/authors/?q=ai:fan.fengli"Xie, Yongjian"https://zbmath.org/authors/?q=ai:xie.yongjianSummary: In this paper, the definition of the substitution of Boole atom in \({R_0}\) algebra and how to get a finite \({R_0}\) algebra by substituting the Boole atoms of a Boolean algebra with some \({R_0}\) algebra are introduced. Thus, the relationship between Booleea algebra and \({R_0}\) algebra is described.Double MS-algebras with demi-pseudocomplementation.https://zbmath.org/1449.060152021-01-08T12:24:00+00:00"Fang, Jie"https://zbmath.org/authors/?q=ai:fang.jie"Wu, Ruichun"https://zbmath.org/authors/?q=ai:wu.ruichun"Sun, Zhongju"https://zbmath.org/authors/?q=ai:sun.zhongjuSummary: In this paper, we initiate an investigation into the class of double MS-algebras with demi-pseudocomplementation, namely the algebra \( (L; \wedge, \vee, {^\circ}, {^+}, {^*}, 0, 1)\) in which \( (L; {^\circ}, {^+})\) is a double MS-algebra, \( (L; {^*})\) is a demi-pseudocomplemented algebra, and the unary operations are linked by the identities \(x^{^\circ *} = x^{*^\circ}\) and \(x^{+ *} = x^{* +}\). We particularly show that such an algebra is properly subdirectly irreducible if and only if its lattice of congruences is either a 3-element chain, or a 4-element chain.The adjoint semigroup of FI algebra.https://zbmath.org/1449.030472021-01-08T12:24:00+00:00"Yang, Wenqi"https://zbmath.org/authors/?q=ai:yang.wenqiSummary: Firstly, the concept of the adjoint semigroup of FI algebra is introduced, which is a commutative order monoid. Secondly, some properties of the adjoint semigroup are discussed, and the relationship between the filter of FI algebra and its adjoint semigroup is obtained.A note on lattices with many sublattices.https://zbmath.org/1449.060042021-01-08T12:24:00+00:00"Czédli, Gábor"https://zbmath.org/authors/?q=ai:czedli.gabor"Horváth, Eszter K."https://zbmath.org/authors/?q=ai:horvath.eszter-kSummary: For every natural number \(n\geq 5\), we prove that the number of subuniverses of an \(n\)-element lattice is \(2^n\), \(13\cdot 2^{n-4}\), \(23\cdot 2^{n-5}\), or less than \(23\cdot 2^{n-5}\). Also, we describe the \(n\)-element lattices with exactly \(2^n\), \(13\cdot 2^{n-4}\), or \(23\cdot 2^{n-5}\) subuniverses.State lattice effect algebras and its basic properties.https://zbmath.org/1449.030492021-01-08T12:24:00+00:00"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.2|zhang.yan.4|zhang.yan.3"Wu, Hongbo"https://zbmath.org/authors/?q=ai:wu.hongboSummary: In the paper, the problems of the state operators on lattice effect algebras are studied. By using the method of internal states on MV-algebras, the state operators are introduced on lattice effect algebras, by which the state lattice effect algebras and state-morphism lattice effect algebras, some basic properties and the connection between state lattice effect algebras and state-morphism lattice effect algebras are investigated.Properties of soft exact sequences.https://zbmath.org/1449.160052021-01-08T12:24:00+00:00"Wu, Yue"https://zbmath.org/authors/?q=ai:wu.yue"Ma, Jing"https://zbmath.org/authors/?q=ai:ma.jingSummary: By using module theory and the basic properties of soft modules, we discuss the decomposition properties of soft homomorphisms. Firstly, we define a single soft homomorphism of soft modules and a soft exact sequence. Secondly, we prove that every soft homomorphism can be decomposed into the composition of an epimorphism and a monomorphism. Finally, we discuss the basic properties of soft exact sequences, and give equivalent conditions for several kinds of simple soft modulus sequence to be exact, and construct a new soft exact sequence by using two soft exact sequences.\(L\)-fuzzy ideals in quantale.https://zbmath.org/1449.060262021-01-08T12:24:00+00:00"Luo, Qingjun"https://zbmath.org/authors/?q=ai:luo.qingjunSummary: In the present paper, by applying \(L\)-fuzzy theory to quantales, we introduce the notations of \(L\)-fuzzy ideals, \(L\)-fuzzy prime ideals and \(L\)-fuzzy primary ideals in quantales, and investigate their properties. Several characterizations of such ideals are presented. The concrete structure of \(L\)-fuzzy ideals generated by \(L\)-fuzzy points is also obtained.Prime strong ideals on posets.https://zbmath.org/1449.060022021-01-08T12:24:00+00:00"Chen, Fangxin"https://zbmath.org/authors/?q=ai:chen.fangxin"Jiang, Guanghao"https://zbmath.org/authors/?q=ai:jiang.guanghao"Tang, Zhaoyong"https://zbmath.org/authors/?q=ai:tang.zhaoyongSummary: In this paper, firstly, the concept of prime strong ideal on posets is introduced and examined. The relationships of prime strong ideal, prime ideal, and strong ideal are explored. Secondly, a sufficient condition for equivalence between strong ideals and prime strong ideals is given. Finally, the result that the image of a (prime) strong ideal under an order isomorphic mapping is still a (prime) strong ideal is obtained.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.The existence of fuzzy Dedekind completion of Archimedean fuzzy Riesz space.https://zbmath.org/1449.460632021-01-08T12:24:00+00:00"Iqbal, Mobashir"https://zbmath.org/authors/?q=ai:iqbal.mobashir"Bashir, Zia"https://zbmath.org/authors/?q=ai:bashir.ziaSummary: The fuzzy Riesz space is an attempt to study vector spaces with fuzzy ordering to model scenarios of more vague nature. The aim of this paper is to prove the existence of fuzzy Dedekind completion, whereas to achieve this goal, other related concepts like fuzzy order convergence, fuzzy positive operators, and their related results are also explored to enrich the theory of fuzzy Riesz spaces.On ideals in De Morgan residuated lattices.https://zbmath.org/1449.030522021-01-08T12:24:00+00:00"Holdon, Liviu-Constantin"https://zbmath.org/authors/?q=ai:holdon.liviu-constantinSummary: In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, \(\odot\)-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.2-absorbing \(\delta\)-primary elements in multiplicative lattices.https://zbmath.org/1449.060102021-01-08T12:24:00+00:00"Nimbhorkar, S. K."https://zbmath.org/authors/?q=ai:nimbhorkar.shriram-khanderao"Nehete, J. Y."https://zbmath.org/authors/?q=ai:nehete.jaya-ySummary: In this paper, we define a 2-absorbing \(\delta\)-primary element and a weakly 2-absorbing \(\delta\)-primary element in a compactly generated multiplicative lattice \(L\). We obtain some properties of these elements. We give a characterization for 2-absorbing \(\delta\)-primary elements. Also we define a \(\delta\)-triple-zero and a free \(\delta\)-triple-zero and prove some results on it.A class of non-matchable distributive lattices.https://zbmath.org/1449.052172021-01-08T12:24:00+00:00"Wang, Xu"https://zbmath.org/authors/?q=ai:wang.xu.5"Zhao, Xuxu"https://zbmath.org/authors/?q=ai:zhao.xuxu"Yao, Haiyuan"https://zbmath.org/authors/?q=ai:yao.haiyuanSummary: In this paper, we consider non-matchable distributive lattices. By introducing the meet-irreducible cell with respect to a perfect matching of a plane elementary bipartite graph and giving its characterizations, we obtain a new class of non-matchable distributive lattices, and extend a result on non-matchable distributive lattices with a cut element.On the maximal spectrum of lattice modules.https://zbmath.org/1449.060282021-01-08T12:24:00+00:00"Phadatare, Narayan"https://zbmath.org/authors/?q=ai:phadatare.narayan"Kharat, Vilas"https://zbmath.org/authors/?q=ai:kharat.vilas-s"Ballal, Sachin"https://zbmath.org/authors/?q=ai:ballal.sachinSummary: Let \(M\) be a lattice module over a \(C\)-lattice \(L\). The maximal spectrum \(\mathrm{Max}(M)\) of \(M\) is the collection of all maximal elements of \(M\). In this paper, we study the topology on \(\mathrm{Max}(M)\) and also establish the interrelations between the topological properties of \(\mathrm{Max}(M)\) and the algebraic properties of \(M\).On the class of subsets of residuated lattice which induces a congruence relation.https://zbmath.org/1449.030442021-01-08T12:24:00+00:00"Harizavi, H."https://zbmath.org/authors/?q=ai:harizavi.habibSummary: In this manuscript, we study the class of special subsets connected with a subset in a residuated lattice and investigate some related properties. We describe the union of elements of this class. Using the intersection of all special subsets connected with a subset, we give a necessary and sufficient condition for a subset to be a filter. Finally, by defining some operations, we endow this class with a residuated lattice structure and prove that it is isomorphic to the set of all congruence classes with respect to a filter.BF-congruence relations in negative non-involutive residuated lattices.https://zbmath.org/1449.060082021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhui"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Jiang, Zhiting"https://zbmath.org/authors/?q=ai:jiang.zhitingSummary: In order to study the structural characteristics of negative non-involutive residuated lattices, the concept of BF-congruence relation in negative non-involutive residuated lattice is introduced and its properties are investigated. The relations between BF-congruence relation and BF-ideal are discussed in a negative non-involutive residuated lattice \(L\). A BF-congruence relation \({\Phi_A}\) is induced base on the BF-ideal \(A\) in \(L\). It is proved that the quotient algebra of \(L\) under \({\Phi_A}\) forms a residuated lattice.\(\delta\)-fuzzy ideals in pseudo-complemented distributive lattices.https://zbmath.org/1449.060182021-01-08T12:24:00+00:00"Alaba, Berhanu Assaye"https://zbmath.org/authors/?q=ai:alaba.berhanu-assaye"Norahun, Wondwosen Zemene"https://zbmath.org/authors/?q=ai:norahun.wondwosen-zemeneSummary: In this paper, we introduce \(\delta\)-fuzzy ideals in a pseudo complemented distributive lattice in terms of fuzzy filters. It is proved that the set of all \(\delta\)-fuzzy ideals forms a complete distributive lattice. The set of equivalent conditions are given for the class of all \(\delta\)-fuzzy ideals to be a sub-lattice of the fuzzy ideals of \(L\). Moreover, \(\delta\)-fuzzy ideals are characterized in terms of fuzzy congruences.Dilworth's decomposition theorem for posets in ZF.https://zbmath.org/1449.030162021-01-08T12:24:00+00:00"Tachtsis, E."https://zbmath.org/authors/?q=ai:tachtsis.eleftheriosDilworth's theorem (DT) is the following statement: If the maximum number of elements in an antichain of a poset \(P\) is a finite number, then it is equal to the minimum number of pairwise disjoint chains into which \(P\) can be decomposed. DT for finite posets is a theorem of ZF. DT is valid in ZFC, but DT does not imply AC in ZF. The Boolean prime ideal theorem (BPI) implies DT in ZF but BPI is strictly weaker than AC in ZF.
In this paper, the author shows DT using the propositional compactness theorem which is equivalent to BPI. Further on, he shows that BPI \(\rightarrow\) DT is not reversible in ZFA.
The author is interested in the strength of DT with respect to variants of AC. He proves that the axiom of choice for well-ordered families of non-empty sets does not imply DT in ZFA. This is done by introducing a new Fraenkel-Mostowski model.
He also shows that DT does not imply Marshall Hall's theorem in ZFA.
Reviewer: Martin Weese (Potsdam)Meet uniform continuous posets.https://zbmath.org/1449.060012021-01-08T12:24:00+00:00"Mao, Xuxin"https://zbmath.org/authors/?q=ai:mao.xuxin"Xu, Luoshan"https://zbmath.org/authors/?q=ai:xu.luoshanSummary: In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are: (1) A uniform complete poset \(L\) is meet uniform continuous iff \(\uparrow (U \cap \downarrow x)\) is a uniform Scott set for each \(x \in L\) and each uniform Scott set \(U\); (2) A uniform complete poset \(L\) is meet uniform continuous iff for each \(x \in L\) and each uniform subset \(S\), one has \(x \wedge \bigvee S = \bigvee \{x \wedge s| {s \in S}\}\). In particular, a complete lattice \(L\) is meet uniform continuous iff \(L\) is a complete Heyting algebra; (3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous; (4) A uniform complete poset \(L\) is meet uniform continuous if \({L^1}\) obtained by adjoining a top element 1 to \(L\) is a complete Heyting algebra; (5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.The measure of fuzzy prime ideals on lattices.https://zbmath.org/1449.060062021-01-08T12:24:00+00:00"Huang, Fei"https://zbmath.org/authors/?q=ai:huang.fei"Liao, Zuhua"https://zbmath.org/authors/?q=ai:liao.zuhuaSummary: In this paper, we mainly deal with the measure of fuzzy prime ideals on lattices in fuzzy algebra. Firstly, the new concept of the measure of fuzzy prime ideals is given and is used to discuss that the fuzzy subset is the degree of fuzzy prime ideals. Secondly, the equivalent description of the measures of fuzzy prime ideals is obtained by using the (strong) level set of fuzzy sets. Finally, we discuss the fuzzy prime ideal degrees of the intersection and the direct product of fuzzy subsets. Meanwhile, the measures of fuzzy prime ideals properties about the homomorphism image and inverse image of the fuzzy subset of lattice are obtained.Notes on locally internal uninorm on bounded lattices.https://zbmath.org/1449.030032021-01-08T12:24:00+00:00"Çaylı, Gül Deniz"https://zbmath.org/authors/?q=ai:cayli.gul-deniz"Ertuğrul, Ümit"https://zbmath.org/authors/?q=ai:ertugrul.umit"Köroğlu, Tuncay"https://zbmath.org/authors/?q=ai:koroglu.tuncay"Karaçal, Funda"https://zbmath.org/authors/?q=ai:karacal.fundaConsidering an arbitrary bounded lattice \(L\), the authors introduce and examine some properties of idempotent and locally internal uninorms on \(L\). Recall that a uninorm on \(L\) is an associative symmetric monotone operation with a neutral element \(e\) (if \(e=1\), then it is a triangular norm, if \(e=0\), we recover a triangular conorm). Locally internal uninorm assigns to arguments \(x\) and \(y\) one of these arguments. Hence, each locally internal operation is idempotent, but not vice-versa, in general. If \(L\) is a chain, then idempotent and locally internal uninorms coincide. The authors add some illustrative examples to show the connection between idempotent and locally internal uninorm. Moreover, they characterize all lattices \(L\) and \(e\) from \(L\) such that each uninorm on \(L\) possessing a neutral element \(e\) is idempotent and locally internal (in fact, then \(L\) is a \(3\) element chain and \(e\) is its interior element).
Reviewer: Radko Mesiar (Bratislava)Congruences and homomorphisms on \(\Omega\)-algebras.https://zbmath.org/1449.060132021-01-08T12:24:00+00:00"Eghosa Edeghagba, Elijah"https://zbmath.org/authors/?q=ai:eghosa-edeghagba.elijah"Šešelja, Branimir"https://zbmath.org/authors/?q=ai:seselja.branimir"Tepavčević, Andreja"https://zbmath.org/authors/?q=ai:tepavcevic.andrejaSummary: The topic of the paper are \(\Omega\)-algebras, where \(\Omega\) is a complete lattice. In this research we deal with congruences and homomorphisms. An \(\Omega\)-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an \(\Omega\)-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce \(\Omega\)-valued congruences, corresponding quotient \(\Omega\)-algebras and \(\Omega\)-homomorphisms and we investigate connections among these notions. We prove that there is an \(\Omega\)-homomorphism from an \(\Omega\)-algebra to the corresponding quotient \(\Omega\)-algebra. The kernel of an \(\Omega\)-homomorphism is an \(\Omega\)-valued congruence. When dealing with cut structures, we prove that an \(\Omega\)-homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an \(\Omega\)-congruence determines a closure system of classical congruences on cut subalgebras. Finally, identities are preserved under \(\Omega\)-homomorphisms.On realizability of graphs as \({\Gamma_S} (L)\) for some lattice \(L\).https://zbmath.org/1449.050792021-01-08T12:24:00+00:00"Malekpour, Shahide"https://zbmath.org/authors/?q=ai:malekpour.shahide"Bazigaran, Behnam"https://zbmath.org/authors/?q=ai:bazigaran.behnamSummary: Let \({\Gamma_S} (L)\) denote the graph associated to the lattice \(L\) with respect to a \(\wedge\)-closed subset \(S\) of \(L\). In this paper, some properties of the graph \({\Gamma_S} (L)\) are presented. Also, we investigate the realizability of \({\Gamma_S} (L)\) for some graphs, especially some split graphs.On principal congruences and the number of congruences of a lattice with more ideals than filters.https://zbmath.org/1449.060052021-01-08T12:24:00+00:00"Czédli, Gábor"https://zbmath.org/authors/?q=ai:czedli.gabor"Mureşan, Claudia"https://zbmath.org/authors/?q=ai:muresan.claudiaThe authors establish the following two main results:
Let \(\lambda\) and \(\kappa\) be cardinal numbers such that \(\kappa\) is infinite and either \(2\leq \lambda\leq \kappa\), or \(\lambda = 2^\kappa\). Then there exists a lattice \( L\) with exactly \(\lambda\) many congruences, \(2^\kappa\) many ideals, however only \(\kappa\) many filters. (Moreover if \(\lambda \geq 2\) is an integer of the form \(2^m\cdot 3^n\) then they can choose \(L\) to be a modular lattice generating one of the minimal modular non-distributive congruence varieties described by \textit{R. Freese} [``Minimal modular congruence varieties'', Notices Am. Math. Soc. 23, \#76T-A181 (1976)], and this \(L\) is even relatively complemented for \(\lambda = 2.\))
It is also proved that if \(P\) is a bounded poset with at least two elements, \(G\) is a group, and \(\kappa\) is an infinite cardinal such that \(\kappa \geq \vert P \vert\) and \(\kappa\geq \vert G \vert\), then there exists a lattice \(L\) of cardinality \(\kappa\) such that (i) the principal congruences of \(L\) yield an ordered set isomorphic to \(P\), (ii) the automorphism group of \(L\) is isomorphic to \(G\), (iii) \(L\) has \(2^\kappa\) many ideals, but (iv) \(L\) has only \(\kappa\) many filters.
The rather delicate and complex ways to verify all these statements are explained in detail in the paper.
Reviewer: Hans Peter Künzi (Rondebosch)\(d\)-fuzzy ideals and injective fuzzy ideals in distributive lattices.https://zbmath.org/1449.060192021-01-08T12:24:00+00:00"Alaba, Berhanu Assaye"https://zbmath.org/authors/?q=ai:alaba.berhanu-assaye"Taye, Mihret Alamneh"https://zbmath.org/authors/?q=ai:taye.mihret-alamneh"Norahun, Wondwosen Zemene"https://zbmath.org/authors/?q=ai:norahun.wondwosen-zemeneSummary: In this paper, we introduce the concept of \(d\)-fuzzy ideals and
injective fuzzy ideals in a distributive lattice with respect to derivation. It is proved that the set of all \(d\)-fuzzy ideals forms a distributive lattice. A set of equivalent conditions are derived for a derivation \(d\) of \(L\) to became injective. Moreover, we proved that the set of all injective fuzzy ideals forms a complete distributive lattice.Lattice of bipolar fuzzy ideals in negative non-involutive residuated lattices.https://zbmath.org/1449.030462021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhui"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Li, Yumao"https://zbmath.org/authors/?q=ai:li.yumaoSummary: In this paper, the problem of bipolar fuzzy ideals is further studied in negative non-involutive residuated lattices. The definition of bipolar fuzzy ideal which is generated by a bipolar fuzzy set is given and its two representation theorems are established. It is proved that the set \(\text bf{BFI} (L)\) which contains all bipolar fuzzy ideals in a negative non-involutive residuated lattice \(L\), under the partial order \(\sqsubseteq\), forms a complete Heyting algebra. This work further expands the way for revealing the structural characteristics of negative non-involutive residuated lattices.Bipolar fuzzy prime ideals in negative non-involutive residuated lattices.https://zbmath.org/1449.030452021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhuiSummary: The problem of bipolar fuzzy ideals is further studied in negative non-involutive residuated lattices. The concept of bipolar fuzzy prime ideal (BF-prime ideal for short) is introduced and its properties are investigated. Some equivalent characterizations of BF-prime ideal are obtained. The BF-prime ideal theorem in a prelinearity negative non-involutive residuated lattice is established. It is proved that the NRL-homomorphism image and the inverse image of a BF-prime ideal are also BF-prime ideals. This work further expands the way for revealing the structural characteristics of negative non-involutive residuated lattices.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 \(\begin{pmatrix}n-1\\ r-1\end{pmatrix}+2n-2\).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.Some methods to obtain t-norms and t-conorms on bounded lattices.https://zbmath.org/1449.030022021-01-08T12:24:00+00:00"Çaylı, Gül Deniz"https://zbmath.org/authors/?q=ai:cayli.gul-denizSummary: In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice \(L\) based on a priori given t-norm acting on \([a,1]\) and t-conorm acting on \([0,a]\) for an arbitrary element \(a\in L\setminus\{0,1\}\). We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice.The superior subalgebra in Boolean algebra.https://zbmath.org/1449.060202021-01-08T12:24:00+00:00"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3"Zhang, Jianzhong"https://zbmath.org/authors/?q=ai:zhang.jianzhong.1|zhang.jianzhongSummary: From new point of view, the subalgebra theory in Boolean algebra is studied further. Given a poset with the smallest element, the concept of superior subalgebra is introduced in Boolean algebra by combining the superior mapping and the Boolean algebra. Some properties of superior subalgebra are discussed. Finally, some equivalent characterizations of superior subalgebra are also obtained.Uniform topological spaces based on normal fuzzy ideals in negative non-involutive residuated lattices.https://zbmath.org/1449.540382021-01-08T12:24:00+00:00"Liu, Chunhui"https://zbmath.org/authors/?q=ai:liu.chunhuiSummary: Topological structure is one of important research contents in the field of logical algebra. In order to describe the topological structure of negative non-involutive residuated lattices, based on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some of their properties are discussed. The following conclusions are proved: (1) every uniform topological space is first-countable, zero-dimensional, disconnected, locally compact and completely regular; (2) a uniform topological space is a \({T_1}\) space iff it is a \({T_2}\) space; (3) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous under the uniform topology, which make the negative non-involutive residuated lattice be topological negative non-involutive residuated lattice. Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices on a topological level.Some types of soft paracompactness via soft ideals.https://zbmath.org/1449.540192021-01-08T12:24:00+00:00"Turanli, Elif"https://zbmath.org/authors/?q=ai:turanli.elif"Demir, İzzettin"https://zbmath.org/authors/?q=ai:demir.izzettin"Özbakir, Oya Bedre"https://zbmath.org/authors/?q=ai:bedre-ozbakir.oyaSummary: In this paper, we introduce the soft \(\mathcal{I}\)-paracompact spaces and the soft \(\mathcal{I}\)-S-paracompact spaces. First, we investigate the relationships between these spaces and soft paracompact spaces. Also, we give some fundamental properties of these spaces. Finally, we prove that soft \(\mathcal{I}\)-S-paracompact spaces are invariant under perfect mappings.Generalized approximation of substructures in quantales by soft relations.https://zbmath.org/1449.060252021-01-08T12:24:00+00:00"Kanwal, Rani Sumaira"https://zbmath.org/authors/?q=ai:kanwal.rani-sumaira"Qurashi, Saqib Mazher"https://zbmath.org/authors/?q=ai:qurashi.saqib-mazher"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammadSummary: The present work is conducted to investigate the relationship among rough sets, soft sets and quantales. The concept of generalized approximation of substructures in quantales by soft relations is introduced, which is an extended notion of a rough quantale and a soft quantale. This paper is focused on studying the rough sets within the context of algebraic structure quantale using soft reflexive and soft compatible relations. Further, we put forward the concepts of aftersets and foresets, which provide a new research idea for soft rough algebraic research. Some basic concepts, operations and related properties with regard to soft binary relations are proposed.\(f\)-orthomorphisms and \(f\)-linear operators on the order dual of an \(f\)-algebra revisited.https://zbmath.org/1449.460032021-01-08T12:24:00+00:00"Jaber, Jamel"https://zbmath.org/authors/?q=ai:jaber.jamelSummary: We give a necessary and sufficient condition on an \(f\)-algebra \(A\) for which orthomorphisms, \(f\)-linear operators, and \(f\)-orthomorphisms on the order dual \(A^\sim\) are the same class of operators.