Recent zbMATH articles in MSC 06Fhttps://zbmath.org/atom/cc/06F2022-05-16T20:40:13.078697ZWerkzeugPFI-algebras and residuated latticeshttps://zbmath.org/1483.030382022-05-16T20:40:13.078697Z"Zhu, Yi Quan"https://zbmath.org/authors/?q=ai:zhu.yiquan"Cao, Xi Wang"https://zbmath.org/authors/?q=ai:cao.xiwang(no abstract)\(DR_0\) algebras: a class of regular residuated lattices via De Morgan algebrashttps://zbmath.org/1483.060152022-05-16T20:40:13.078697Z"Zhang, Xiao Hong"https://zbmath.org/authors/?q=ai:zhang.xiaohong"Wei, Ping"https://zbmath.org/authors/?q=ai:wei.ping(no abstract)On the insertion of \(n\)-powershttps://zbmath.org/1483.060172022-05-16T20:40:13.078697Z"Almeida, Jorge"https://zbmath.org/authors/?q=ai:almeida.jorge"Klíma, Ondřej"https://zbmath.org/authors/?q=ai:klima.ondrejSummary: In algebraic terms, the insertion of \(n\)-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality \(1\le x^n\). We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity \(x^n=1\). In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under \(n\)-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from \(1\le x^n\) in which both sides are regular elements with respect to the upper bound.From ordered semigroups to ordered hypersemigroupshttps://zbmath.org/1483.060182022-05-16T20:40:13.078697Z"Kehayopulu, Niovi"https://zbmath.org/authors/?q=ai:kehayopulu.nioviFor a hyperoperation \(\circ\) on a nonempty set \(H\), the operation \(*\) on the set \(\mathcal P^*(H)\) of nonempty subsets of \(H\) is defined by \(A * B := \bigcup(a \circ b\colon (a ,b) \in A \times B)\). Given an order \(\le\) on \(H\), the preorder \(\preceq\) on \(\mathcal P^*(H)\) is defined by \(A \preceq B :\equiv (\forall a \in A) (\exists b \in B) a \le b\). A hypersemigroup \((H, \circ)\) is a set \(H\) with an hyperoperation \(\circ\) on it such that the operation \(*\) satisfies the condition \(\{x\} * (y \circ z) = (x \circ y) * \{z\}\); then \(*\) is associative. An ordered hypersemigroup is a hypersemigroup equipped with an order relation \(\le\) such that \(a \le b\) implies that \(a \circ c \preceq b \circ c\) and \(c \circ a \preceq c \circ b\) for every \(c \in H\).
In this paper, some of the author's previous results on regular and intraregular ordered semigroups are adjusted to ordered hypersemigroups, and consequences for hypersemigroups without order are obtained. Some of presented results and other information can be found also in other papers by the author, say [PU.M.A., Pure Math. Appl. 25, No. 2, 151--156 (2015; Zbl 1374.20073); Lobachevskii J. Math. 39, No. 1, 121--128 (2018; Zbl 1387.20053)]. In the proofs, the author tries to use sets rather than their elements ``to show the pointless character of the results''.
Reviewer: Jānis Cīrulis (Riga)Chains of Archimedean ordered semigroupshttps://zbmath.org/1483.060192022-05-16T20:40:13.078697Z"Tang, Jian"https://zbmath.org/authors/?q=ai:tang.jian"Xie, Xiangyun"https://zbmath.org/authors/?q=ai:xie.xiang-yun(no abstract)A class of regular congruences on ordered semigroupshttps://zbmath.org/1483.060202022-05-16T20:40:13.078697Z"Xie, Xiang Yun"https://zbmath.org/authors/?q=ai:xie.xiang-yun"Guo, Xiao Jiang"https://zbmath.org/authors/?q=ai:guo.xiaojiang(no abstract)Closed filters of quantaleshttps://zbmath.org/1483.060212022-05-16T20:40:13.078697Z"Liu, Zhi Bin"https://zbmath.org/authors/?q=ai:liu.zhibin(no abstract)Prequantale congruence and its propertieshttps://zbmath.org/1483.060222022-05-16T20:40:13.078697Z"Wang, Shun Qin"https://zbmath.org/authors/?q=ai:wang.shunqin"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin(no abstract)On the Riesz structures of a lattice ordered abelian grouphttps://zbmath.org/1483.060232022-05-16T20:40:13.078697Z"Lenzi, Giacomo"https://zbmath.org/authors/?q=ai:lenzi.giacomoSummary: A Riesz structure on a lattice ordered abelian group \(G\) is a real vector space structure where the product of a positive element of \(G\) and a positive real is positive. In this paper we show that for every cardinal \(k\) there is a totally ordered abelian group with at least \(k\) Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same \(l\)-group with strong unit. This gives a solution to a problem posed by \textit{P. Conrad} in 1975 [J. Aust. Math. Soc., Ser. A 20, 332--347 (1975; Zbl 0317.06014)]. Finally we apply the main result to MV-algebras and Riesz MV-algebras.A structure theorem in BCI-algebrashttps://zbmath.org/1483.060242022-05-16T20:40:13.078697Z"Huang, Yi-sheng"https://zbmath.org/authors/?q=ai:huang.yisheng(no abstract)Normed BCK-algebrashttps://zbmath.org/1483.060252022-05-16T20:40:13.078697Z"Peng, Jiayin"https://zbmath.org/authors/?q=ai:peng.jiayin(no abstract)Riesz and pre-Riesz monoidshttps://zbmath.org/1483.130122022-05-16T20:40:13.078697Z"Zafrullah, Muhammad"https://zbmath.org/authors/?q=ai:zafrullah.muhammadA directed partially ordered cancellative divisibility monoid $M$ is said to be a Riesz monoid if for all $x, y_1, y_2 \geq 0$ in $M,$ $x \leq y_1 +y_2$ $\implies x = x_1 + x_2$ where $0\leq x_i \leq y_i.$ In this paper authors explore the necessary and sufficient conditions under which a Riesz monoid $M$ with $M^+ = \{x \geq 0 | x \in M\} = M$ generates a Riesz group. A directed p.o. monoid $M$ is called as a $\Omega$-pre-Riesz if $M^+ = M$ and for all $x_1, x_2,\ldots, x_n \in M,$ $glb(x_1,x_2,\ldots, x_n) = 0$ or there is $r \in \Omega$ such that $0 < r \leq x_1, x_2,\ldots, x_n,$ for some subset $\Omega$ of $M.$ In this paper some examples of $\Omega$-pre-Riesz monoids of $*$-ideals of different types are provided. First it is shown that if $M$ is the monoid of nonzero (integral) ideals of a Noetherian domain $D$ and $\Omega$ the set of invertible ideals, $M$ is $\Omega$-pre-Riesz if and only $D$ is a Dedekind domain. Authors also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain.
Reviewer: T. Tamizh Chelvam (Tirunelveli)On Rayner structureshttps://zbmath.org/1483.130392022-05-16T20:40:13.078697Z"Krapp, Lothar Sebastian"https://zbmath.org/authors/?q=ai:krapp.lothar-sebastian"Kuhlmann, Salma"https://zbmath.org/authors/?q=ai:kuhlmann.salma"Serra, Michele"https://zbmath.org/authors/?q=ai:serra.micheleThe article ``On Rayner structures'' by Lothar Sebastian Krapp, Salma Kuhlmann and Michele Serra explores the algebraic and combinatorial properties of generalised power series fields. More specifically, given the field \(k((G))\) of \(k\)-valued power series in a totally ordered abelian group \(G\) (which can be realised as the space of k-valued functions on \(G\) with well ordered support), and a set \(\mathcal{F}\) of well ordered subsets of \(G\), the article explores the \emph{\(k\)-hull} of \(\mathcal{F}\), and establishes necessary conditions for this to satisfy appropriate algebraic properties.
The paper begins by giving a list of algebraic and set theoretic properties (Conditions 2.1) labelled (S1)--(S6), (A1)--(A5) that can be satisfied by the set \(\mathcal{F}\), and recalls from a previous work of Rayner that the \(k\)-hull \(k((\mathcal{F}))\) of \(\mathcal{F}\) is a subfield of \(k((G))\) in the event that \(\mathcal{F}\) satisfies an appropriate collection of these conditions. Explicitly, \(k((\mathcal{F}))\) is an additive subgroup when it satisfies conditions (S2), (S3), and (S5), it is a subring when it also satisfies (A3) and (A4), and it is a subfield when it also satisfies (A1). However, these are merely sufficient conditions, and the aim of the paper is to establish necessary conditions for these properties to hold.
The first result of the article, Proposition 3.4, states that provided \(k\neq\mathbb{F}_2\), \(k((\mathcal{F}))\) is indeed an additive subgroup of \(k((G))\) if and only if \(\mathcal{F}\) is closed under taking unions and subsets, and contains the singleton {0}, i.e. if and only if \(\mathcal{F}\) satisfies (S2), (S3) and (S5). This strengthens Rayner's original result (Theorem 3.1(i)), showing that the original condition is indeed necessary. The result does not hold if \(k=\mathbb{F}_2\), as demonstrated by Example 3.6.
The authors go on to demonstrate a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subring of \(k((G))\) in Proposition 3.9, provided k is a sufficiently large field. This result gives a stronger condition than that originally stated by Rayner, since it only requires \(\mathcal{F}\) to satisfy (S2), (S3), (S5) and (A2), while (A3) and (A4) are unnecessary.
The remainder of the paper focuses on field structure. The aim is to find necessary and sufficient conditions for \(k((\mathcal{F}))\) to be a field, a \emph{Hahn field} and a \emph{Rayner field}. Briefly, a Hahn field is a subfield of \(k((G))\) containing all polynomials, and a Rayner field is a \(k\)-hull \(k((\mathcal{F}))\) where \(\mathcal{F}\) satisfies (S2), (S3), (S5), (A1), (A3) and (A4), which is a subfield of \(k((G))\) by Rayner's original theorem.
The final main results of the paper are Proposition 3.15 and Theorem 3.18. Proposition 3.15 gives a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subfield and a Hahn field in terms of the Conditions 2.1, at least in the case where \(k\) has characteristic 0. Theorem 3.18 states that if \(k((\mathcal{F}))\) is a Raynor field then it is a Hahn field, and that the converse holds when \(k\) has characteristic 0. The article concludes by stating that the \(k\)-hull \(k((\mathcal{F}))\) is a Rayner field if and only if it is a Hahn field, if and only if it satisfies all of Conditions 2.1.
Overall, this paper should be of interest to anyone who is concerned with power series and generalisations thereof, but since it is a short article with very understandable proofs, I would say that it is accessible to anyone algebraically minded, and certainly worth reading.
Reviewer: Adam Jones (Manchester)Regular semigroups whose full regular subsemigroups form a chainhttps://zbmath.org/1483.201022022-05-16T20:40:13.078697Z"Guo, Xiaojiang"https://zbmath.org/authors/?q=ai:guo.xiaojiang"Jun, Young Bae"https://zbmath.org/authors/?q=ai:jun.young-bae(no abstract)An abstract theory of physical measurementshttps://zbmath.org/1483.810152022-05-16T20:40:13.078697Z"Resende, Pedro"https://zbmath.org/authors/?q=ai:resende.pedroSummary: The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a fundamental process via which a finite amount of classical information is produced. This translates into an algebraic and topological definition of \textit{measurement space} that caters for the distinction between quantum and classical measurements and allows a notion of observer to be derived.