Recent zbMATH articles in MSC 06F05https://www.zbmath.org/atom/cc/06F052022-05-16T20:40:13.078697ZWerkzeug\(DR_0\) algebras: a class of regular residuated lattices via De Morgan algebrashttps://www.zbmath.org/1483.060152022-05-16T20:40:13.078697Z"Zhang, Xiao Hong"https://www.zbmath.org/authors/?q=ai:zhang.xiaohong"Wei, Ping"https://www.zbmath.org/authors/?q=ai:wei.ping(no abstract)On the insertion of \(n\)-powershttps://www.zbmath.org/1483.060172022-05-16T20:40:13.078697Z"Almeida, Jorge"https://www.zbmath.org/authors/?q=ai:almeida.jorge"Klíma, Ondřej"https://www.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://www.zbmath.org/1483.060182022-05-16T20:40:13.078697Z"Kehayopulu, Niovi"https://www.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://www.zbmath.org/1483.060192022-05-16T20:40:13.078697Z"Tang, Jian"https://www.zbmath.org/authors/?q=ai:tang.jian"Xie, Xiangyun"https://www.zbmath.org/authors/?q=ai:xie.xiang-yun(no abstract)A class of regular congruences on ordered semigroupshttps://www.zbmath.org/1483.060202022-05-16T20:40:13.078697Z"Xie, Xiang Yun"https://www.zbmath.org/authors/?q=ai:xie.xiang-yun"Guo, Xiao Jiang"https://www.zbmath.org/authors/?q=ai:guo.xiaojiang(no abstract)Riesz and pre-Riesz monoidshttps://www.zbmath.org/1483.130122022-05-16T20:40:13.078697Z"Zafrullah, Muhammad"https://www.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)Regular semigroups whose full regular subsemigroups form a chainhttps://www.zbmath.org/1483.201022022-05-16T20:40:13.078697Z"Guo, Xiaojiang"https://www.zbmath.org/authors/?q=ai:guo.xiaojiang"Jun, Young Bae"https://www.zbmath.org/authors/?q=ai:jun.young-bae(no abstract)