Recent zbMATH articles in MSC 06F99https://www.zbmath.org/atom/cc/06F992022-05-16T20:40:13.078697ZWerkzeugFrom 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)