Soft ordered semigroups. (English) Zbl 1191.06009

If \(S\) and \(A\) are two nonempty sets, the pair \(({\mathcal F},A)\) is called a soft set over \(S\) if \(\mathcal F\) is a mapping of \(A\) into the set of all subsets of \(S\) [D. Molodtsov, Comput. Math. Appl. 37, No. 4–5, 19–31 (1999; Zbl 0936.03049)]. An ordered semigroup \(S\) is called a soft ordered semigroup if there is a nonempty set \(A\) and a soft set \((\mathcal F, A)\) over \(S\) satisfying the following property: If \(x\in A\) such that \({\mathcal F} (x)\not=\emptyset\), then \({\mathcal F} (x)\) is a subsemigroup of \(S\). An ordered semigroup \(S\) is called \(l\)-idealistic (resp. \(r\)-idealistic) soft ordered semigroup if there is a nonempty set \(A\) and a soft set \(({\mathcal F}, A)\) over \(S\) such that \({\mathcal F}(x)\) is a left (resp. right) ideal of \(S\) for every \(x\in A\). For the definition of homomorphism between ordered semigroups given in the introduction of the paper we refer to [N. Kehayopulu and M. Tsingelis, Semigroup Forum 50, No. 2, 161–177 (1995; Zbl 0823.06010)].


06F05 Ordered semigroups and monoids
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