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Soft ordered semigroups. (English) Zbl 1191.06009
If $S$ and $A$ are two nonempty sets, the pair $\left(ℱ,A\right)$ is called a soft set over $S$ if $ℱ$ 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 $\left(ℱ,A\right)$ over $S$ satisfying the following property: If $x\in A$ such that $ℱ\left(x\right)\ne \varnothing$, then $ℱ\left(x\right)$ 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 $\left(ℱ,A\right)$ over $S$ such that $ℱ\left(x\right)$ 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)].
MSC:
 06F05 Ordered semigroups and monoids