Intra-regular, left quasi-regular and semisimple fuzzy ordered semigroups. (Q2870888)

An intra-regular ordered semigroup is an ordered semigroup \(S\) satisfying the condition \(a\in (Sa^2S]\) for all \(a\in S\), while it is called left (resp. right) quasi-regular if \(a\in (SaSa]\) (resp. \(a\in (aSaS]\)) for all \(a\in S\). If \(a\in (SaSaS]\) for all \(a\in S\), then \(S\) is a semisimple ordered semigroup. These structures were investigated by the author in some of her previous works.NEWLINENEWLINE In this paper the author characterizes the intra-regular and left quasi-regular ordered semigroups in terms of fuzzy sets. More precisely, she proves that an ordered semigroup \(S\) is intra-regular and left quasi-regular if and only if for every fuzzy subset \(f\) of \(S\) we have \(f\preceq 1\circ f^2\circ 1\circ f\). Similarly, an ordered semigroup \(S\) is intra-regular and right quasi-regular if and only if for every fuzzy subset \(f\) of \(S\) we have \(f\preceq f\circ 1\circ f^2\circ 1\). It is also proved that an ordered semigroup \(S\) is intra-regular and semisimple if and only if for every fuzzy subset \(f\) of \(S\) we have \(f\preceq 1\circ f^2\circ 1\circ f\circ 1\).