Weaky regular \(*\)-semigroups
A regular \(*\)-semigroup is a semigroup \(S\) endowed with a supplementary operation \(*\) satisfying: (1) \(xx^*=x\), for every \(x\in S\); (2) \((x^*)^*=x\), for every \(x\in S\); (3) \((xy)^*=y^*x^*\), for every \(x,y\) in \(S\). It has been proved by \textit{M. Yamada} [Semigroup Forum 24, 173-187 (1982; Zbl 0479.20030)] that a regular semigroup is a regular \(*\)-semigroup if and only if it has at least one \(P\)-system. (A subset \(F\) of \(E_S\), where \(S\) is a regular semigroup, is a \(P\)-system of \(S\) if: (4) for every \(a\in S\), there exists a unique \(a^*\in V(a)\) such that \(a^*a\) and \(aa^*\) belong to \(F\); (5) \(a^*Fa\subseteq F\), for every \(a\in S\); (6) \(F^2\subseteq E_S\).) In this paper an analogous result for weakly regular \(*\)-semigroups is proved. (A weakly regular \(*\)-semigroup is a regular semigroup \(S\) equipped with a unary operation \(*\) satisfying (1), (2) and (3') where: (3') \((xx^*yy^*)=yy^*xx^*\), for every \(x,y\) in \(S\).) It is proved that a regular semigroup is a weakly regular \(*\)-semigroup if and only if it has at least one weakly \(P\)-system. (A weakly \(P\)-system of a regular semigroup \(S\) is a subset \(F\) of \(E_S\) satisfying (4), (5') and (6) where: (5') \(fFf\subseteq F\), for every \(f\in F\)).
scientific article; zbMATH DE number 1309719
scientific article