Structure and characteristics of left Clifford semigroups
A regular semigroup \(S\) is called left Clifford semigroup (left \(C\)- semigroup) if \(aS \subseteq Sa\) for all \(a\in S\). The \(\xi\)-product \(S = T\times_ \xi I\) of semigroups \(T\) and \(I\) is defined as particular product defined on \(S = \bigcup_{\alpha\in Y}S_ \alpha\), where \(Y\) is a semilattice, \(S_ \alpha = T_ \alpha \times I_ \alpha\), \(T = \bigcup_{\alpha \in Y} T_ \alpha\) is a strong semilattice of semigroups \(T_ \alpha\), \(I = \bigcup_{\alpha \in Y} I_ \alpha\) is a semilattice of left zero bands \(I_{\alpha}\), and \(\xi\) is a special map from \(S\) to the transformation semigroup over \(I\). It is proved that the \(\xi\)-product \(T \times_ \xi I\), where \(T\) is a strong semilattice of groups \(G_ \alpha\), is a left \(C\)-semigroup and conversely any left \(C\)-semigroup is precisely the \(\xi\)-product of the above semigroups. Moreover, some characteristics for left \(C\)-semigroups are obtained, some special classes of left \(C\)-semigroups are studied, and questions how the \(\xi\)-product becomes a spined product and the semilattice decomposition becomes a strong semilattice decomposition are discussed.
scientific article; zbMATH DE number 98074