Weakly \(E\)-unitary locally inverse semigroups.
If a completely simple semigroup \(T\) acts on a regular semigroup \(K\) by endomorphisms on the left, then \(\{(a,t)\mid{^{t^0}a}=a\}\) endowed with the multiplication \((a,t)(b,u)=(a\cdot{^tb},tu)\) is a regular semigroup which is called a restricted semidirect product of \(K\) by \(T\). If \(T\) has the Rees matrix representation \(T={\mathcal M}(G;I,\Lambda;P)\) and the group \(G\) acts on the regular semigroup \(K\) by automorphisms on the left, then \(K\times T\) endowed with the multiplication \((a,(i,g,\lambda))(b,(j,h,\mu))=(a\cdot{^{^{gp}\lambda j}b},(i,gp_{\lambda j}h,\mu))\) is a semigroup called a Pastijn product of \(K\) by \(T\). It is proved that the notion of a Pastijn product is self-dual. The authors show that if \(K\) is a regular semigroup and \(T\) a completely simple semigroup, then every restricted semidirect product of \(K\) by \(T\) is isomorphic to a Pastijn product of a regular subsemigroup of \(K\) by \(T\), and, each Pastijn product of \(K\) by \(T\) is embeddable into a restricted semidirect product of a power of \(K\) by \(T\). A locally inverse semigroup \(S\) is weakly \(E\)-unitary if and only if the least completely simple congruence on \(S\) is idempotent pure. It is shown that each weakly \(E\)-unitary locally inverse semigroup can be embedded into a restricted semidirect product [Pastijn product] of a normal band \(B\) and a completely simple semigroup. If in particular the maximal subsemilattices of \(S\) are pairwise disjoint, then \(B\) can be chosen to be a semilattice.
scientific article; zbMATH DE number 1985874