Clifford semigroups and seminear-rings of endomorphisms. (Q1026891)

A `seminear-ring' is a triple \((U,+,\cdot)\), where \(U\) is a nonempty set such that both \((U,+)\) and \((U,\cdot)\) are semigroups, and the left distributive law is satisfied, i.e., \(a\cdot(b+c)=a\cdot b+a\cdot c\) for all \(a,b,c\in U\). An element \(d\) of \(U\) is a `distributive' element if \((a+b)\cdot d=a\cdot d+b\cdot d\) for all \(a,b\in U\). A seminear-ring is `distributively generated' (dg, for short) if there exists a (multiplicative) subsemigroup \((T,\cdot)\) of distributive elements in \(U\) that generates \(U\) as a seminear-ring. The main purpose of the paper is to prove the theorem: Let \(\Lambda=\{0,1,\dots,n\}\) be a semilattice with \(n\geq\cdots\geq 1\geq 0\). Let \(\{G_i\}_{i\in\Lambda}\) be a family of isomorphic groups and consider \(S=\bigcup_{i\in\Lambda}G_i\), a Clifford semigroup defined by isomorphisms. Then the dg seminear-ring \(E(S)\) generated by \(\text{End}(S)\), the set of all endomorphisms of \(S\), has a Clifford semigroup structure that forms its additive semigroup \((E(S),+)\). Moreover, \(E(S)=\bigcup_{i\in\Lambda}(E(G_i),+)\), where each \(E(G_i)\) is a dg near-ring generated by a set of endomorphisms of \(G_i\) of a specific type.