Covers of semigroups and function semirings. (Q2016028)

Let \(S\) be a medial semigroup. Thus, for all \(a,b,c,d\in S\), \(a+b+c+d=a+c+b+d\). Let \(\mathbf C=\{C_\alpha\mid\alpha\in A\}\) be a cover of \(S\), namely, each \(C_\alpha\) is a subsemigroup of \(S\) and \(\bigcup\mathbf C=S\). The set \(\mathcal S(\mathbf C)=\{f\colon S\to S\mid f|_{C_\alpha}\) is an endomorphism of \(C_\alpha\) for all \(\alpha\in A\}\) is a semiring in the nearring \(M(S)\) of all self-maps on \(S\). Conversely, given a semiring \(T\) in \(M(S)\), the set \(\mathcal C(T)\) of subsemigroups \(B\) of \(S\) with \(t|_B\) an endomorphism of \(B\) for all \(t\in T\) forms a cover of \(S\). In fact, \(\mathcal S\) and \(\mathcal T\) define a Galois correspondence between the set \(\Gamma\) of all covers of \(S\) and the set \(\Lambda\) of all semirings in \(M(S)\). Using this Galois correspondence, some maximal semirings in \(M(S)\) are identified for certain semigroups \(S\). For example, let \(S\) be a commutative semigroup which is a partition of torsion bounded abelian groups \(\mathbf C=\{A_i\}_{i\in Y}\) with \(Y\) a chain. It is shown that \(\mathcal S(\mathbf C)\) is maximal as a semiring in \(M(\mathcal S)\) if and only if \(|Y|=1\) (Theorem 2.1.2). An example to illustrate this result is given. Further conditions on \(\mathbf C\) are considered in the paper.