Medial permutable semigroups of the first kind. (Q2481328)

A semigroup \(S\) is said to be `permutable' if its congruences permute and if it satisfies the identity \(axyb=ayxb\), for all \(a,b,x,y\in S\), it is called `medial'. \textit{C. Bonzini} and \textit{A. Cherubini} [Semigroups, Proc. Conf., Szeged/Hung. 1981, Colloq. Math. Soc. János Bolyai 39, 21-39 (1985; Zbl 0633.20042)] proved that medial permutable semigroups can be of five possible types. Their results give an immediate description of two of them and later \textit{C. Bonzini} [Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 117, No. 4-6, 355-368 (1983; Zbl 0795.20050)] studied another case and its dual. In this paper, the author considers the remaining type, namely the case when \(S\) is a semilattice of a nil semigroup and a rectangular Abelian group \(S_1\) such that \(a\in S_1aS_1\), for all \(a\in S\). Such a semigroup is said to be of the `first kind'. Any non-Archimedean commutative permutable semigroup is such an example. Let \(S\) be a semigroup, \(I\) an ideal of \(S\) and \(\varphi\colon s\to s'\) an isomorphism from \(S\) onto a semigroup \((S',+)\) such that \(S\cap S'=I\) and \(\varphi\) fixes all elements of \(I\). Consider the set \(S''=S\cup S'\) with an operation \(*\) that extends both the operations on \(S\) and on \(S'\) and \(x*y'=xy\) and \(y'*x=(yx)'\), for \(x\in S\) and \(y'\in S'\). In this manner one obtains a groupoid called a `left reflection' of \(S\) with respect to \(I\). `Right reflections' are defined dually. A semigroup is said to be `left commutative' if it satisfies the identity \(abc=bac\); `right commutativity' is defined dually. The aim of this paper is to show that the study of medial permutable semigroups of the first kind can be reduced to studying certain one sided commutative permutable semigroups. In fact, a semigroup is a medial permutable semigroup of the first kind if and only if it is either a left reflection of a left commutative permutable semigroup of the same kind or, dually, a right reflection of a right commutative permutable semigroup also of this type.