Embedding any countable semigroup in a 2-generated congruence-free semigroup
Let (S,.) be a semigroup and A, B sets which are mutually disjoint and disjoint from S. Let \(\alpha: A\times S\to A\), \((a,s)\to a\triangleright s\) and \(\beta: S\times B\to B\), \((s,b)\to s\triangleleft b\) be right and left semigroup actions, respectively. Let \(P=(p_{a,b})\) be an \(A\times B\)-matrix with entries in \(S\cup A\cup B\) such that \(p_{a\triangleright s,b}=p_{a,s\triangleleft b}\) for all \(a\in A\), \(b\in B\) and \(s\in S\). Then \({\mathcal C}(S;\alpha,\beta;P)\) is the semigroup generated by the elements of \(S\cup A\cup B\) subject to the defining relations \(ab=p_{a,b}\), \(as=a\triangleright s\), \(sb=s\triangleleft b\), \(st=s.t\), for all \(a\in A\), \(b\in B\), s,t\(\in S\). One can show that the mapping \(S\to {\mathcal C}(S;\alpha,\beta;P)\), \(s\to s\), is an embedding. If S contains an identity, then a modification of the above construction yields a monoid \({\mathcal C}^ 1(S;\alpha,\beta;P)\) so that again \(S\to {\mathcal C}^ 1(S;\alpha,\beta;P)\) is a monoid embedding. Properties satisfied by \(\alpha\), \(\beta\) or P translate into properties enjoyed by \({\mathcal C}(S;\alpha,\beta;P)\) or \({\mathcal C}^ 1(S;\alpha,\beta;P)\). Using the above construction the author proves the following remarkable results: 1. any [countable] semigroup can be embedded in a [2-generated] congruence free bisimple semigroup with identity, 2. any [countable] semigroup without idempotents can be embedded in a [2-generated] congruence free semigroup without idempotents.
congruence free semigroups without idempotents
scientific article; zbMATH DE number 4150514
scientific article