On varieties of semilattice-ordered semigroups.
\((S,\cdot,\vee)\) is a semilattice ordered semigroup if \((S,\cdot)\) is a semigroup, \((S,\vee)\) a semilattice and \(a(b\vee c)=ab\vee ac\), \((a\vee b)c=ac\vee bc\) hold for all \(a,b,c\in S\). Section 2 gives examples which provide motivation. For any semigroup \(S\), \(F(S)\) consists of the finite nonempty subsets of \(S\) and then \((F(S),\cdot,\cup)\) is a semilattice ordered semigroup. If \(X^+\) is the free semigroup on the set \(X\), then \((F(X^+),\cdot,\cup)\) is the free semilattice ordered semigroup on \(X\). Section 3 of the paper brings a case study for what will be established more in general in the main results of Section 4. If \(S_r\) is the semigroup free on the set \(X\) in the semigroup variety determined by \(x^r=x\), then the authors find the fully invariant congruence \(\rho\) on \((F(S_r),\cdot,\cup)\) such that \(F(S_r)/\rho\) is free on \(X\) in the variety of semilattice ordered semigroups satisfying \(x^r=x\). Here \(\rho\) is obtained in terms of a transparently defined closure operation on the set of subsets of \(S_r\). More in general, given a fully invariant congruence \(\rho\) on \(X^+\), the authors define so-called \(\rho\)-admissible closure operations on \(F(X^+)/\rho\) which in turn determine fully invariant congruences on \((F(X^+),\cdot,\cup)\). Proceeding this way, they find all fully invariant congruences on \((F(X^+),\cdot,\cup)\), \(X\) countably infinite, which correspond to semilattice ordered semigroup varieties whose semigroup reducts belong to the semigroup variety corresponding to \(\rho\).
scientific article; zbMATH DE number 5014166