The lattice of congruences on direct products of cyclic semigroups and certain other semigroups

Abstract

SynopsisLet W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.