Semimodularity and bisimple ω-semigroups

Abstract

Let S be a completely 0-simple semigroup and let Λ( S ) be the lattice of congruences on S . G. Lallement ( 2 ) has described necessary and sufficient conditions on S for Λ( S ) to be modular, and has shown that Λ( S ) is always semimodular . This result may be stated: If S is 0-bisimple and contains a primitive idempotent, then Λ( S ) is semimodular.