Topologically congruence-free compact semigroups

Abstract

Abstract A classical result of semigroup theory says that any finite congruence-free semigroup S (i.e., S has exactly two congruences) without zero such that card ( S ) > 2 is a simple group. We shall show that an analogous result holds for any infinite topologically congruence-free compact semigroup (a compact semigroup A is topologically congruence-free if the set of its algebraic congruences ρ for which A / ρ is a compact semigroup, is equal to { 1 A , A × A } ). In fact, every such semigroup must be a metric Lie group with cardinality c . Also, we prove that all topologically congruence-free compact semigroups with zero are (unfortunately) finite.