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 {1A,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.
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