Acts with identities in the congruence lattice

We prove that for any act X over a finite semigroup S , the congruence lattice Con X embeds the lattice Eq M of all equivalences of an infinite set M if and only if X is infinite. Equivalently: for an act X over a finite semigroup S , the lattice Con X satisfies a non-trivial identity if and only if X is finite. Similar statements are proved for an act with zero over a completely 0-simple semigroup M 0 ( G , I , Λ , P ) where | G | , | I | < ∞ . We construct examples that show that the assumption | G | , | I | < ∞ is essential.