Acts with identities in the congruence lattice

We prove that for any act X over a finite semigroup S, the congruence lattice \({{\,\mathrm{Con}\,}}X\) embeds the lattice \({{\,\mathrm{Eq}\,}}M\) 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 \({{\,\mathrm{Con}\,}}X\) 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 \({\mathcal {M}}^0(G,I,\Lambda ,P)\) where \(|G|,|I| <\infty \). We construct examples that show that the assumption \(|G|,|I| <\infty \) is essential.