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.

我们证明对于有限半群S上的任何动作X，同余格\({{\,\mathrm{Con}\,}}X\)嵌入了格\({{\,\mathrm{Eq}\, }}M\)的无限集M的所有等价当且仅当X是无限的。等效地：对于有限半群S上的动作X ，当且仅当X是有限时，格\({{\,\mathrm{Con}\,}}X\)满足非平凡恒等式。对于完全 0-单半群\({\mathcal {M}}^0(G,I,\Lambda ,P)\)上的零行为，类似的陈述被证明，其中\(|G|,|I| < \infty \). 我们构建的例子表明假设\(|G|,|I| <\infty \)是必不可少的。