Abstract
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.
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The authors would like to thank the referee for his valuable comments and suggestions which allowed them to eliminate defects and to correct one proof.
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Communicated by Presented by E. W. Kiss.
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Research supported by a grant of the Center Fund. Appl. Math. of Lomonosov MSU.
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Kozhuhov, I.B., Pryanichnikov, A.M. Acts with identities in the congruence lattice. Algebra Univers. 83, 16 (2022). https://doi.org/10.1007/s00012-022-00773-6
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DOI: https://doi.org/10.1007/s00012-022-00773-6