On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups
We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitskiĭ showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitskiĭ result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.
Keywordslattice semigroup sublattice variety
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- 12.Repnitskii V. B., “Nilpotency of algebras and identities on subalgebra lattices,” in: Eds. S. Kublanovsky, A. Mikhalev, J. Ponizovski, Semigroups with Applications, Including Semigroup Rings, Walter de Gruyter, Berlin, 1998, pp. 315–328.Google Scholar