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Decidability of elementary theories of lattices of subsemigroups. (English. Russian original) Zbl 0709.03004

Sib. Math. J. 31, No. 1, 21-26 (1990); translation from Sib. Mat. Zh. 31, No. 1(179), 27-33 (1990).
Let \({\mathcal E}\) be an elementary language, Sub be the operator of taking the lattices of subsemigroups. The variety of semigroups \({\mathcal M}\) is said to be \({\mathcal E} Sub\)-solvable if the elementary theory \({\mathcal E} Sub {\mathcal M}\) of the class Sub \({\mathcal M}=\{Sub A|\) \(A\in {\mathcal M}\}\) is solvable. The main result of the paper is: Theorem. The variety of semigroups \({\mathcal M}\) is \({\mathcal E} Sub\)-solvable iff \({\mathcal M}\subset [xyz=xz]\) or \({\mathcal M}\subseteq [xyz=x]\) \([x^ ny=y\), \(xy=yx]\) for some \(n\geq 1\).
Reviewer: M.Tetruašvili

MSC:

03B25 Decidability of theories and sets of sentences
20M07 Varieties and pseudovarieties of semigroups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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