Bases of identities for semigroups of bounded rank transformations of a set

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Abstract

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup Tk(X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and {pipe}X{pipe} ∈ {3, 4} or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).

Original languageEnglish
Pages (from-to)451-481
Number of pages31
JournalIsrael Journal of Mathematics
Volume191
Issue number1
DOIs
StatePublished - 1 Sep 2012

ASJC Scopus subject areas

  • Mathematics (all)

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