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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 T k (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 |X| ∈ «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).

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, PoB 653, Beer-Sheva, 84105, Israel

    G. Mashevitzky

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  1. G. Mashevitzky
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Correspondence to G. Mashevitzky.

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Mashevitzky, G. Bases of identities for semigroups of bounded rank transformations of a set. Isr. J. Math. 191, 451–481 (2012). https://doi.org/10.1007/s11856-012-0009-0

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  • DOI: https://doi.org/10.1007/s11856-012-0009-0

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