Abstract
We present a method for proving that a semigroup is finitely based (FB) and find some new sufficient conditions under which a monoid is FB. As an application, we find a class of finite aperiodic monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm. The method results in a short proof of the theorem of E. Lee that every monoid that satisfies \(xt_1xyt_2y \approx xt_1yxt_2y\) and \(xyt_1xt_2y \approx yxt_1xt_2y\) is FB. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of \(n\)-testable languages is FB if and only if \(n \le 3\).
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References
Ashikhmin, D.N., Volkov, M.V.: The finite basis problem for Kiselman monoids. Preprint, available under arXiv:1411.0223 [math.GR]
Blanchet-Sadri, F.: Equations and dot-depth one. Semigroup Forum 47, 305–317 (1993)
Blanchet-Sadri, F.: Equations and monoid varieties of dot-depth one and two. Theor. Comput. Sci. 123, 239–258 (1994)
Edmunds, C.C.: On certain finitely based varieties of semigroups. Semigroup Forum 15(1), 21–39 (1977)
Eilenberg, S.: Automata, Languages and Machines, Vol. B. Academic Press, New York (1976)
Jackson, M.G.: Finiteness properties of varieties and the restriction to finite algebras. Semigroup Forum 70, 159–187 (2005)
Jackson, M.G., Sapir, O.B.: Finitely based, finite sets of words. Int. J. Algebr. Comput. 10(6), 683–708 (2000)
Lee, E.W.H.: Finitely generated limit varieties of aperiodic monoids with central idempotents. J. Algebr. Appl. 8(6), 779–796 (2009)
Lee, E.W.H.: Maximal Specht varieties of monoids. Mosc. Math. J. 12, 787–802 (2012)
Perkins, P.: Bases for equational theories of semigroups. J. Algebr. 11, 298–314 (1969)
Pin, J.-E.: Varietes de Languages Formels (Masson, 1984)(in French); English translation Varieties of Formal Languages (North Oxford Academic, 1986 and Plenum, 1986)
Sapir, O. B.: Non-finitely based monoids. Accepted to Semigroup Forum, available under arXiv:1402.5409v6 [math.GR]
Sapir, O. B.: The finite basis problem for words with at most two non-linear variables. Preprint, available under arXiv:1403.6430 [math.GR]
Sapir, O. B.: Finitely based sets of 2-limited block-2-simple words (in preparation)
Sapir, O.B.: Finitely based words. Int. J. Algebr. Comput. 10(4), 457–480 (2000)
Shevrin, L.N., Volkov, M.V.: Identities of semigroups. Russ. Math. (Iz. VUZ) 29(11), 1–64 (1985)
Volkov, M.V.: A general finite basis condition for system of semigroup identities. Semigroup Forum 86, 181–191 (1990)
Volkov, M.V.: Reflexive relations, extensive transformations and piecewise testable languages of a given height. Int. J. Algebr. Comput. 14(5,6), 817–827 (2004)
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The author thanks Gili Golan, Edmond Lee, Wenting Zhang and an anonymous referee for helpful comments and suggestions.
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Communicated by Marcel Jackson.
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Sapir, O. Finitely based monoids. Semigroup Forum 90, 587–614 (2015). https://doi.org/10.1007/s00233-015-9709-1
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DOI: https://doi.org/10.1007/s00233-015-9709-1