Abstract
We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid \({\mathcal{K}_n}\) are nonfinitely based for each \({n \geq 3}\). This result holds also for the case when \({\mathcal{K}_n}\) is considered as an involution semigroup under either of its natural involutions.
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Auinger K.: Pseudovarieties generated by Brauer-type monoids. Forum Math. 26, 1–24 (2014)
Auinger K., Dolinka I., Pervukhina T.V., Volkov M.V.: Unary enhancements of inherently non-finitely based semigroups. Semigroup Forum 89, 41–51 (2014)
Auinger K., Dolinka I., Volkov M.V.: Matrix identities involving multiplication and transposition. J. European Math. Soc. 14, 937–969 (2012)
Auinger K., Dolinka I., Volkov M.V.: Equational theories of semigroups with involution. J. Algebra 369, 203–225 (2012)
Bokut’, L.A., Lee, D.V.: A Gröbner–Shirshov basis for the Temperley–Lieb–Kauffman monoid. Izv. Ural. Gos. Univ. Mat. Mekh. No. 7 (36): 47–66 (2005) (Russian)
Borisavljević M., Došen K., Petrić Z.: Kauffman monoids. J. Knot Theory Ramifications 11, 127–143 (2002)
Brauer R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 857–872 (1937)
Brown, T. C.: On locally finite semigroups. Ukrain. Mat. Ž. 20, 732–738 (1968) (Russian; Engl. translation Ukrainian Math. J. 20, 631–636)
Brown T.C.: An interesting combinatorial method in the theory of locally finite semigroups. Pacific J. Math. 36, 285–289 (1971)
Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Springer, Berlin (1981)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I Amer. Math. Soc., Providence (1961)
Došen K., Petrić Z.: Self-adjunctions and matrices. J. Pure Appl. Algebra 184, 7–39 (2003)
Hall T.E.: Regular semigroups: amalgamation and the lattice of existence varieties. Algebra Universalis 28, 79–102 (1991)
Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Kauffman L.: An invariant of regular isotopy. Trans. Amer. Math. Soc. 318, 417–471 (1990)
Lau K.W., FitzGerald D.G.: Ideal structure of the Kauffman and related monoids. Comm. Algebra 34, 2617–2629 (2006)
Mal’cev, A.I.: Multiplication of classes of algebraic systems. Sibirsk. Mat. Ž. 8, 346–365 (1967) (Russian; Engl. translation Siberian Math. J. 8, 254–267)
Perkins P.: Bases for equational theories of semigroups. J. Algebra 11, 298–314 (1969)
Sapir, M.V.: Problems of Burnside type and the finite basis property in varieties of semigroups. Izv. Akad. Nauk SSSR Ser. Mat. 51, 319–340 (1987) (Russian; Engl. translation Math. USSR–Izv. 30, 295–314)
Sapir M.V., Volkov M.V.: On the join of semigroup varieties with the variety of commutative semigroups. Proc. Amer. Math. Soc. 120, 345–348 (1994)
Shneerson L.M.: On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation. Semigroup Forum 39, 17–38 (1989)
Temperley H.N.V., Lieb E.H.: Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the ‘percolation’ problem. Proc. Roy. Soc. London, Ser. A 322, 251–280 (1971)
Volkov M.V.: The finite basis problem for finite semigroups. Sci. Math. Jpn. 53, 171–199 (2001)
Volkov, M.V.: A nonfinitely based semigroup of triangular matrices. In: P.G. Romeo, J. Meakin, A.R. Rajan (eds.), Semigroups, Algebra and Operator Theory, Springer Proceedings in Mathematics & Statistics, Vol. 142. Springer, Berlin, 2015
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Presented by M. Jackson.
Dedicated to Brian Davey on the occasion of his 65th birthday
Yuzhu Chen, Xun Hu, Yanfeng Luo have been partially supported by the Natural Science Foundation of China (projects no. 10971086, 11371177). M. V. Volkov acknowledges support from the Presidential Programme “Leading Scientific Schools of the Russian Federation”, project no. 5161.2014.1, the Russian Foundation for Basic Research, project no. 14-01-00524, and the Ministry of Education and Science of the Russian Federation, project no. 1.1999.2014/K.
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Auinger, K., Chen, Y., Hu, X. et al. The finite basis problem for Kauffman monoids. Algebra Univers. 74, 333–350 (2015). https://doi.org/10.1007/s00012-015-0356-x
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DOI: https://doi.org/10.1007/s00012-015-0356-x