IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

Mikhail V. Volkov     (Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000, Russian Federation)

Abstract


We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.

Keywords


Brandt semigroup, Semigroup identity, Identity basis, Finite basis problem

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References


Belyaev V.V., Sesekin N.F. Free subsemigroups in soluble groups. Ural. Gos. Univ. Mat. Zap., 1981. Vol. 12, No. 3. P. 13–18. (In Russian)

Brandt H. Übereine Verallgemeinerung des Gruppenbegriffes. Math. Ann., 1927. Vol. 96, No. 1. P. 360–366. DOI: 10.1007/BF01209171

Burris S., Sankappanavar H.P. A Course in Universal Algebra. Berlin–Heidelberg–New York: Springer-Verlag, 1981. xvi+276 p.

Clifford, A.H. Matrix representations of completely simple semigroups. Amer. J. Math., 1942. Vol. 64, No. 1. P. 327–342. DOI: 10.2307/2371687

Clifford A.H., Preston G.B. The Algebraic Theory of Semigroups, Vol. I. 2nd ed. Providence, RI: Amer. Math. Soc., 1964. xvi+224 p.

Cohen D.E. On the laws of a metabelian variety. J. Algebra, 1967. Vol. 5, No. 3. P. 267–273. DOI: 10.1016/0021-8693(67)90039-7

Hall T.E., Kublanovskii S.I., Margolis S., Sapir M.V., Trotter P.G. Algorithmic problems for finite groups and finite 0-simple semigroups. J. Pure Appl. Algebra, 1997. Vol. 119, No. 1. P. 75–96. DOI: 10.1016/S0022-4049(96)00050-3

Howie J.M. Fundamentals of Semigroup Theory. 2nd ed. Oxford: Clarendon Press, 1995. xvi+352 p.

Isbell J.R. Two examples in varieties of monoids. Proc. Cambridge Philos. Soc., 1970. Vol. 68, No. 2. P. 265–266. DOI: 10.1017/S0305004100046065

Kad’ourek J. On bases of identities of finite inverse semigroups with solvable subgroups. Semigroup Forum, 2003. Vol. 67, No. 3. P. 317–343. DOI: 10.1007/s00233-001-0005-x

Kad’ourek J. On finite completely simple semigroups having no finite basis of identities. Semigroup Forum, 2018. Vol. 97, No. 1. P. 154–161. DOI: 10.1007/s00233-017-9907-0

Kalicki J. On comparison of finite algebras. Proc. Amer. Math. Soc. , 1952. Vol. 3, No. 1. P. 36–40. DOI: 10.2307/2032452

Kleiman E.I. On bases of identities of Brandt semigroups. Semigroup Forum, 1977. Vol. 13, No. 3. P. 209–218. DOI: 10.1007/BF02194938

Kleiman Ju.G. On a basis of the product of varieties of groups. Math. USSR. Izv., 1973. Vol. 7, No. 1. P. 91–94. DOI: 10.1070/IM1973v007n01ABEH001927

Lee E.W.H. Finite basis problem for semigroups of order five or less: generalization and revisitation. Studia Logica, 2013. Vol. 101, No. 1. P. 95–115. DOI: 10.1007/s11225-012-9369-z

Lee E.W.H., Volkov M.V. On the structure of the lattice of combinatorial Rees–Sushkevich varieties. Semigroups and Formal Languages. Hackensack, NJ: World Sci. Publ., 2007. P. 164–187. DOI: 10.1142/9789812708700_0012

Mashevitzky G.I. Identities in Brandt semigroups. Polugruppovye mnogoobrazija i polugruppy endomorfizmov [Semigroup varieties and semigroups of endomorphisms]. Leningrad: Leningrad State Pedagogical Institute, 1979. P. 126–137. (In Russian)

Mel’nik I.I. On varieties and lattices of varieties of semigroups. Issledovaniya po algebre [Investigations in algebra]. Saratov: Saratov State Univ., 1970. Vol. 2. P. 47–57. (In Russian)

Munn W.D. Matrix representations of semigroups. Proc. Cambrdige Philos. Soc. , 1957. Vol. 53, No. 1. P. 5–12. DOI: 10.1017/S0305004100031935

Neumann B.H. Identical relations in groups. I. Math. Ann. , 1937. Vol. 114, No. 1. P. 506–525. DOI: 10.1007/BF01594191

Neumann H. Varieties of groups. Berlin–Heidelberg–New York: Springer–Verlag, 1967. xii+192 p.

Oates S., Powell M.B. Identical relations in finite groups. J. Algebra, 1964. Vol. 1, No. 1. P. 11–39. DOI: 10.1016/0021-8693(64)90004-3

Petrich M. Inverse semigroups. New York: John Wiley & Sons, 1984. xii+674 p.

Reilly N.R. The interval \([\mathbf{B}_2, \mathbf{NB}_2]\) in the lattice of Rees–Sushkevich varieties. Algebra Universalis, 2008. Vol. 59, No. 3-4. P. 345–363. DOI: 10.1007/s00012-008-2091-z

Sapir M.V. Problems of Burnside type and the finite basis property in varieties of semigroups. Math. USSR. Izv. , 1988. Vol. 30, No. 2. P. 295–314. DOI: 10.1070/IM1988v030n02ABEH001012

Shevrin L.N., Sukhanov E.V. Structural aspects of the theory of varieties of semigroups. Soviet Math. (Iz. VUZ) , 1989. Vol. 33, No. 6. P. 1–34.

Trahtman A.N. An identity basis of the five-element Brandt semigroup. Ural. Gos. Univ. Mat. Zap. , 1981. Vol. 12, No. 3. P. 147–149. (In Russian)

Trahtman A.N. The finite basis problem for semigroups of order less than six. Semigroup Forum, 1983. Vol. 27. P. 387–389. DOI: 10.1007/BF02572749

Trahtman A.N. Finiteness of identity bases of 5-element semigroups. Polugruppy i ikh gomomorfizmy [Semigroups and their Homomorphisms]. Leningrad: Leningrad State Pedagogical Institute, 1991. P. 76–97. (In Russian)

Volkov M.V. On the identity bases of Brandt semigroups. Ural. Gos. Univ. Mat. Zap., 1985. Vol. 14, No. 1. P. 38–42. (In Russian)

Volkov M.V. The finite basis problem for finite semigroups. Sci. Math. Japon., 2001, Vol. 53, No. 1. P. 171–199.

Volkov M.V. On a question by Edmond W. H. Lee. Proc. Ural State Univ., 2005. No. 36 (Mathematics and Mechanics, No. 7). P. 167–178.




DOI: http://dx.doi.org/10.15826/umj.2019.2.008

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