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ON RESIDUALLY FINITE VARIETIES OF INVOLUTION SEMIGROUPS

Part of: Semigroups

Published online by Cambridge University Press:  16 December 2010

IGOR DOLINKA
IGOR DOLINKA*
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia (email: [email protected])
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Abstract

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We prove that the variety consisting of all involutory inflations of normal bands is the unique maximal residually finite variety consisting of combinatorial semigroups with involution.

Keywords

involution semigroupvarietyresidually finitesubdirectly irreducible

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 2 , October 2010 , pp. 181 - 198
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Andersen, O., ‘Ein Bericht über die Strukture abstrakter Halbgruppen’, Thesis, Hamburg, 1952.Google Scholar
[2]Auinger, K., ‘Free products of combinatorial strict inverse semigroups’, Pacific J. Math. 164 (1994), 201–227.CrossRefGoogle Scholar
[3]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, Graduate Texts in Mathematics (Springer, New York, 1981).CrossRefGoogle Scholar
[4]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. I (American Mathematical Society, Providence, RI, 1961).Google Scholar
[5]Crvenković, S. and Dolinka, I., ‘Varieties of involution semigroups and involution semirings: a survey’, Bull. Soc. Math. Banja Luka 9 (2002), 7–47.Google Scholar
[6]Crvenković, S., Dolinka, I. and Vinčić, M., ‘Equational bases for some 0-direct unions of semigroups’, Studia Sci. Math. Hungar. 36 (2000), 423–431.Google Scholar
[7]Descalço, L. and Ruškuc, N., ‘Properties of the subsemigroups of the bicyclic monoid’, Czechoslovak Math. J. 58(133) (2008), 311–330.CrossRefGoogle Scholar
[8]Dolinka, I., ‘Remarks on varieties of involution bands’, Comm. Algebra 28 (2000), 2837–2852.CrossRefGoogle Scholar
[9]Dolinka, I., ‘All varieties of normal bands with involution’, Period. Math. Hungar. 40 (2000), 109–122.CrossRefGoogle Scholar
[10]Dolinka, I., ‘On the lattice of varieties of involution semigroups’, Semigroup Forum 62 (2001), 438–459.CrossRefGoogle Scholar
[11]Dolinka, I., ‘Subdirectly irreducible bands with involution’, Acta Sci. Math. (Szeged) 67 (2001), 535–554.Google Scholar
[12]Dolinka, I., ‘Varieties of involution semilattices of Archimedean semigroups’, Publ. Math. Debrecen 66 (2005), 439–447.CrossRefGoogle Scholar
[13]Dolinka, I., ‘On identities of finite involution semigroups’, Semigroup Forum 80 (2010), 105–120.CrossRefGoogle Scholar
[14]Fajtlowicz, S., ‘Equationally complete semigroups with involution’, Algebra Universalis 1 (1972), 355–358.CrossRefGoogle Scholar
[15]Gerhard, J. A., ‘Subdirectly irreducible idempotent semigroups’, Pacific J. Math. 39 (1971), 669–676.CrossRefGoogle Scholar
[16]Gerhard, J. A., ‘Some subdirectly irreducible idempotent semigroups’, Semigroup Forum 5 (1973), 362–369.CrossRefGoogle Scholar
[17]Golubov, E. A. and Sapir, M. V., ‘Varieties of finitely approximable semigroups’, Izv. Vysš. Učebn. Zaved. Mat. 11(246) (1982), 21–29 (in Russian); English transl. Soviet Math. (Iz. VUZ) 26 (1982), 25–36.Google Scholar
[18]Howie, J. M., Fundamentals of Semigroup Theory (Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[19]Kleiman, E. I., ‘On basis of identities of Brandt semigroups’, Semigroup Forum 13 (1977), 209–218.CrossRefGoogle Scholar
[20]Kublanovskiĭ, S. I., ‘Finite approximability of prevarieties of semigroups with respect to predicates’, in: Modern Algebra (Gos. Ped. Inst., Leningrad, 1980), pp. 58–88 (in Russian).Google Scholar
[21]Lallement, G., ‘On monoids presented by a single relation’, J. Algebra 32 (1974), 370–388.CrossRefGoogle Scholar
[22]McKenzie, R., ‘Residually small varieties of semigroups’, Algebra Universalis 13 (1981), 171–201.Google Scholar
[23]McKenzie, R., ‘A note on residually small varieties of semigroups’, Algebra Universalis 17 (1983), 142–149.Google Scholar
[24]Mel’nik, I. I., ‘Varieties and lattices of varieties of semigroups’, in: Studies in Algebra, Vol. 2 (ed. Vagner, V. V.) (Izdat. Saratov. Univ., Saratov, 1970), pp. 47–57 (in Russian).Google Scholar
[25]Ol’shanskiĭ, A. Yu., ‘Varieties of finitely approximable groups’, Izv. Akad. Nauk SSSR, Ser. Mat. 33 (1969), 915–927 (in Russian); English transl. Math. USSR-Izv. 3 (1969), 867–877.Google Scholar
[26]Petrich, M., Introduction to Semigroups (Merrill, Columbus, OH, 1973).Google Scholar
[27]Petrich, M., ‘All subvarieties of a certain variety of semigroups’, Semigroup Forum 7 (1974), 104–152.CrossRefGoogle Scholar
[28]Petrich, M., Inverse Semigroups (Wiley, New York, 1984).Google Scholar
[29]Putcha, M. S., ‘Semilattice decompositions of semigroups’, Semigroup Forum 6 (1973), 12–34.CrossRefGoogle Scholar
[30]Reilly, N. R., ‘Free combinatorial strict inverse semigroups’, J. Lond. Math. Soc. (2) 39 (1989), 102–120.CrossRefGoogle Scholar
[31]Sapir, M. V., ‘Problems of Burnside type and the finite basis property in varieties of semigroups’, Izv. Akad. Nauk SSSR, Ser. Mat. 51 (1987), 319–340 (in Russian); English transl. Math. USSR-Izv. 30 (1987), 295–314.Google Scholar
[32]Sapir, M. V. and Shevrin, L. N., ‘Residually small varieties of semigroups and groups’, Izv. Vysš. Učebn. Zav. Mat. 10(317) (1988), 41–49 (in Russian); English transl. Soviet Math. (Iz. VUZ) 32 (1988), 57–73.Google Scholar
[33]Sapir, M. V. and Sukhanov, E. V., ‘Varieties of periodic semigroups’, Izv. Vysš. Učebn. Zav. Mat. 4(227) (1981), 48–55 (in Russian); English transl. Soviet Math. (Iz. VUZ) 25 (1981), 53–63.Google Scholar
[34]Scheiblich, H. E., ‘Projective and injective bands with involution’, J. Algebra 109 (1987), 281–291.Google Scholar