A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.
@article{bwmeta1.element.doi-10_2478_s11533-010-0080-x,
author = {Edmond Lee},
title = {Finite basis problem for 2-testable monoids},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1-22},
zbl = {1221.20040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0080-x}
}
Edmond Lee. Finite basis problem for 2-testable monoids. Open Mathematics, Tome 9 (2011) pp. 1-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0080-x/
[1] Almeida J., Finite Semigroups and Universal Algebra, Series in Algebra, 3, World Scientific, Singapore, 1994 | Zbl 0844.20039
[2] Burris S., Sankappanavar H.P., A Course in Universal Algebra, Grad. Texts in Math., 78, Springer, New York-Heidelberg-Berlin, 1981 | Zbl 0478.08001
[3] Edmunds C.C., On certain finitely based varieties of semigroups, Semigroup Forum, 1977, 15(1), 21–39 http://dx.doi.org/10.1007/BF02195732 | Zbl 0375.20057
[4] Edmunds C.C., Varieties generated by semigroups of order four, Semigroup Forum, 1980, 21(1), 67–81 http://dx.doi.org/10.1007/BF02572537 | Zbl 0449.20060
[5] 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, 119(1), 75–96 http://dx.doi.org/10.1016/S0022-4049(96)00050-3 | Zbl 0880.20040
[6] Lee E.W.H., Identity bases for some non-exact varieties, Semigroup Forum, 2004, 68(3), 445–457 http://dx.doi.org/10.1007/s00233-003-0029-5 | Zbl 1059.20054
[7] Lee E.W.H., Subvarieties of the variety generated by the five-element Brandt semigroup, Internat. J. Algebra Comput., 2006, 16(2), 417–441 http://dx.doi.org/10.1142/S0218196706002998 | Zbl 1103.20051
[8] Lee E.W.H., On identity bases of exclusion varieties for monoids, Comm. Algebra, 2007, 35(7), 2275–2280 http://dx.doi.org/10.1080/00927870701328722 | Zbl 1125.20044
[9] Lee E.W.H., Combinatorial Rees-Sushkevich varieties are finitely based, Internat. J. Algebra Comput., 2008, 18(5), 957–978 http://dx.doi.org/10.1142/S0218196708004755 | Zbl 1160.20057
[10] Lee E.W.H., On the variety generated by some monoid of order five, Acta Sci. Math. (Szeged), 2008, 74(3–4), 509–537
[11] Lee E.W.H., Hereditarily finitely based monoids of extensive transformations, Algebra Universalis, 2009, 61(1), 31–58 http://dx.doi.org/10.1007/s00012-009-0001-7 | Zbl 1184.20048
[12] Lee E.W.H., Combinatorial Rees-Sushkevich varieties that are Cross, finitely generated, or small, Bull. Aust. Math. Soc., 2010, 81(1), 64–84 http://dx.doi.org/10.1017/S0004972709000616 | Zbl 1192.20044
[13] Lee E.W.H., Varieties generated by 2-testable monoids (submitted) | Zbl 1274.20057
[14] Lee E.W.H., Li J.R., Minimal non-finitely based monoids (submitted)
[15] Lee E.W.H., Volkov M.V., On the structure of the lattice of combinatorial Rees-Sushkevich varieties, In: Semigroups and Formal Languages, Lisboa, 2005, World Scientific, Singapore, 2007, 164–187
[16] Lee E.W.H., Volkov M.V., Limit varieties generated by completely 0-simple semigroups, Internat. J. Algebra Comput. (in press)
[17] Luo Y.F., Zhang W.T., On the variety generated by all semigroups of order three (submitted) | Zbl 1243.20071
[18] Perkins P., Bases for equational theories of semigroups, J. Algebra, 1969, 11(2), 298–314 http://dx.doi.org/10.1016/0021-8693(69)90058-1
[19] Sapir M.V., Problems of Burnside type and the finite basis property in varieties of semigroups, Math. USSR, Izv., 1988, 30(2), 295–314 http://dx.doi.org/10.1070/IM1988v030n02ABEH001012
[20] Shevrin L.N., Volkov M.V., Identities of semigroups, Soviet Math. (Iz. VUZ), 1985, 29(11), 1–64 | Zbl 0629.20029
[21] Torlopova N.G., Varieties of quasi-orthodox semigroups, Acta Sci. Math. (Szeged), 1984, 47(3–4), 297–301 (in Russian) | Zbl 0577.20052
[22] Trahtman A.N., The finite basis question for semigroups of order less than six, Semigroup Forum, 1983, 27(1), 387–389 http://dx.doi.org/10.1007/BF02572749
[23] Trahtman A.N., Some finite infinitely basable semigroups, Ural. Gos. Univ. Mat. Zap., 1987, 14(2), 128–131 (in Russian)
[24] Trahtman A.N., Identities of a five-element 0-simple semigroup, Semigroup Forum, 1994, 48(3), 385–387 http://dx.doi.org/10.1007/BF02573687
[25] Trahtman A.N., Identities of locally testable semigroups, Comm. Algebra, 1999, 27(11), 5405–5412 http://dx.doi.org/10.1080/00927879908826762 | Zbl 0939.20058
[26] Volkov M.V., The finite basis question for varieties of semigroups, Math. Notes, 1989, 45(3), 187–194
[27] Volkov M.V., The finite basis problem for finite semigroups, Sci. Math. Jpn., 2001, 53(1), 171–199 | Zbl 0990.20039
[28] Zhang W.T., Some Studies on Varieties Generated by Finite Semigroups and Their Subvarieties Lattices, PhD thesis, Lanzhou University, Gansu, China, 2009 (in Chinese)