On the rank of the Rees–Sushkevich varieties

Kublanovskiĭ, S.

doi:10.1090/S1061-0022-2012-01214-8

On the rank of the Rees–Sushkevich varieties
HTML articles powered by AMS MathViewer

by S. I. Kublanovskiĭ
Translated by: B. M. Bekker

St. Petersburg Math. J. 23 (2012), 679-730

DOI: https://doi.org/10.1090/S1061-0022-2012-01214-8

Published electronically: April 13, 2012

PDF | Request permission

Abstract:

A specific numerical characteristic of a variety of semigroups, the rank, is introduced. It is proved that the Rees–Sushkevich varieties with the same derivative, i.e., containing the same $0$-simple semigroups, are determined by their rank uniquely up to permutation identities. As a consequence, answers to several well-known questions are obtained. In particular, a description is given for the Rees–Sushkevich varieties satisfying finiteness conditions (finiteness of the base of identities or of the lattice of subvarieties, generation by a finite semigroup or by a completely $0$-simple semigroup, the condition of maximality, minimality, finite width, etc.). Some applications of an algorithmic nature are presented. In particular, it is shown that a Rees–Sushkevich variety defined by a finite set of identities or by a finite semigroup has a decidable (polynomially decidable) equational theory if and only if its derivative has the same property. This holds true for combinatorial varieties.

References

L. N. Shevrin, B. M. Vernikov, and M. V. Volkov, Lattices of semigroup varieties, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (2009), 3–36 (Russian, with English and Russian summaries); English transl., Russian Math. (Iz. VUZ) 53 (2009), no. 3, 1–28. MR 2581451, DOI 10.3103/S1066369X09030013
M. V. Sapir and E. V. Suhanov, Varieties of periodic semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1981), 48–55 (Russian). MR 624862
S. V. Pleshcheva and I. V. Verteshi, Complexity of the problem of identity verification in a certain finite $0$-simple semigroup, Ural. Gos. Univ. Mat. Zap. 43 (2006), 72–102. (Russian)
V. L. Murskiĭ, Some examples of varieties of semigroups, Mat. Zametki 3 (1968), 663–670 (Russian). MR 231930
G. I. Mashevitskiĭ, On identities in varieties of completely simple semigroups over abelian groups, Modern Algebra, Leningrad. Ped. Inst., Leningrad, 1978, pp. 81–89. (Russian)
M. V. Volkov, Bases of identities of Brandt semigroups, Ural. Gos. Univ. Mat. Zap. 14 (1985), no. 1, Issled. Algebr. Sist. po Svoĭst. ikh Podsist., 38–42, i–ii (Russian). MR 867373
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
Sheila Oates and M. B. Powell, Identical relations in finite groups, J. Algebra 1 (1964), 11–39. MR 161904, DOI 10.1016/0021-8693(64)90004-3
T. E. Hall, S. I. Kublanovskii, S. Margolis, M. V. Sapir, and P. G. Trotter, Algorithmic problems for finite groups and finite $0$-simple semigroups, J. Pure Appl. Algebra 119 (1997), no. 1, 75–96. MR 1456096, DOI 10.1016/S0022-4049(96)00050-3
Edmond W. H. Lee, Subvarieties of the variety generated by the five-element Brandt semigroup, Internat. J. Algebra Comput. 16 (2006), no. 2, 417–441. MR 2228521, DOI 10.1142/S0218196706002998
Edmond W. H. Lee, Combinatorial Rees-Sushkevich varieties are finitely based, Internat. J. Algebra Comput. 18 (2008), no. 5, 957–978. MR 2440721, DOI 10.1142/S0218196708004755
Edmond W. H. Lee, Combinatorial Rees-Sushkevich varieties that are Cross, finitely generated, or small, Bull. Aust. Math. Soc. 81 (2010), no. 1, 64–84. MR 2584924, DOI 10.1017/S0004972709000616
S. A. Malyshev, Permutational varieties of semigroups whose lattice of subvarieties is finite, Modern algebra, Leningrad. Univ., Leningrad, 1981, pp. 71–76 (Russian). MR 662692
G. Pollák, On the consequences of permutation identities, Acta Sci. Math. (Szeged) 34 (1973), 323–333. MR 322084
D. Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387–400. MR 2893
Norman R. Reilly, Varieties generated by completely 0-simple semigroups, J. Aust. Math. Soc. 84 (2008), no. 3, 375–403. MR 2453687, DOI 10.1017/S1446788708000311
S. Bakulin, On identities of indicator Burnside semigroups, arXiv:1009.2337v1 [math.GR].
Steve Seif, The Perkins semigroup has co-NP-complete term-equivalence problem, Internat. J. Algebra Comput. 15 (2005), no. 2, 317–326. MR 2142086, DOI 10.1142/S0218196705002293
Steve Seif and Csaba Szabó, Computational complexity of checking identities in 0-simple semigroups and matrix semigroups over finite fields, Semigroup Forum 72 (2006), no. 2, 207–222. MR 2216090, DOI 10.1007/s00233-005-0510-4
S. Kublanovsky, On the Rees–Sushkevich variety, arXiv:1009.2982v1 [math.GR].
—, Cyclically regular semigroups, arXiv:1103.3246v1 [math.GR].
P. M. Cohn, Universal algebra, 2nd ed., Mathematics and its Applications, vol. 6, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. MR 620952
B. M. Vernikov, Upper-modular elements of the lattice of semigroup varieties, Algebra Universalis 59 (2008), no. 3-4, 405–428. MR 2470588, DOI 10.1007/s00012-008-2108-7