Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T18:13:35.194Z Has data issue: false hasContentIssue false
  • English
  • Français

COMBINATORIAL REES–SUSHKEVICH VARIETIES THAT ARE CROSS, FINITELY GENERATED, OR SMALL

Part of: Semigroups Varieties

Published online by Cambridge University Press:  13 August 2009

EDMOND W. H. LEE
EDMOND W. H. LEE*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set ℱ of finitely generated varieties constitutes an incomplete sublattice and the set 𝒮 of small varieties constitutes a strict incomplete sublattice of ℱ. Consequently, a combinatorial Rees–Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set Σ of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity 𝒪(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ.

Keywords

semigroups0-simplevarietiesRees–SushkevichCrossfinitely basedfinitely generatedsmall

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups 08B15: Lattices of varieties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 64 - 84
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Almeida, J., Finite Semigroups and Universal Algebra (World Scientific, Singapore, 1994).Google Scholar
[2]Auinger, K. and Szendrei, M. B., ‘On identity bases of epigroup varieties’, J. Algebra 220 (1999), 437–448.CrossRefGoogle Scholar
[3]Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra (Springer, New York, 1981).CrossRefGoogle Scholar
[4]Edmunds, C. C., ‘Varieties generated by semigroups of order four’, Semigroup Forum 21 (1980), 67–81.CrossRefGoogle Scholar
[5]Hall, T. E., Kublanovsky, S. I., Margolis, S., Sapir, M. V. and Trotter, P. G., ‘Algorithmic problems for finite groups and finite 0-simple semigroups’, J. Pure Appl. Algebra 119 (1997), 75–96.CrossRefGoogle Scholar
[6]Kublanovsky, S. I., ‘On the Rees–Sushkevich variety’, Unpublished manuscript.Google Scholar
[7]Kublanovsky, S. I., Lee, E. W. H. and Reilly, N. R., ‘Some conditions related to the exactness of Rees–Sushkevich varieties’, Semigroup Forum 76 (2008), 87–94.CrossRefGoogle Scholar
[8]Lee, E. W. H., ‘Identity bases for some non-exact varieties’, Semigroup Forum 68 (2004), 445–457.CrossRefGoogle Scholar
[9]Lee, E. W. H., ‘Subvarieties of the variety generated by the five-element Brandt semigroup’, Internat. J. Algebra Comput. 16 (2006), 417–441.CrossRefGoogle Scholar
[10]Lee, E. W. H., ‘On the complete join of permutative combinatorial Rees–Sushkevich varieties’, Int. J. Algebra 1 (2007), 1–9.CrossRefGoogle Scholar
[11]Lee, E. W. H., ‘Combinatorial Rees–Sushkevich varieties are finitely based’, Internat. J. Algebra Comput. 18 (2008), 957–978.CrossRefGoogle Scholar
[12]Lee, E. W. H. and Reilly, N. R., ‘The intersection of pseudovarieties of central simple semigroups’, Semigroup Forum 73 (2006), 75–94.CrossRefGoogle Scholar
[13]Lee, E. W. H. and Reilly, N. R., ‘Centrality in Rees–Sushkevich varieties’, Algebra Universalis 58 (2008), 145–180.CrossRefGoogle Scholar
[14]Lee, E. W. H. and Volkov, M. V., ‘On the structure of the lattice of combinatorial Rees–Sushkevich varieties’, in: Semigroups and Formal Languages (eds. André, J. M.et al.) (World Scientific, Singapore, 2007), pp. 164–187.CrossRefGoogle Scholar
[15]Lee, E. W. H. and Volkov, M. V., ‘Limit varieties generated by completely 0-simple semigroups’, Preprint.Google Scholar
[16]Malyshev, S. A., ‘Permutational varieties of semigroups whose lattice of subvarieties is finite’, in: Modern Algebra (Leningrad Univ., Leningrad, 1981), pp. 71–76 (in Russian).Google Scholar
[17]Mashevitzky, G. I., ‘On bases of completely simple semigroup identities’, Semigroup Forum 30 (1984), 67–76.CrossRefGoogle Scholar
[18]Mashevitzky, G. I., ‘Varieties generated by completely 0-simple semigroups’, in: Semigroups and their Homomorphisms (ed. Lyapin, E. S.) (Ross. Gos. Ped. Univ., Leningrad, 1991), pp. 53–62 (in Russian).Google Scholar
[19]Mashevitzky, G. I., ‘On the finite basis problem for completely 0-simple semigroup identities’, Semigroup Forum 59 (1999), 197–219.CrossRefGoogle Scholar
[20]Rees, D., ‘On semi-groups’, Proc. Cambridge Philos. Soc. 36 (1940), 387–400.CrossRefGoogle Scholar
[21]Reilly, N. R., ‘The interval [B2,NB2] in the lattice of Rees–Sushkevich varieties’, Algebra Universalis 59 (2008), 345–363.CrossRefGoogle Scholar
[22]Sapir, M. V., ‘On Cross semigroup varieties and related questions’, Semigroup Forum 42 (1991), 345–364.CrossRefGoogle Scholar
[23]Sushkevich, A. K., ‘Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit’, Math. Ann. 99 (1928), 30–50 (in German).Google Scholar
[24]Tarjan, R., ‘Depth-first search and linear graph algorithms’, SIAM J. Comput. 1 (1972), 146–160.CrossRefGoogle Scholar
[25]Trahtman, A. N., ‘Identities of a five-element 0-simple semigroup’, Semigroup Forum 48 (1994), 385–387.CrossRefGoogle Scholar
[26]Vernikov, B. M. and Volkov, M. V., ‘Lattices of nilpotent semigroup varieties. II’, Izv. Ural. Gos. Univ. Mat. Mekh. 1 (1998), 13–33 (in Russian).Google Scholar
[27]Volkov, M. V., ‘The finite basis problem for finite semigroups’, Sci. Math. Jpn. 53 (2001), 171–199.Google Scholar
[28]Volkov, M. V., ‘On a question by Edmond W. H. Lee’, Izv. Ural. Gos. Univ. Mat. Mekh. 7 (2005), 167–178.Google Scholar