Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T08:43:55.744Z Has data issue: false hasContentIssue false

Canonical Varieties of Completely Regular Semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Mario Petrich
Affiliation:
21420 Bol, Brač Croatia
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.

Completely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L(CR). Let B denote the variety of bands and L(B) the lattice of its subvarieties. The mapping VVB is a complete homomorphism of L(CR) onto L(B). The congruence induced by it has classes that are intervals, say VB = [VB, VB] for VL(CR). Here VB = VB. We characterize VB in several ways, the principal one being an inductive way of constructing bases for v-irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties BG, OBG and B.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Gerhard, J. A. and Petrich, M., ‘Certain characterizations of varieties of bands’, Proc. Edinburgh Math. Soc. (2) 31 (1988), 301319.CrossRefGoogle Scholar
[2]Gerhard, J. A. and Petrich, M., ‘Varieties of bands revisited’, Proc. London Math. Soc. (3) 58 (1989), 323350.CrossRefGoogle Scholar
[3]Pastijn, F. and Petrich, M., ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607633.CrossRefGoogle Scholar
[4]Pastijn, F. J., ‘The lattice of completely regular semigroup varieties’, J. Aust. Math. Soc. 49 (1990), 2442.CrossRefGoogle Scholar
[5]Petrich, M., ‘Varieties of orthodox bands of groups’, Pacific J. Math. 58 (1975), 209217.CrossRefGoogle Scholar
[6]Petrich, M. and Reilly, N. R., ‘Operators related to E-disjunctive and fundamental completely regular semigroups’, J. Algebra 134 (1990), 127.CrossRefGoogle Scholar
[7]Petrich, M. and Reilly, N. R., Completely regular semigroups, Vol. 1, Canadian Math. Soc. Series of Monographs and Advanced Texts 23 (Wiley, New York, 1999).Google Scholar
[8]Petrich, M. and Reilly, N. R., Completely regular semigroups, Vol. II (in preparation).Google Scholar
[9]Polák, L., ‘On varieties of completely regular semigroups I’, Semigroup Forum 32 (1985), 97123.CrossRefGoogle Scholar
[10]Polák, L., ‘On varieties of completely regular semigroups II’, Semigroup Forum 36 (1987), 253284.CrossRefGoogle Scholar
[11]Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Izv. Vyssh. Uchebn. Zaved. Matem. 11 (1982), 8285 (Russian).Google Scholar
[12]Reilly, N. R., ‘Completely regular semigroups’, in: Lattices, Semigroups and Universal Algebra (ed. Almeida, J.) (Plenum Press, New York, 1990) pp. 225242.CrossRefGoogle Scholar
[13]Reilly, N. R. and Zhang, S., ‘Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands’, Algebra Universalis 44 (2000), 217239.CrossRefGoogle Scholar
[14]Trotter, P. G., ‘Subdirect decomposition of the lattice of varieties of completely regular semigroups’, Bull. Austral. Math. Soc. 39 (1989), 343351.CrossRefGoogle Scholar
[15]Trotter, P. G. and Weil, P., ‘The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue’, Algebra Universalis 37 (1997), 491526.CrossRefGoogle Scholar