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Canonical Varieties of Completely Regular Semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Mario Petrich
Mario Petrich
Affiliation:
21420 Bol, Brač Croatia
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Abstract

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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 V → V ∩ B 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 V ∈ L(CR). Here VB = V ∩ B. 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.

Keywords

completely regular semigroupvarietybandcongruenceintervalPolák's theoremladder.

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 87 - 104
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), 301–319.CrossRefGoogle Scholar
[2]Gerhard, J. A. and Petrich, M., ‘Varieties of bands revisited’, Proc. London Math. Soc. (3) 58 (1989), 323–350.CrossRefGoogle Scholar
[3]Pastijn, F. and Petrich, M., ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607–633.CrossRefGoogle Scholar
[4]Pastijn, F. J., ‘The lattice of completely regular semigroup varieties’, J. Aust. Math. Soc. 49 (1990), 24–42.CrossRefGoogle Scholar
[5]Petrich, M., ‘Varieties of orthodox bands of groups’, Pacific J. Math. 58 (1975), 209–217.CrossRefGoogle Scholar
[6]Petrich, M. and Reilly, N. R., ‘Operators related to E-disjunctive and fundamental completely regular semigroups’, J. Algebra 134 (1990), 1–27.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), 97–123.CrossRefGoogle Scholar
[10]Polák, L., ‘On varieties of completely regular semigroups II’, Semigroup Forum 36 (1987), 253–284.CrossRefGoogle Scholar
[11]Rasin, V. V., ‘Varieties of orthodox Clifford semigroups’, Izv. Vyssh. Uchebn. Zaved. Matem. 11 (1982), 82–85 (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. 225–242.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), 217–239.CrossRefGoogle Scholar
[14]Trotter, P. G., ‘Subdirect decomposition of the lattice of varieties of completely regular semigroups’, Bull. Austral. Math. Soc. 39 (1989), 343–351.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), 491–526.CrossRefGoogle Scholar