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Identities for existence varieties of regular semigroups

Published online by Cambridge University Press:  17 April 2009

T.E. Hall
T.E. Hall
Affiliation:
Mathematics DepartmentMonash UniversityClayton, Victoria 3168Australia
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Abstract

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A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 59 - 77
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Birjukov, A.P., ‘Varieties of idempotent semigroups’, Algebra i Logika 9 (1970), 255–273. (Russian).Google Scholar
[2]Clifford, A.H. and Preston, G.B., The Algebraic Theory of Semigroups Vol. I (Math. Surveys of the American Math. Soc. 7, Providence, R.I., 1961).Google Scholar
[3]Fennemore, C.F., ‘All varieties of bands I and II’, Math. Nachr 48 (1971), 237–252 and 253–262.CrossRefGoogle Scholar
[4]FitzGerald, D.G., ‘On inverses of products of idempotents in regular senhigroups’, J. Austral. Math. Soc. 13 (1972), 335–337.CrossRefGoogle Scholar
[5]Gerhard, J.A., ‘The lattice of equational classes of idempotent semigroups’, J. Algebra 15 (1970), 195–224.CrossRefGoogle Scholar
[6]Hall, T.E., ‘On regular semigroups’, J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[7]Hall, T.E., ‘Amalgamation for inverse and generalized inverse semigroups’, Trans. Amer. Math. Soc. (to appear).Google Scholar
[8]Hall, T.E. and Jones, P.R., ‘On the lattice of varieties of bands of groups’, Pacific J. Math. 91 (1980), 327–337.CrossRefGoogle Scholar
[9]Howie, J.M., An Introduction to Semigroup Theory, London Math. Soc. Monographs 7 (Academic Press, London, New York, 1976).Google Scholar
[10]Kadourek, Jiři and Szendrei, Maria B., ‘A new approach in the theory of orthodox semigroups’, (preprint).Google Scholar
[11]Lallement, G., ‘Structure theorems for regular semigroups’, Semigroup Forum 4 (1972), 95–123.CrossRefGoogle Scholar
[12]Petrich, Mario, Inverse Semigroups (John Wiley and Sons, 1984).Google Scholar
[13]Petrich, Mario and Reilly, Norman R., ‘The join of the varieties of strict inverse semigroups and rectangular bands’, Glasgow Math. J. 25 (1984), 59–74.CrossRefGoogle Scholar
[14]Reilly, N.R. and Scheiblich, H.E., ‘Congruences on regular semigroups’, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar