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Saturated and epimorphically closed varieties of semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

P. M. Higgins
P. M. Higgins
Affiliation:
Department of Mathematics Monash University Clayton, Victoria 3168, Australia
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Abstract

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We establish a necessary condition (E) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition (S) for a variety to consist entirely of saturated semigroups. Condition (S) is shown to be sufficient for heterotypical varieties and a stronger condition (S′) is shown to be sufficient for homotypical varieties.

Keywords

semigroupvarietyepimorphismdiminionsaturatedhomotypicalheterotypical

MSC classification

Secondary: 20M07: Varieties and pseudovarieties of semigroups 20M05: Free semigroups, generators and relations, word problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 153 - 175
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Bulaszewska, A. and Krempa, J., ‘On epimorphisms in the category of all associative rings’, Bull Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975).Google Scholar
[2]Burgess, W., ‘The meaning of mono and epi in some familiar categories’, Canad. Math. Bull. 8 (1965).CrossRefGoogle Scholar
[3]Chrislock, J. L., ‘A certain class of identities on semigroups’, Proc. Amer. Math. Soc. 21 (1969), 189–190.CrossRefGoogle Scholar
[4]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys No. 7 (Amer. Math. Soc., Providence, R.I., Vol. I, 1961; Vol II, 1967).Google Scholar
[5]Drbohlav, K., ‘A note on epimorphisms in algebraic categories’, Comment. Math. Univ. Carolinae 4 (1963).Google Scholar
[6]Evans, T., ‘An embedding theorem for semigroups’, Amer. J. Math. 76 (1954), 399–413.CrossRefGoogle Scholar
[7]Gardner, B. J., ‘Epimorphisms of regular rings’, Comment. Math. Univ. Carolinae 16 (1975).Google Scholar
[8]Gardner, B. J., ‘A note on ring epimorphisms and polynomial identities’, Comment. Math. Univ. Carolinae 20 (1979).Google Scholar
[9]Gardner, B. J., ‘Some aspects of T-nilpotence’, Pacific J. Math. 53, (1974).CrossRefGoogle Scholar
[10]Hall, T. E., ‘Inverse semigroup varieties with the amalgamation property’, Semigroup Forum 16 (1978), 37–51.CrossRefGoogle Scholar
[11]Hall, T. E. and Jones, P. R., ‘Epis are onto for finite regular semigroups’, Proc. Edinburgh Math. Soc., to appear.Google Scholar
[12]Higgins, P. M., ‘The determination of all absolutely closed varieties of semigroups’, Proc. Amer. Math. Soc., to appear.Google Scholar
[13]Higgins, P. M., ‘Epis are onto for generalised inverse semigroups’, Semigroup Forum 23 (1981), 255–259.CrossRefGoogle Scholar
[14]Higgins, P. M., ‘The commutative varieties of semigroups for which epis are onto’, Proc. Edinburgh Math. Soc., to appear.Google Scholar
[15]Higgins, P. M., ‘A semigroup with an epimorphically embedded subband’, submitted.Google Scholar
[16]Howie, J. M., An introduction to semigroup theory, London Math. Soc. Monographs 7 (Academic Press, 1976).Google Scholar
[17]Howie, J. M. and Isbell, J. R., ‘Epimorphisms and dominions II’, J. Algebra 6 (1967), 7–21.CrossRefGoogle Scholar
[18]Isbell, J. R., ‘Epimorphisms and dominions’, Proceedings of the Conference on Categorical Algebra, La Jolla, 1965 (Lange and Springer, Berlin, 1966, 232–246).Google Scholar
[19]Khan, N. M., ‘Epimorphisms, dominions and varieties of semigroups’, Semigroup Forum 25 (1982), 331–337.CrossRefGoogle Scholar
[20]Khan, N. M., ‘On saturated and epimorphically closed permutative varieties and consequences of permutation identities’, submitted.Google Scholar
[21]Munn, W. D., ‘Semigroups satisfying mininal conditions’, Proc. Glasgow Math. Assoc. 3 (1957), 145–152.CrossRefGoogle Scholar
[22]Tamura, T., ‘The theory of construction of finite semigroups III. Finite unipotent semigroups’, Osaka Math. J. 10 (1958), 191–204.Google Scholar