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Congruences contained in an equivalence on a semigroup

Part of: Algebraic structures Semigroups

Published online by Cambridge University Press:  09 April 2009

P. R. Jones
P. R. Jones
Affiliation:
Mathematics Department University of Western AustraliaNedlands, Western Australia 6009, Australia
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Abstract

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The main theorem of this paper shows that the lattice of congruences contained is some equivalence π on a semigroup S can be decomposed into a subdirect product of sublattices of the congruence lattices on the ‘prinipal π-facotrsρ of S—the semigroups formed by adjoining zeroes to the π-classes—whenever these are well-defined. The theorem is then applied to various equavalences and classes of semigroups to give some new results and alternative proofs of known ones.

MSC classification

Secondary: 20M10: General structure theory 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 2 , March 1980 , pp. 162 - 176
Copyright
Copyright © Australian Mathematical Society 1980

References

Baird, G. R. (1972), ‘On the lattice of congruences on a band’, J. Aust. Math. Soc. 14, 49–58.Google Scholar
Birkhoff, G. (1967), Lattice theory, 3rd ed. (Amer. Math. Soc., Providence, R.I).Google Scholar
Clifford, A. H. and Preston, G. B. (1961), Algebraic theory of semigroups, Vol. I, Math. Surveys 7 (Amer. Math. Soc., Providence, R.I.).Google Scholar
Clifford, A. H. and Preston, G. B. (1967), Algebraic theory of semigroups, Vol. II, Math. Surveys 7 (Amer. Math. Soc., Providence, R.I.).CrossRefGoogle Scholar
Crawley, P. and Dilworth, R. P. (1973), Algebraic theory of lattices (Prentice-Hall, New Jersey).Google Scholar
Dean, R. A. and Oehmke, R. H. (1964), ‘Idempotent semigroups with distributive right congruence lattices’, Pacific J. Math. 14, 1187–1209.Google Scholar
Eberhart, C. and Williams, W. (1978). ‘Semimodularity in lattices of congruences’, J. Algebra 52, 73–87.Google Scholar
Hall, T. E. (1972), ‘Congruences and Green's relations on regular semigroups’, Glasgow Math. J. 13, 167–175.Google Scholar
Howie, J. M. (1976), An introduction to semi group theory (Academic Press, London).Google Scholar
Howie, J. M. and Lallement, G. (1966), ‘Certain fundamental congruences on a regular semigroup’, Proc. Glascow Math. Assoc. 7, 145–159.Google Scholar
Johnston, K. G. (1978), ‘Congruence lattices on Rees matrix semigroups’, Semigroup Forum 15, 247–262.Google Scholar
Jones, P. R. (1978a), ‘Semimodular inverse semigroups’, J. London Math. Soc. 17, 446–456.Google Scholar
Jones, P. R. (1978b), ‘Distributive inverse semigroups’, J. London Math. Soc. 17, 457–466.Google Scholar
Lallement, G. (1966), ‘Congruences et équivalences de Green sur un demi-groupe regulier’, C. R. Acad. Sci. Paris, Sér. A 262, 613–616.Google Scholar
Lallement, G. (1967), ‘Demi-groupes réguliers’, Ann. Mat. Pura Appl. 77, 47–129.Google Scholar
Maeda, F. and Maeda, S. (1970), Theory of symmetric lattices (Springer, Berlin).Google Scholar
Meakin, J. (1975), ‘One-sided congruences on inverse semigroups,’ Trans. Amer. Math. Soc. 206, 67–82.Google Scholar
Munn, W. D. (1964), ‘A certain sublattice of the lattice of congruences on a regular semigroup’, Math. Proc. Cambridge Philos. Soc. 60, 385–391.Google Scholar
Petrich, M. (1973), Introduction to semi groups (Merrill Publishing Co., Columbus, Ohio).Google Scholar
Scheiblich, H. E. (1970), ‘Semimodularity and bisimple w-semigroups, Proc. Edinburgh Math. Soc. 17, 79–81.Google Scholar
Spitznagel, C. (1973a), ‘The lattice of congruences on a band of groups’ Glasgow math. J. 14, 189–197.Google Scholar
Spitznagel, C. (1973b), ‘θmodular bands of groups’, Trans. Amer. Math. Soc. 177, 469–482.Google Scholar