Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T03:29:34.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice isomorphisms of orthodox semigroups

Published online by Cambridge University Press:  17 April 2009

Katherine G. Johnston  and
F.D. Cleary
Katherine G. Johnston
Affiliation:
Department of MathematicsRoyal Melbourne Institute of TechnologyP.O. Box 2476V Melbourne, Vic. 3001, Australia
F.D. Cleary
Affiliation:
Department of MathematicsRoyal Melbourne Institute of TechnologyP.O. Box 2476V Melbourne, Vic. 3001, Australia
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.

It is shown that the set of all orthodox subsemigroups of an orthodox semigroup forms a lattice. This lattice is a join-sublattice of the lattice of all semigroups, but is not in general a meet-sublattice. We obtain results concerning lattice isomorphisms between orthodox semigroups, several of which include known results for inverse semigroups as special cases.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 2 , April 1992 , pp. 201 - 213
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Ault, J.E., ‘Semigroups with bisimple and simple ω–subsemigroups’, Semigroup Forum 9 (1975), 318–333.CrossRefGoogle Scholar
[2]Clifford, A.H. and Preston, G.B., Algebraic theory of semigroups, Math. Surveys 7 (Amer. Math. Soc., Providence, R.I., Vol. I, 1961, Vol. II, 1967).CrossRefGoogle Scholar
[3]Crawley, P. and Dilworth, R.P., Algebraic theory of lattices (Prentice-Hall, Englewood Cliffs, N.J., 1973).Google Scholar
[4]Hall, T.E., ‘On the natural ordering of J-classes and of idempotents in a regular semigroup’, Glasgow Math. J. 11 (1970), 167–168.CrossRefGoogle Scholar
[5]Hall, T.E., ‘On regular semigroups’, J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[6]Hall, T.E., ‘Almost commutative bands’, Glasgow Math. J. 13 (1972), 176–178.CrossRefGoogle Scholar
[7]Howie, J.M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
[8]Johnston, K.G., ‘Lattice isomorphisms of modular inverse semigroups’, Proc. Edinburgh Math. Soc. 31 (1988), 441–446.CrossRefGoogle Scholar
[9]Johnston, K.G. and Jones, P.R., ‘The lattice of full regular subsemigroups of a regular semigroup’, Proc. Royal Soc. Edinburgh 98A (1984), 203–214.CrossRefGoogle Scholar
[10]Jones, P.R., ‘Distributive inverse semigroups’, J. London Math. Soc. 17 (1978), 457–466.CrossRefGoogle Scholar
[11]Jones, P.R., ‘Lattice isomorphisms of distributive inverse semigroups’, Quart. J. Math. Oxford Ser. 30 (1979), 301–314.CrossRefGoogle Scholar
[12]Jones, P.R., ‘Inverse semigroups determined by their lattices of inverse subsemigroups’, J. Austral. Math. Soc. (Ser. A) 30 (1981), 321–346.CrossRefGoogle Scholar
[13]Reilly, N.R. and Scheiblich, H.E., ‘Congruences on regular semigroups’, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar
[14]Shevrin, L.N., ‘Lattice properties of idempotent semigroups I’, Siberian Math. J. 6 (1965), 459–474. (in Russian).Google Scholar
[15]Shevrin, L.N. and Ovsyannikov, A.J., ‘Semigroups and their subsemigroup lattices’, Semigroup Forum 27 (1983), 1–154.CrossRefGoogle Scholar
[16]Suzuki, M., Structure of a group and the structure of its lattice of subgroups (Springer-Verlag, Berlin, Heidelberg, New York, 1956).CrossRefGoogle Scholar