Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T20:49:10.338Z Has data issue: false hasContentIssue false
  • English
  • Français

On categorical semigroups

Published online by Cambridge University Press:  17 April 2009

R.A.R. Monzo
R.A.R. Monzo
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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.

A semigroup S is called categorical if every ideal I of S has the property that abc є I (a, b, c є S) implies ab є I or be bc є I Necessary and sufficient conditions for an orthodox semigroup to be categorical are found and then used to characterize those bands which appear as an isomorphic copy of the band of idempotents of some orthodox categorical semigroup and to simplify the proof of a theorem of Mario Petrich. The structure of commutative categorical semigroups is found modulo the structure of abelian groups and categorical semilattices.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 61 - 68
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Baird, G.R., “On semigroups and uniform partial bands”, Semigroup Forum 4 (1972), 185–188.CrossRefGoogle Scholar
[2]Clifford, A.H., “Semigroups admitting relative inverses”, Ann. of Math. 42 (1941), 1037–1049.Google Scholar
[3]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Vol. 1 (Math. Surveys 7 (I), Amer. Math. Soc., Providence, Rhode Island, 1961).Google Scholar
[4]Hall, T.E., “On orthodox semigroups and uniform and antiuniform bands”, J. Algebra 16 (1970), 204–217.CrossRefGoogle Scholar
[5]McMorris, F.R. and Satyanarayana, M., “Categorical semigroups”, Proc. Amer. Math. Soc. 33 (1972), 271–277.CrossRefGoogle Scholar
[6]Monzo, R.A.R., “Categorical semigroups”, Semigroup Forum 6 (1973), 59–68.CrossRefGoogle Scholar
[7]Petrich, Mario, “On a class of completely semisimple inverse semigroups”, Proc. Amer. Math. Soc. 24 (1970), 671–676.CrossRefGoogle Scholar
[8]Petrich, Mario, “Regular semigroups satisfying certain conditions on idempotents and ideals”, Trans. Amer. Math. Soc. 170 (1972), 245–268.CrossRefGoogle Scholar