Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-26T15:07:01.691Z Has data issue: false hasContentIssue false
  • English
  • Français

On semigroups with involution

Published online by Cambridge University Press:  17 April 2009

D. Easdown  and
W.D. Munn
D. Easdown
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
W.D. Munn
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland United Kingdom
Rights & Permissions [Opens in a new window]

Extract

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 with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S,

.

It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 1 , August 1993 , pp. 93 - 100
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups 1 and 2 (Amer. Math. Soc., Providence, 1961 and 1967).Google Scholar
[2]Domanov, O.I., ‘The semisimplicity and identities of semigroup algebras in inverse semigroups’, (in Russian), Mat. Issled. (Rings and Modules) 38 (1976), 123–127.Google Scholar
[3]Munn, W.D., ‘Semiprimitivity of inverse semigroup algebras’, Proc. Roy. Soc. Edinburgh Ser. A 93 (1982), 83–98.CrossRefGoogle Scholar
[4]Okniński, J. and Putcha, M.S., ‘Complex representations of matrix semigroups’, Trans. Amer. Math. Soc. 323 (1991), 563–581.CrossRefGoogle Scholar
[5]Passman, D.S., Infinite group rings (Dekker, New York, 1971).Google Scholar
[6]Penrose, R., ‘A generalized inverse for matrices’, Proc. Cambridge Philos. Soc. 51 (1955), 406–413.CrossRefGoogle Scholar
[7]Shehadah, A. A., ‘Formal complexity of inverse semigroup rings’, Bull. Austral. Math. Soc. 41 (1990), 343–346.CrossRefGoogle Scholar