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A Class of Irreducible matrix representations of an Arbitrary Inverse Semigroup

Published online by Cambridge University Press:  18 May 2009

W. D. Munn
W. D. Munn
Affiliation:
The University Glasgow
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By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 1 , January 1961 , pp. 41 - 48
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

1.Andersen, O., Ein Bericht über die Struktur abstrakter Halbgruppen, Thesis, Hamburg (1952).Google Scholar
2.Clifford, A. H., Semigroups admitting relative inverses, Ann of Math. (2), 42 (1941), 1037–1049.CrossRefGoogle Scholar
3.Clifford, A. H., Bands of semigroups, Proc. Amer. Math. Soc., 5 (1954), 499–504.CrossRefGoogle Scholar
4.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Mathematical Surveys of the American Mathematical Society (Providence, R. I., 1961).CrossRefGoogle Scholar
5.Croisot, R., Demi-groupes inversifs et demi-groupes réunions de demi-groupes simples, Ann. Sci. École Norm. Sup. (3), 70 (1953), 361–379.CrossRefGoogle Scholar
6.Green, J. A., On the structure of semigroups, Ann. of Math. (2), 54 (1951), 163–172.CrossRefGoogle Scholar
7.Liber, A. E., On the theory of generalised groups, Doklady Akad. Nauk SSSR(N.S.), 97 (1954), 25–28 (Russian).Google Scholar
8.Munn, W. D., On semigroup algebras, Proc. Cambridge Phil. Soc., 51 (1955), 1–15.CrossRefGoogle Scholar
9.Munn, W. D., Irreducible matrix representations of semigroups. To appear in Quart. J. Math. Oxford Ser. (2).Google Scholar
10.Munn, W. D. and Penrose, R., A note on inverse semigroups, Proc Cambridge Phil. Soc., 51 (1955), 396–399.CrossRefGoogle Scholar
11.Preston, G. B., Inverse semi-groups, J. London Math. Soc., 29 (1954), 396–403.CrossRefGoogle Scholar
12.Preston, G. B., Representations of inverse semi-groups, J. London Math. Soc., 29 (1954), 411–419CrossRefGoogle Scholar
13.Rees, D., On the group of a set of partial transformations, J. London Math. Soc., 22 (1947), 281–284.CrossRefGoogle Scholar
14.Stoll, R. R., Homomorphisms of a semigroup onto a group, Amer. J. Math., 73 (1951), 475–481.CrossRefGoogle Scholar
15.Vagner, V. V., Generalised groups, Doklady Akad. Nauk SSSR (N.S.), 84 (1952), 1119–1122.Google Scholar