Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-17T23:30:58.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularity and products of idempotents in endomorphism monoids of projective acts

Part of: Semigroups

Published online by Cambridge University Press:  26 February 2010

Sydney Bulman-Fleming
Sydney Bulman-Fleming
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada.
Get access

Extract

That the monoid of all transformations of any set and the monoid of all endomorphisms of any vector space over a division ring are regular (in the sense of von Neumann) has been known for many years (see [6] and [16], respectively). A common generalization of these results to the endomorphism monoid of an independence algebra can be found in [13]. It also follows from [13] that the endomorphism monoid of a free G-act is regular, where G is any group. In the present paper we use a version of the wreath product construction of [8], [9] to determine the projective right S-acts (S any monoid) whose endomorphism monoid is regular.

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.

Information

Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 354 - 367
Copyright
Copyright © University College London 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Aizenstat, A. Ya.. Defining relations of finite symmetric semigroups. Mat. Sb. N.S., 45 (87) (1958), 261–280. (Russian)Google Scholar
2.Aizenstat, A. Ya. The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sibirsk. Mat. Z., 3 (1962), 161–169. (Russian)Google Scholar
3.Anderson, F. W. and Fuller, K. R.. Rings and Categories of Modules (Springer-Verlag, New York-Heidelberg-Berlin, 1974).CrossRefGoogle Scholar
4.Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups, Vol II (Amer. Math. Soc, Providence, R. I., 1967).Google Scholar
5.Dawlings, R. J. H.. Products of idempotents in the semigroup of singular endomorphisms of a finite-dimensional vector space. Proc. Royal Soc. Edinburgh, 91A (1981), 123–133.CrossRefGoogle Scholar
6.Doss, C. G.. Certain equivalence relations in transformation semigroups. M. A. Thesis, directed by D. D. Miller (University of Tennessee, 1955).Google Scholar
7.Erdös, J. A.. On products of idempotent matrices. Glasgow Math. J., 8 (1967), 118–122.CrossRefGoogle Scholar
8.Fleischer, V. G.. On the wreath product of monoids with categories. ENSV TA (Proc. Acad. Sci. Estonian SSR, Physics, Mathematics), 35 (1986), 237–243. (Russian)Google Scholar
9.Fleischer, V. and Knauer, U.. Endomorphism monoids of acts are wreath products of monoids with small categories. In: Semigroups, Theory and Application, Oberwohlfach, 1986. Lecture Notes in Math., vol 1320, pp. 84–86 edited by Jürgensen, H., Lallement, G. and Weinert, H. J. (Springer-Verlag, New York-Heidelberg-Berlin, 1988).CrossRefGoogle Scholar
10.Fountain, J.. Products of integer matrices. Math. Proc. Camb. Phil. Soc, 110 (1991), 431–441.CrossRefGoogle Scholar
11.Fountain, J. and Lewin, A.. Products of idempotent endomorphisms of an independence algebra of finite rank. Proc. Edinburgh Math. Soc, 35 (1992), 493–500.CrossRefGoogle Scholar
12.Fountain, J. and Lewin, A.. Products of idempotent endomorphisms of an independence algebra of infinite rank. Math. Proc. Camb. Phil. Soc, 114 (1993), 303–319.CrossRefGoogle Scholar
13.Gould, V. A. R.. Independence algebras. Algebra Universalis, 33 (1995), 294–318.CrossRefGoogle Scholar
14.Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc, 41 (1966), 707–716.CrossRefGoogle Scholar
15.Howie, J. M.. Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc, 17 (Series II) (1971), 223–236.CrossRefGoogle Scholar
16.Johnson, R. E. and Kiokemeister, F.. The endomorphisms of the total operator domain of an infinite module. Trans. Amer. Math. Soc, 62 (1947), 404–430.CrossRefGoogle Scholar
17.Knauer, U.. Projectivity of acts and Morita equavalence of monoids. Semigroup Forum 3 (1972), 359–370.CrossRefGoogle Scholar
18.Knauer, U. and Normak, P.. Hereditary endomorphism monoids of projective acts. Manuscripta Math., 70 (1991), 133–143.CrossRefGoogle Scholar
19.Laffey, T. J.. Products of idempotent matrices. Linear and Multilinear Algebra, 14 (1983), 309–314.CrossRefGoogle Scholar
20.Reynolds, M. A. and Sullivan, R. P.. Products of idempotent linear transformations. Proc. Royal Soc Edinburgh, 100A (1985), 123–138.CrossRefGoogle Scholar
21.Skornjakov, L. A.. Regularity of the wreath product of monoids. Semigroup Forum, 18 (1979), 83–86.CrossRefGoogle Scholar