Abstract
In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.
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References
Aizenštat, A. Ya: Defining relations of finite symmetric semigroups. Mat. Sb. N.S.45 (87), 261–280 (1958). (Russian).
Aizenštat, A. Ya: The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sibirsk. Mat. Z.3, 161–169 (1962). (Russian).
Bulman-Fleming, S.: Regularity and products of idempotents in endomorphism monoids of projective acts. Mathematika42, 354–367 (1995).
Dawlings, R. J. H.: Products of idempotents in the semigroup of singular endomorphisms of a finite-dimensional vector space. Proc. Royal Soc. Edinburgh91A, 123–133 (1981).
Erdos, J. A.: On products of idempotent matrices. Glasgow Math. J.8, 118–122 (1967).
Fleischer, V. G.: On the wreath product of monoids with categories. ENSV TA (Proc. Acad. Sci. Estonian SSR, Physics, Mathematics)35, 237–243 (1986). (Russian).
Fleischer, V., Knauer, U.: Endomorphism monoids of acts are wreath products of monoids with small categories. In: H. Jürgensen, G. Lallement, H. J. Weinert (eds.) Semigroups, Theory and Application, Oberwolfach, 1986. Lect. Notes Math.1320, pp. 84–86. New York-Heidelberg-Berlin: Springer. 1988.
Fountain, J.: Products of integer matrices. Math. Proc. Camb. Phil. Soc.110, 431–441 (1991).
Fountain, J., Lewin, A.: Products of idempotent endomorphisms of an independence algebra of finite rank. Proc. Edinburgh Math. Soc.35, 493–500 (1992).
Fountain, J., Lewin, A.: Products of idempotent endomorphisms of an independence algebra of infinite rank. Math. Proc. Camb. Phil. Soc.114, 303–310 (1993).
Howie, J. M.: The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc.41, 707–716 (1966).
Howie, J. M.: Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc.17 (Series II), 223–236 (1971).
Howie, J. M.: Some subsemigroups of inifinite full transformation semigroups. Proc. Royal Soc. Edinburgh88A, 159–167 (1981).
Laffey, T. J.: Products of idempotent matrices. Linear and Multilinear Algebra14, 309–314 (1983).
Reynolds, M. A., Sullivan, R. P.: Products of idempotent linear transformations. Proc. Royal Soc. Edinburgh100A, 123–138 (1985).
Sierpinski, W.: Cardinal and ordinal numbers. PWN-Polish Scientific Publishers, Warsaw, 1965.
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Research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494
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Bulman-Fleming, S., Fountain, J. Products of idempotent endomorphisms of free acts of infinite rank. Monatshefte für Mathematik 124, 1–16 (1997). https://doi.org/10.1007/BF01320734
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DOI: https://doi.org/10.1007/BF01320734