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Finite semilattices whose non-invertible endomorphisms are products of idempotents

Published online by Cambridge University Press:  14 November 2011

M. E. Adams ,
Sydney Bulman-Fleming ,
Matthew Gould  and
Amy Wildsmith
M. E. Adams
Affiliation:
Department of Mathematics, State University of New York, New Paltz, NY 12561, U.S.A.
Sydney Bulman-Fleming
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
Matthew Gould
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A.
Amy Wildsmith
Affiliation:
Department of Mathematics, David Lipscomb University, Nashville, TN 37204, U.S.A.
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Abstract

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1193 - 1198
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Aîzenštat, A. Ya.. Defining relations of finite symmetric semigroups. Mat. Sb. N.S. 45(87) (1958), 261–280 (in Russian).Google Scholar
2Aîzenštat, A. Ya.. The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sibirsk. Mat. Zh. 3 (1962), 161–169 (in Russian).Google Scholar
3Dawlings, R. J. H.. Products of idempotents in the semigroup of singular endomorphisms of a finite dimensional vector space. Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 123–133.CrossRefGoogle Scholar
4Erdos, J. A.. On products of idempotent matrices. Glasgow Math. J. 8 (1967), 118–122.CrossRefGoogle Scholar
5Grätzer, G.. General Lattice Theory (Basel and Stuttgart: Birkhauser, 1978).CrossRefGoogle Scholar
6Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
7Howie, J. M.. Properties of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17(2) (1971), 223–236.CrossRefGoogle Scholar
8Howie, J. M. and Schein, B. M.. Products of idempotent order-preserving transformations. J. London Math. Soc. 7 (1973), 357–366.CrossRefGoogle Scholar
9Magill, K. D.. Semigroups of functions generated by idempotents. J. London Math. Soc. 44 (1969), 236–242.CrossRefGoogle Scholar
10Magill, K. D.. The semigroup of endomorphisms of a Boolean ring. J. Austral. Math. Soc. 11 (1970), 411–416.CrossRefGoogle Scholar
11O'Meara, K.. Products of idempotents in regular rings. Glasgow Math. J. 28 (1986), 143–152.CrossRefGoogle Scholar
12Reynolds, M. A. and Sullivan, R. P.. Products of idempotent linear transformations. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 123–138.CrossRefGoogle Scholar
13Thornton, M. C.. Semigroups of isotone selfmaps on partially ordered sets. J. London Math. Soc. 14 (1976), 545–553.CrossRefGoogle Scholar