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Combinatorial results relating to products of idempotents in finite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

John M. Howie ,
Ewing L. Lusk  and
Robert B. McFadden
John M. Howie
Affiliation:
Department of Mathematics, University of St Andrews, St Andrews KY16 9SS, Scotland, U.K.
Ewing L. Lusk
Affiliation:
Argonne National Laboratory, Argonne, IL 60439, U.S.A.
Robert B. McFadden
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, U.S.A.
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Synopsis

Each singular element α of the full transformation semigroup on a finite set is generated by the idempotents of defect one. The length of the shortest expression of α as a product of such idempotents is given by the gravity function g(α).We use certain consequences of a result by Tatsuhiko Saito to explore connections between the defect and the gravity of α, and then determine the number of elements that have maximum gravity. Finally, we obtain formulae for the number of elements of small gravity. Such elements must have defect 1, and we determine their number within each ℋ-class. Many of the results obtained were suggested, and all have been verified, by programs written in PROLOG, a logic programming language very well suited for algebraic calculations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 289 - 299
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Cohen, Daniel I. A.. Basic Techniques of Combinatorial Theory (New York: John Wiley & Sons, 1978).Google Scholar
2Howie, John M..Products of idempotents in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A. 86 (1980), 243–254.CrossRefGoogle Scholar
3Howie, John M. and McFadden, Robert B.. Idempotent rank in finite full transformation semigroups (to be submitted).Google Scholar
4Howie, John M., Robertson, Edmund F. and Schein, Boris M..A combinatorial property of finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 319–328.CrossRefGoogle Scholar
5Iwahori, Nobuko.A length formula in a semigroup of mappings. J Fac. Sci. Univ. Tokyo, Sect.1A Math. 24 (1977), 255–260.Google Scholar
6Saito, Tatsuhiko. Products of idempotents in finite full transformation semigroups. Semigroup Forum (to appear).Google Scholar
7Tainiter, M..A characterization of idempotents in semigroups. J. Combin. Theory 5 (1968), 370–373.CrossRefGoogle Scholar