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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
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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