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

Abstract

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 or 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 K-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. © 1990, Royal Society of Edinburgh. All rights reserved.