New results on the conjecture of Rhodes and on the topological conjecture

Margolis, S.W. and J.E. Pin, New results on the conjecture of Rhodes and on the topological conjecture, Journal of Pure and Applied Algebra 80 (1992) 305-313. The Conjecture of Rhodes, originally called the ‘type II conjecture’ by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases. 1. The conjecture of Rhodes and the topological conjecture In this paper, all semigroups (respectively monoids, groups) are finite except in the case of free monoids or free groups. If M is a monoid, E(M) (respectively Reg(M)) denotes the set of idempotents (respectively regular elements) of M. If x E M, x”’ denotes the unique idempotent of the subsemigroup of M generated by X. A block-group monoid is a monoid in which every Z-class and every Z-class contain at most one idempotent. A number of equivalent conditions are given in [7]. For instance, a monoid M is a block-group monoid if and only if, for every regular $-class D of S, the semigroup D” is a Brandt semigroup, or if and only if 0022-4049/92/$05.00