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

By Jean-Eric Pin and Stuart Margolis

Abstract

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. La conjecture de Rhodes, appelée à l'origine la conjecture de "Type II" par Rhodes, donne un algorithme pour calculer le noyau d'un semigroupe fini. Cette conjecture a des applications importantes et est l'un des problèmes les plus attractifs sur les semigroupes finis. On savait que la conjecture de Rhodes était conséquence d'une autre conjecture sur la topologie pro-groupe du monoïde libre. Dans cet article, on montre que la conjecture topologique et la conjecture de Rhodes sont en fait équivalentes à une troisième conjecture que nous prouvons dans plusieurs cas particuliers significatifs

Topics: Finite semigroups, MR 20M05 (20M07)
Publisher: Elsevier
Year: 1992
OAI identifier: oai:HAL:hal-00019888v1
Provided by: Hal-Diderot
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