Solvable Monoids with Commuting Idempotents

The notion of Abelian kernel of a finite monoid extends the notion of derived subgroup of a finite group. In this line, an extension of the notion of solvable group to monoids is quite natural: they are the monoids such that the chain of Abelian kernels ends with the submonoid generated by the idempotents. We prove in this paper that the finite idempotent commuting monoids satisfying this property are precisely those whose subgroups are solvable. Introduction The computability of the kernel KH(M) of a finite monoid M relative to a pseudovariety H of groups is closely related to the decidability of Mal’cev products where the second factor is a pseudovariety of groups. In fact, for a decidable pseudovariety V of monoids and a pseudovariety H of groups, being able to compute KH(M), for any finite monoid M , automatically guarantees the decidability of the pseudovariety of monoids V ©m H = {M finite monoid | KH(M) ∈ V} which is well-known to be the Mal’cev product of V and H. As the Mal’cev product of pseudovarieties of monoids interests many researchers, the importance of computing kernels is out of question. The popularity of the kernel notion comes from a conjecture of J. Rhodes that proposed an algorithm to perform the computation of the kernel relative to the pseudovariety of all finite groups. It is known as the Rhodes Type II Conjecture and survived as a conjecture almost 20 years. It became a theorem after independent and deep work of Ash [7] and Ribes and Zalesskĭı [22]. The history and some consequences of the Type II Conjecture may be found in [15]. The results of Ash and of Ribes and Zalesskĭı that led to its proof have since then been extended in various directions [6, 16, 3, 4] and several connections between both results have also been found [11, 16, 3, 4]. The Abelian counterpart of the Rhodes Type II Conjecture was solved by the first author in [9] and the algorithm there obtained to compute the Abelian kernel of a finite The author gratefully acknowledges support of FCT through the Centro de Matemática da Universidade do Porto and the FCT and POCTI Project POCTI/32817/MAT/2000 which is funded in cooperation with the European Community Fund FEDER and of project INTAS 99-1224. The author gratefully acknowledges support of FCT and FEDER, within the project “Álgebra e Aplicações”, POCTI/32440/MAT/1999, of Centro de Álgebra da Universidade de Lisboa.