
One of the milestones in the theory of semigroups and automata is the KrohnRhodes Theorem. It states that every finite semigroup S divides a wreath product of finite simple groups, each of them divisor of S, and finite aperiodic semigroups, i. e. semigroups with trivial maximal subgroups.
The smallest number of groups in any KohnRhodes decomposition is called the group complexity of the semigroup. Since there is no obvious way to compute the complexity of a finite semigroup in general, the decidability of this number is one of the most important open problems in finite semigroup theory and the search for the solution has led to the development of many tools and ideas that are useful in finite semigroup theory and of independent interest.
At this point, the notion of the kernel of a semigroup came on the scene. It was introduced by Rhodes and Tilson in their seminal paper of 1972 as the set of elements related to the indentity under every relational morphism between the semigroup and a group, and its computability would have important consequences in the solution of the complexity problem. In fact, one of the main results in Rhodes and Tilsons's paper is a description of the regular elements of the kernel of a semigroup. That result led to the Rhodes' conjecture, originally called the Type II Conjecture: the kernel of a semigroup is the smallest subsemigroup containing the idempotents and closed under weak conjugation. This conjecture attracted the attention of many semigroup theorists during about two decades before being solved. It was proved independently by Ash, and Ribes and Zalesskii using diferents methods.
Like many problems with a no immediate solution, once they are solved, not only a great number of consequences spring from their solution, but also new questions related with them can be set out. A first natural step is to extend the definition of kernel of a semigroup to an arbitrary variety of finite groups F: the Fkernel of a finite semigroup S is the subsemigroup of S consisting of all elements of S such that relate to the identity under every relational morphism of S with a group in F. The generalised Rhode's Type II Conjecture asks if the Fkernel associated with a given variety of finite groups F is computable for every semigroup.
Since generalised Rhodes' Type II Conjecture is completely general, it is natural to hope that some argument might exist that would prove it, but up to now no one seems to have an inkling of how such a proof might proceed. Failing that, one could try to reduce the problem to a question about some restricted class of finite semigroups. Indeed, in the important case where F is extension closed, a theorem of Ribes and Zalesskii allows us to conclude that the Fkernel is computable if its regular elements are computable.
The main result of this thesis is meant to provide a decisive step towards verifying the generalised Rhodes' Type II Conjecture for extension closed varieties.
