It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the “strong Dubreil-Jacotin condition” forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name “epigroup” in the paper of Blyth and Almeida S...

In this paper we determine the automorphism groups of profinite braid with four or more strings in terms Grothendieck-Teichm\"uller group.

We generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms.

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projecti...

We associate the iterated block product of a bimachine with a deterministic Turing machine. This allows us to introduce new algebraic notions to study the behavior of the Turing machine. Namely, we introduce double semidirect products through matrix multiplication of upper triangular matrices with coefficients in certain semigroups, which leads in turn to the study of the iterations of bimachin...

For predual categories C andD we establish isomorphisms between opfibrations representing local varieties of languages in C, local pseudovarieties of D-monoids, and finitely generated profinite D-monoids. The global sections of these opfibrations are shown to correspond to varieties of languages in C, pseudovarieties of D-monoids, and profinite equational theories of D-monoids, respectively. As...

Before embarking on the proof, we first explain Theorem 1 in a bit more detail, in particular identifying the two categories that are relevant to our discussion. So let Gk denote the category whose objects are profinite group extensions of Gk by a pro-p group, and whose morphisms are outer open Gkhomomorphisms. Thus, an object of Gk is a profinite group G together with a continuous surjection π...

A group Γ is called boundedly generated (BG) if it is the settheoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations. We use this to reprove and generalise Rapinchuk’s theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension ...

The syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of A(Bn), the affine nearsemiring over a Brandt semigroup Bn. It is ascertained that both the semigroup reducts of A(Bn) are syntactic semigroups.

Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups.