We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated implicit operators, subword complexity and the associated entropy.

‎The aim of this paper is to establish the equivalence between the concepts‎ ‎of an $S$-metric space and a cone $S$-metric space using some topological‎ ‎approaches‎. ‎We introduce a new notion of a $TVS$-cone $S$-metric space using‎ ‎some facts about topological vector spaces‎. ‎We see that the known results on‎ ‎cone $S$-metric spaces (or $N$-cone metric spaces) can be directly obtained‎ from...

it is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. we introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. we deal with vietoris and hausdorff continuity concepts.stability of microperiodi...

Let G be a non-finite profinite group and let G− Setsdf be the canonical site of finite discrete G-sets. Then the category R G, defined by Devinatz and Hopkins, is the category obtained by considering G − Setsdf together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R G is a site when equipped with the pretopology of epimorphic covers. Als...

Let G be a non-finite profinite group and let G− Setsdf be the canonical site of finite discrete G-sets. Then the category R G, defined by Devinatz and Hopkins, is the category obtained by considering G− Setsdf together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R G is a site when equipped with the pretopology of epimorphic covers. Also...

Surprising as it may be at first sight, there are a number of connections between the theories of finite semigroups and dynamical systems, both viewed in a broad sense. For instance in symbolic dynamics, ideas or analogies from the theory of finite automata find a natural setting for application in sofic systems [10, 24] and, even though not usually formulated in dynamical terms, the dynamical ...

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. resolve a serious deficit in existing theory by constructing good notion of presheaves, with suitable sheafification functor. Using sheafification, we develop general modules ring, give explicit constructions infinite products and introduce an analogue skyscraper sheaves. These results underlie re...

In their more general form, our equations are of the form u → v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman’s theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.)...

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...