arXiv subjects

Elton Pasku

Publications and source records attributed to Elton Pasku.

8 recordsLinked to original sources

arXiv2025

A groupoid approach to the study of fuzzy topological spaces

The definition of the complement of a fuzzy subset is algebraic in nature and when it is used in the context of fuzzy topological spaces it does not share any similarity with the usual property of topological spaces that the complement of an open subset is closed. To tackle this inconsistency, we associate to any fuzzy topological space a topological space and use its fundamental groupoid equipped with the Lasso topology to give a topological characterization for the complementation of fuzzy subsets.

math.GN↗

arXiv2023

Fuzzy semigroups via semigroups

The theory of fuzzy semigroups is a branch of mathematics that arose in early 90's as an effort to characterize properties of semigroups by the properties of their fuzzy subsystems which include, fuzzy subsemigroups and their alike, fuzzy one (resp. two) sided ideals, fuzzy quasi-ideals, fuzzy bi-ideals etc. To be more precise, a fuzzy subsemigroup of a given semigroup $(S,\cdot)$ is just a $\wedge$-prehomomorphism $f$ of $(S,\cdot)$ to $([0,1],\wedge)$. Variations of this, which correspond to the other before mentioned fuzzy subsystems, can be obtained by imposing certain properties to $f$. It turns out from the work of Kuroki, Mordeson, Malik and that of many of their descendants, that fuzzy subsystems play a similar role to the structure theory of semigroups that play their non fuzzy analogues. The aim of the present paper is to show that this similarity is not coincidental. As a first step to this, we prove that there is a 1-1 correspondence between fuzzy subsemigroups of $S$ and subsemigroups of a certain type of $S\times I$. Restricted to fuzzy one sided ideals, this correspondence identifies the above fuzzy subsystems to their analogues of $S\times I$. Using these identifications, we prove that the characterization of the regularity of semigroups in terms of fuzzy one sided ideals and fuzzy quasi-ideals can be obtained as an implication of the corresponding non fuzzy analogue.

math.GM↗

arXiv2021

An answer to the Whitehead asphericity question

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. We give a positive answer to this question by proving that if $\cP=(\mathbf{x}, \mathbf{r})$ is an aspherical presentation of the trivial group, and $r_{0} \in \mathbf{r}$ a fixed relation, then $\cP_{1}=(\mathbf{x}, \mathbf{r}_{1})$ is aspherical where $\mathbf{r}_{1}=\mathbf{r} \setminus \{r_{0}\}$.

math.AT↗

arXiv2020

On a connection between fuzzy subgroups and $F$-inverse covers of inverse monoids

We define two categories, the category $\mathfrak{F}\mathfrak{G}$ of fuzzy subgroups, and the category $\mathfrak{F}\mathfrak{C}$ of $F$-inverse covers of inverse monoids, and prove that $\mathfrak{F}\mathfrak{G}$ fully embeds into $\mathfrak{F}\mathfrak{C}$. This shows that, at least from a categorical viewpoint, fuzzy subgroups belong to the standard mathematics as much as they do to the fuzzy one.

math.GM↗

arXiv2019

A semigroup theoretic approach to Whitehead's asphericity question

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. This is a long standing open problem which has attracted a lot of attention. Related to it, throughout the years there have been given several useful characterizations of asphericity which are either combinatorial or topological in nature. The aim of this paper is two fold. First, it brings in methods from semigroup theory to give a new combinatorial characterization of asphericity in terms of what we define here to be the weak dominion of a submonoid of a monoid, and uses this to give a sufficient and necessary condition under which a subpresentation of an aspherical group presentation is aspherical.

math.AT↗

arXiv2015

A note on the relation $\mathcal{J}$ in $le$-semigroups

We prove that if $S$ is a $le$-semigroup in which left ideal elements commute (condition which is called $\mathbfΛ$), then any $\mathcal{J}$-class satisfying the Green condition is a subsemigroup of $S$. As a corollary of this we show that semisimple $le$-semigroups satisfying $\mathbfΛ$ are precisely those that decompose as a semilattice of left simple $\vee e$-semigroups which are in addition semisimple, intra-regular and satisfy $\mathbfΛ$.

math.RA↗

arXiv2014

The Eilenberg-Mac Lane cohomology of an inverse monoid and the maximum group image

The aim of this paper is to see at what extent homological properties of an inverse monoid are determined from those of its maximum group image. We provide several evidences that the maximum group image contains vital homological information which can be used to study certain properties of the monoid itself. For instance, we prove that an inverse monoid $S$ is of type $FP_{\infty}$, if and only if it contains a minimal idempotent and its maximum group image is of the same type. Regarding cohomological dimensions, we show that the cohomological dimension of a free Clifford monoid and that of its maximum group image agree and are equal to one. Also we define the index of a full submonoid of an inverse monoid in terms of their maximum group images and show that if the index is finite then, the monoid is of type $\text{FP}_{\infty}$ if and only if its submonoid is of the same type.

math.GR↗

arXiv2013

The universal semigroup of a $Γ$-semigroup

Given a $Γ$-semigroup $S$, we construct a semigroup $Σ$ in such a way that one sided ideals and quasi-ideals of $S$ can be regarded as one sided ideals and quasi-ideals respectively of $Σ$. This correspondence and other properties of $Σ$, allow us to obtain several results for $S$ without having the need to work directly with it, but solely employing well known results of semigroup theory. For example, we obtain the Green's theorem for $Γ$-semigroups found in \cite{PT}, as a corollary of the usual Green's theorem in semigroups. Also we prove that, if $S$ is a $Γ$-semigroup and $γ_{0} \in Γ$ such that $S_{γ_{0}}$ is a completely simple semigroup, then for every $γ\in Γ$, $S_γ$ is completely simple too.

math.GR↗