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M. Shahryari

Publications and source records attributed to M. Shahryari.

At least 19 recordsLinked to original sources
arXiv2024

New classes of groups which are equational domains

A group is CSA, if all of its maximal abelian subgroups are malnormal. It is known that every non-abelian CSA group is an equational domain. We generalize this result in two directions: we show that for a non-nilpotent group $G$ and a fixed positive integer $k$, if all maximal elements in the set of class $k$ nilpotent subgroups of $G$ are malnormal, then $G$ is an equational domain. Also, we prove that if a group $G$ is not locally nilpotent and if every maximal locally nilpotent subgroup of $G$ is malnormal, then $G$ is an equational domain.

math.GR↗
arXiv2023

On conjugate separability of nilpotent subgroups

We study groups, all maximal nilpotent subgroups of class at most $k$ in which are malnormal. We show that such groups share many similar properties with the ordinary CSA groups. Similarly, we introduce the class of {\em nilpotency transitive} groups and we show that in presence of a special residuality condition, these two concepts are the equivalent. As a result, we see that the theory of CSA and CT groups is a small part of a more general idea.

math.GR↗
arXiv2023

The Haar measure of a profinite $n$-ary group

We prove that every profinite $n$-ary group $(G, f)=\Gf$ has a unique Haar measure $m_p$ and further for every measurable subset $A\subseteq G$, we have $$ m_p(A)=m(A)=(n-1)m^{\ast}(A) $$ where $m$ and $m^{\ast}$ are the normalized Haar measures of the profinite groups $(G, \bullet)$ and the Post cover $G^{\ast}$, respectively.

math.GR↗
arXiv2021

On profinite polyadic groups

We study the structure of profinite polyadic groups and we prove that a polyadic topological group $(G, f)$ is profinite, if and only if, it is compact, Hausdorff, totally disconnected. More generally, for a pseudo-variety (or a formation) of finite groups $\mathfrak{X}$, we define the class of $\mathfrak{X}$-polyadic groups, and we show that a polyadic group $(G, f)$ is pro-$\mathfrak{X}$, if and only if, it is compact, Hausdorff, totally disconnected and for every open congruence $R$, the quotient $(G/R, f_R)$ is $\mathfrak{X}$-polyadic.

math.GR↗
arXiv2020

A note on surjunctive groups

In this article, we prove that a semidirect product of a locally finite group with a surjunctive group is also surjunctive. We also prove that a surjunctive-by-locally finite group is again surjunctive.

math.GR↗
arXiv2019

A note on cellular automata

For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.

math.GR↗
arXiv2018

On the equationally Artinian groups

In this article, we study the property of being equationally Artinian in groups. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form $G[t]$ by a normal subgroup which is a finite union of radicals, is again equationally Artnian. This provides a large class of examples of equationally Artinian groups.

math.GR↗
arXiv2016

Comapactness Conditions in Universal Algebraic Geometry

In this article, the properties of being equational noetherian, $q_ω$ and $u_ω$-compactness, and equational Artinian are studied from the perspective of the Zariski topology. The equational conditions on the relative free algebras of arbitrary varieties are also investigated.

math.RA↗
arXiv2016

Dehn functions and the space of marked groups

In the space of marked group, we suppose that a sequence $(G_i, X_i)$ converges to $(G,X)$, where $G$ is finitely presented. We obtain an inequality which connects Dehn functions of $G_i$s and $G$. As a result, we show that if a sequence $(K, X_i)$ converges to a hyperbolic marked group, then $K$ is already hyperbolic.

math.GR↗
arXiv2016

Algebraic sets with fully characteristic radicals

We obtain a necessary and sufficient condition for an algebraic set in a group to have a fully characteristic radical. As a result, we see that if the radical of a system of equation $S$ over a group $G$ is fully characteristic, then there exists a class $\mathfrak{X}$ of subgroups of $G$ such that elements of $S$ are identities of $\mathfrak{X}$.

math.GR↗
arXiv2015

Equations in polyadic groups

Systems of equations and their solution sets are studied in polyadic groups. We prove that a polyadic group $(G, f)=\mathrm{der}_{θ, b}(G, \cdot)$ is equational noetherian, if and only if the ordinary group $(G, \cdot)$ is equational noetherian. The structure of coordinate polyadic group of algebraic sets in equational noetherian polyadic groups are also determined.

math.GR↗
arXiv2015

On the Equational Artinian Algebras

Equational Artinian algebras were introduced in our previous work: {\em Equational conditions in universal algebraic geometry, to appear in Algebra and Logic, 2015}. In this note, we define the notion of {\em radical topology with respect to an algebra $A$} and using the well-known König lemma in graph theory, we show that the algebra $A$ is equational Artinian iff this topology is noetherian. This completes the analogy between equational noetherian and equational Artinian algebras.

math.GR↗
arXiv2014

A combinatorial approach to rational exponential groups

We give a suitable definition of the concept of rational complex and prove that every rational exponential group is the fundamental group of some such a complex. In this framework, we prove that the variety of rational exponential groups is a Schreier variety.

math.GR↗
arXiv2014

Topo-Groups and a Tychonoff Type Theorem

In this article, we introduce an interesting topology-like concept concerning groups (and with almost the same method it can be defined for other algebraic systems). Given an arbitrary group $G$, we define a {\em topo-system} on $G$ as a set of subgroups satisfying certain conditions like a topology on a set. We will call such a group, a {\em topo-group}. These topo-groups are not rare and as we will see, there are many examples of topo-groups. We investigate fundamental notions concerning topo-groups and we see that many basic concepts of topology can be introduced in the frame of topo-groups. A {\em filter of subgroups} will be defined in such way that we will be able to formulate a Tychonoff type theorem for the direct product of {\em topo-compact} topo-groups.

math.GR↗
arXiv2014

On Logically Cyclic Groups

A group $G$ is called logically cyclic, if it contains an element $s$ such that every element of $G$ can be defined by a first order formula with parameter $s$. The aim of this paper is to investigate the structure of such groups.

math.GR↗