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M. H. Shahzamanian

Publications and source records attributed to M. H. Shahzamanian.

18 recordsLinked to original sources
arXiv2026

Homogeneous Weights on Semigroup Algebras of Finite Commutative Semigroups

The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero--nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.

math.RT↗
arXiv2026

Simplicity of Augmentation Submodules in Monoids with 0-Minimal Ideals of Rank Greater than Two

In this paper, we construct explicit families of transformation monoids whose augmentation submodules are simple and whose associated 0-minimal J-classes have rank greater than two. These examples provide new monoids with simple augmentation submodules and non-complete associated graphs. We also establish a connection between the sandwich matrix of a 0-minimal J-class of rank two and the simplicity of the corresponding augmentation module, yielding a criterion that determines simplicity directly from the rank of this matrix for this class of monoids.

math.RT↗
arXiv2025

The determinant of \(\lll\)-smooth semigroups

This paper continues the investigation of non-zero determinants associated with finite semigroups containing a pair of non-commuting idempotents, as initiated in~\cite{Sha-Det2}. We focus on a class of semigroups, called \( \lll \)-smooth semigroups, that allow meaningful structural analysis despite the absence of \( \ll \)-transitivity. Within this framework, we develop a method for computing contracted semigroup determinants, building on and extending the approach carried over from~\cite{Sha-Det2}. These computations are motivated by applications in coding theory, particularly by the potential extending the MacWilliams theorem for codes over semigroup algebras.

math.GR↗
arXiv2025

The Layered Catalan Monoids: Structure and Determinants

In this paper, we introduce and study a class of monoids, called Layered Catalan Monoids (\( {LC}_n \)), which satisfy the structural conditions for $\ll$-smoothness as defined in~\cite{Sha-Det2}. These monoids are defined by specific identities inspired by Catalan monoids. We establish their canonical forms and compute their determinant, proving that it is non-zero for \(1 \leq n \leq 7\) but vanishes for \(n \geq 8\).

math.GR↗
arXiv2025

A step to compute the determinant of finite semigroups not in ECom

The purpose of this paper is to begin studying the computation of the nonzero determinant of semigroups within the class of finite semigroups that possesses a pair of non-commutative idempotents. This paper focuses on a class of these semigroups introduced as $\ll$-smooth semigroups. This computation is applicable in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

math.GR↗
arXiv2024

The determinant of finite semigroups of the pseudovariety ECOM

The purpose of this paper is to compute the non-zero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

math.GR↗
arXiv2021

Identities of the Jones Monoid $\mathcal{J}_5$

Jones monoids $\mathcal{J}_n$, for $1< n$, is a family of monoids relevant in knot theory. The purpose of this paper is to characterize of the identities satisfied by the Jones monoid $\mathcal{J}_5$.

math.GR↗
arXiv2021

The quiver of an affine monoid

If $R$ is a finite commutative ring, then the affine monoid of $R$ is the monoid of all affine mappings $x\mapsto ax+b$ on $R$. Alternatively, it is the semidirect product of the multiplicative monoid of $R$ with the additive group of $R$. In this paper we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.

math.RT↗
arXiv2020

The rank of variants of nilpotent pseudovarieties

We investigate the rank of pseudovarieties defined by several of the variants of nilpotency conditions for semigroups in the sense of Mal'cev. For several of them, we provide finite bases of pseudoidentities. We also show that the Neumann-Taylor variant does not have finite rank.

math.GR↗
arXiv2018

Simplicity of augmentation submodules for transformation monoids

For finite permutation groups, simplicity of the augmentation submodule is equivalent to $2$-transitivity over the field of complex numbers. We note that this is not the case for transformation monoids. We characterize the finite transformation monoids whose augmentation submodules are simple for a field $\mathbb{F}$ (assuming the answer is known for groups, which is the case for $\mathbb C$, $\mathbb R$, and $\mathbb Q$) and provide many interesting and natural examples such as endomorphism monoids of connected simplicial complexes, posets, and graphs (the latter with simplicial mappings).

math.CO↗
arXiv2017

Nilpotency and strong nilpotency for finite semigroups

Nilpotent semigroups in the sense of Mal'cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, $\mathsf{MN}$, which has finite rank. The semigroup identities that define nilpotent semigroups, lead us to define strongly Mal'cev nilpotent semigroups. Finite strongly Mal'cev nilpotent semigroups constitute a non-finite rank pseudovariety, $\mathsf{SMN}$. The pseudovariety $\mathsf{SMN}$ is strictly contained in the pseudovariety $\mathsf{MN}$ but all finite nilpotent groups are in $\mathsf{SMN}$. We show that the pseudovariety $\mathsf{MN}$ is the intersection of the pseudovariety $\mathsf{BG_{nil}}$ with a pseudovariety defined by a $κ$-identity. We further compare the pseudovarieties $\mathsf{MN}$ and $\mathsf{SMN}$ with the Mal'cev product of the pseudovarieties $\mathsf{J}$ and $\mathsf{G_{nil}}$.

math.GR↗
arXiv2016

The pro-nilpotent group topology on a free group

In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.

math.GR↗
arXiv2016

The congruence $η^{\ast}$ on semigroups

In this paper we define a congruence $η^{\ast}$ on semigroups. For the finite semigroups $S$, $η^{\ast}$ is the smallest congruence relation such that $S/η^{\ast}$ is a nilpotent semigroup (in the sense of Malcev). In order to study the congruence relation $η^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/η^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/η^{\ast}$ is a Clifford semigroup. We show that, for every non-null principal factor $A/B$ of $S$, there is a special principal factor $C/D$ such that every element of $A\setminus B$ is $η^{\ast}$-equivalent with some element of $C\setminus D$. We call the principal factor $C/D$, the $η^{\ast}$-root of $A/B$. All $η^{\ast}$-roots are $\textbf{CS}$-diagonal. If certain elements of $S$ act in the special way on the $\textbf{R}$-classes of a $\textbf{CS}$-diagonal principal factor then it is not an $η^{\ast}$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

math.GR↗
arXiv2012

A description of a class of finite semigroups that are near to being Malcev nilpotent

In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is \B\ if and only if it is nilpotent. Our main result is a description of \B\ finite semigroups $S$ in terms of their associated graph ${\mathcal N}_{S}$. In particular, $S$ has a largest nilpotent ideal, say $K$, and $S/K$ is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.

math.GR↗
arXiv2012

The Non-Nilpotent Graph of a Semigroup

We associate a graph ${\mathcal N}_{S}$ with a semigroup $S$ (called the upper non-nilpotent graph of $S$). The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case $S$ is a group this graph has been introduced by A. Abdollahi and M. Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of $S$ determined by the associated graph. It is shown that if a finite semigroup $S$ has empty upper non-nilpotent graph then $S$ is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected ${\mathcal N}_{S}$-components of a semigroup $S$ are complete (with at least two elements) then $S$ is a band that is a semilattice of its connected components and, moreover, $S$ is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle $C_{n}$ on $n$ vertices (with $n\geq 5$), are not the upper non-nilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okniński, Riley and the first named author on (Malcev) nilpotent semigroups.

math.GR↗
arXiv2012

Roughness in Cayley Graphs

In this paper, rough approximations of Cayley graphs are studied and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition called pseudo-Cayley graphs containing Cayley graphs is proposed. Rough approximation is expanded to pseudo-Cayley graphs. Also, rough vertex pseudo-Cayley graphs and rough pseudo-Cayley graphs are introduced. Some theorems are provided, form which some properties such as connectivity and optimal connectivity are derived. This approach opens a new research field in sciences such as data networks.

math.GR↗