arXiv subjects

Ali Madanshekaf

Publications and source records attributed to Ali Madanshekaf.

11 recordsLinked to original sources

arXiv2025

Strongly Hopfian and Co-Hopfian Acts over Monoids: Structure and Characterizations

In this paper, we introduce and explore new classes of S-acts over a monoid S, namely, strongly Hopfian and strongly co-Hopfian acts, as well as their weaker counterparts, Hopfian and co-Hopfian acts. We investigate the relationships between these newly defined structures and well-studied classes of S-acts, including Noetherian, Artinian, injective, projective, quasi-injective, and quasi-projective acts. A key result shows that, under certain conditions, a quasi-projective (respectively quasi-injective) S-act that is strongly co-Hopfian (respectively strongly Hopfian) is also strongly Hopfian (respectively strongly co-Hopfian). Moreover, we provide a variety of examples and structural results concerning the behavior of subacts and quotient acts of strongly Hopfian and strongly co-Hopfian S-acts, further elucidating the internal structure and interrelationships within this extended framework.

math.RT↗

arXiv2024

Application of Hohle's Square Roots on Hoop Algebras

Square root is a useful tool to study the properties of (ordered) algebraic structures. In this article, we are going to employ this tool to study hoop algebras. To do so, we define square root and make the first attempt to explore the significance properties of this concept in this setting. Then, due to the key role of square roots in obtaining new hoop algebras, we apply them on the filters of hoop algebras, and show that the formation of square roots on quotient structures of hoop algebras by their filters is well-behaved. In addition, a new class of hoop algebras having square roots, so called good hoop algebras, is introduced and its relationships with other classes of ordered algebras such as Boolean algebras and Godel algebras are explored. Several examples are provided as well. Ultimately, it is shown that the class of all bounded hoop algebras with square roots is a variety.

math.RA↗

arXiv2024

Equivalence of multiset-based consequence relations

The pioneering work of Blok and Jónsson and its further development by Galatos and Tsinakis initiated an abstract study of consequence relations using the tools of module theory, where consequence relations over all types of syntactic objects are put on an equal footing. However, the assumption that in a consequence relation the premises form merely a set, as opposed to a more complicated structure, is still retained. An attempt to extend this framework to account for inferentially substructural generalizations of consequence relations, where the premises have the structure of a finite multiset, was recently made by Cintula, Gil-Férez, Moraschini, and Paoli. In this paper, we develop a different inferentially substructural generalization of the work of Galatos and Tsinakis, where we instead assume that the premises have the structure of a set of finite multisets. This leads a somewhat smoother framework which, unlike that of Cintula et al., covers the original theory of Galatos and Tsinakis as a special case.

math.LO↗

arXiv2020

Weak Topologies on Toposes

This paper deals with the notion of weak Lawvere-Tierney topology on a topos. Our motivation to study such a notion is based on the observation that the composition of two Lawvere-Tierney topologies is no longer idempotent, when seen as a closure operator. For a given topos $\mathcal{E}$, in this paper we investigate some properties of this notion. Among other things, it is shown that the set of all weak Lawvere-Tierney topologies on $\mathcal{E}$ constitutes a complete residuated lattice provided that $\mathcal{E}$ is (co)complete. Furthermore, when the weak Lawvere-Tierney topology on $\mathcal{E}$ preserves binary meets we give an explicit description of the (restricted) associated sheaf functor on $\mathcal{E}$.

math.CT↗

arXiv2017

Action preserving (weak) topologies on the category of presheaves

Let $\mathcal{C}$ be a finitely complete small category. In this paper, first we construct two weak (Lawvere-Tierney) topologies on the category of presheaves. One of them is established by means of a subfunctor of the Yoneda functor and the other one, is constructed by an admissible class on $\mathcal{C}$ and the internal existential quantifier in the presheaf topos $\widehat{\mathcal{C}}$. Moreover, by using an admissible class on $\mathcal{C},$ we are able to define an action on the subobject classifier $Ω$ of $\widehat{\mathcal{C}}$. Then we find some necessary conditions for that the two weak topologies and also the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ to be action preserving maps. Finally, among other things, we constitute an action preserving weak topology on $\widehat{\mathcal{C}}$.

math.CT↗

arXiv2016

Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos

Let $j$ be a Lawvere-Tierney topology (a topology, for short) on an arbitrary topos $\mathcal{E}$, $B$ an object of $\mathcal{E}$, and $j_B = j\times 1_B$ the induced topology on the slice topos $\mathcal{E}/B$. In this manuscript, we analyze some properties of the pullback functor $Π_B:\mathcal{E}\rightarrow \mathcal{E}/B$ which have deal with topology. Then for a left cancelable class $\mathcal{M}$ of all $j$-dense monomorphisms in a topos $\mathcal{E}$, we achieve some necessary and sufficient conditions for that $(\mathcal{M} , \mathcal{M}^{\perp})$ is a factorization system in $\mathcal{E}$, which is related to the factorization systems in slice topoi $\mathcal{E}/B,$ where $B$ ranges over the class of objects of $\mathcal{E}$. Among other things, we prove that an arrow $f : X\rightarrow B$ in $\mathcal{E}$ is a $j_B$-sheaf whenever the graph of $f$, is a section in $\mathcal{E}/B$ as well as the object of sections $S(f)$ of $f$, is a $j$-sheaf in $\mathcal{E}$. Furthermore, we introduce a class of monomorphisms in $\mathcal{E}$, which we call them $j$-essential. Some equivalent forms of those and some of their properties are presented. Also, we prove that any presheaf in a presheaf topos has a maximal essential extension. Finally, some similarities and differences of the obtained result are discussed if we put a (productive) weak topology $j$, studied by some authors, instead of a topology.

math.CT↗

arXiv2015

Weak Factorization System for Actions of Po-monoids on Posets

Let $S$ be a pomonoid. In this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in {\bf Pos}-$S.$ We show that if the identity element of $S$ is the bottom element, then $(\mathcal{C_D}, \mathcal{E_S})$ is a weak factorization system in {\bf Pos}-$S,$ where $\mathcal{C_D}$ and $\mathcal{E_S}$ are the class of down-closed embedding $S$-poset maps and the class of all split $S$-poset epimorphisms, respectively. Among other things, we use a fibrewise notion of complete posets in the category {\bf Pos}-$S/B$ under a particular case where $B$ has trivial action. We get a necessary condition for regular injective objects in {\bf Pos}-$S/B$. Finally, we characterize them under a spacial case, where $S$ is, a pogroup and conclude $(Emb, Top)$ is a weak factorization system in {\bf Pos}-$S$.

math.CT↗

arXiv2015

On the Generators in the Category of Actions of Pomonoids on Posets and its Slices

Let $S$ be a pomonoid, in this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. First, we characterize some pomonoids on which all projectives in this category are generator or free. Then, we study regular injectivity and weakly regularly $d$-injectivity which lead to some homological classification results for pomonoids. Among other things, we get some relationships between regular injectivity in the slice category {\bf Pos}-$S/B_S$ and generators or cyclic projectives in {\bf Pos}-$S$.

math.RT↗

arXiv2015

Characterization of Pomonoids by Properties of Generators

The study of flatness properties of ordered monoids acting on posets was initiated by S.M. Fakhruddin in the 1980's. Although there exist many papers which investigate various properties of $S$-posets (posets equipped with a compatible right action of an ordered monoid $S$) from free to torsion free, among them generators, there seems to be known very little. In 2008, Laan characterized generators in the category {\bf Pos}-$S$ of all $S$-posets with monotone action-preserving maps between them. His characterization is similar to the case of acts over monoids. We attempt here to collect the knowledge on generators in the category {\bf Pos}-$S$ and to apply this to proceed on the questions of homological classification of ordered monoids, that is results of the type: all generators in the category {\bf Pos}-$S$, satisfy a flatness property if and only if $S$ has a certain property.

math.RT↗

arXiv2014

Coproduct Cancellation on \textbf{Act}-$S$

The themes of cancellation, internal cancellation, substitution have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. In this paper, we introduce and study cancellation problem in the theory of acts over monoids. We show that if $A$ is an $S$-act and $A={\dot\bigcup_{i\in I}}A_i$ is the unique decomposition of $A$ into indecomposable subacts $A_i, i\in I$ such that the set $P=\{{\rm Card} [i] \mid i\in I\}$ is finite, then $A$ is cancellable if and only if the equivalence class $[i]=\{j\in I \mid A_i\cong A_j\}$ is finite, for every $i\in I$. Likewise, we prove that every $S$-act is cancellable if and only if it is internally cancellable. Thus, the concepts cancellation and internal cancellation coincide here.

math.GR↗

arXiv2014

Nakayama's Lemma on $\textbf{Act}-S$

A crucial lemma on module theory is Nakayama's lemma \cite{AF}. In this article, we shall investigate some forms of Nakayama's lemma in the category of right acts over a given monoid $S$ with identity 1. More precisely, among other things, we show that equality $AI=A$ for some proper ideal $I$ of $S$ implies $A=\{θ\}$, when $A$ is a finitely generated quasi-strongly faithful $S$-act with unique zero element $θ$ and $S$ is a monoid in which its unique maximal right ideal $\mathfrak{M}$ is two-sided. Furthermore, as an application of Nakayama's lemma we prove Krull intersection theorem for $S$-acts. Finally, as a consequence, we shall see a homological classification form of this lemma, i.e, we prove if $S$ is a commutative monoid then every projective $S$-act is free if and only if $E(S)=\{1\}$, which $E(S)$ is the set of all idempotents of $S$.

math.GR↗