https://cgasa.sbu.ac.ir/ Categories and General Algebraic Structures with Applications en daily 1 Thu, 01 Jan 2026 00:00:00 +0330 Thu, 01 Jan 2026 00:00:00 +0330 https://cgasa.sbu.ac.ir/article_106674.html https://cgasa.sbu.ac.ir/article_106295.html Constellations are partial algebras in the sense that they possess a partial product, and a unary operation modelling domain.  They were first used to give an ESN-style theorem for left restriction semigroups in terms of so-called inductive constellations.  Here, we consider constellations in which elements have a suitable notion of inverse, giving the notion of a D-inverse constellation.  We show that there is a categorical isomorphism between the category of ordered groupoids and the category of D-inverse constellations.  This may be viewed as a generalisation of the ESN theorem, which relates the category of inductive groupoids to the category of inverse semigroups. https://cgasa.sbu.ac.ir/article_104940.html In this paper we generalize fibrations by $\mathcal{H}$-fibrations, the maps which homotopically lift homotopies. We replace the equalities in the definition of covering homotopy property with the homotopy relation so that we can first get an expression of the concept of covering homotopy property in the homotopy category. After introducing $\mathcal{H}$-fibrations, we will have a homotopy expression of some concepts related to fibration, such as path lifting, lifting function and unique path lifting property, to generalize some results in fibration. In particular, we show that an $\mathcal{H}$-fibration has homotopical path lifting property and also prove that a map is an $\mathcal{H}$-fibration if and only if it has a homotopical lifting function. https://cgasa.sbu.ac.ir/article_105052.html  In \cite{cap}, the concept of $\cap$-structure space is defined and it is studied from an algebraic and topological points of view. Indeed, the $\cap$-structure  is considered as a model for all algebraic substructures such as subgroups, subrings and submodules, ideals, etc. Moreover, the elements of these $\cap$-structures are seen as an open set, and from this point of view, another goal is to relate some  algebraic properties to some topological properties. The present article follows the same points of view of \cite{cap}. In particular, similar to algebraic homomorphisms, $\cap$-structural homomorphisms  are defined and investigated in $\cap$-structure spaces. In addition, we examine some classical results related to homomorphisms. In this regard, similar to lattice theory, we define the congruence relation on $\cap$-structure spaces and give some facts about them, and then we generalize the isomorphism theorems of algebraic structure to $\cap$-structure spaces.    https://cgasa.sbu.ac.ir/article_105269.html In this paper, we delve into the lattice of filters of a triangle algebra. Moreover, we establish the prime filter theorem, and investigate the algebraic structure of the set of co-annihilators of a triangle algebra. In addition, we explore the concept of  pure filter within the framework of  triangle algebras. Furthermore, we describe the topological properties of the prime filter space of a triangle algebra by equipping the lattice of prime filters with the Zariski topology. Thanks to the notion of pure filters in triangle algebras, we also provide a characterization of the open stable sets with respect to the stable topology, a topology that is coarser than the Zariski topology. https://cgasa.sbu.ac.ir/article_105134.html We introduce $\mathcal{M}$-spans for a class $\mathcal{M}$ of morphisms in a category $\mathcal{C}$. Using the equivalence class of $\mathcal{M}$-spans under a given equivalence relation, we give the notion of an $\mathcal{M}$-relation in $\mathcal{C}$. We first show under what conditions, $\mathcal{C}$-objects together with $\mathcal{M}$-relations form a category, called the category of $\mathcal{M}$-relations and we construct a quotient of the span category as a byproduct. Then we investigate the connection between $\mathcal{M}$-relation categories and quotient span categories. We establish when a category of $\mathcal{M}$-relations is isomorphic to a quotient span category. Finally several illustrative examples are given. https://cgasa.sbu.ac.ir/article_105208.html  For a frame $L$, $\mathcal{R}^+(L)$ denotes the nonnegative real valued continuous functions on $L$. We define the concept of $z$-ideals in this  semiring and  give a characterization of  its   $z$-ideals in terms of cozero elements of $L$. Also, we show that there is a one-one correspondence between  $z$-ideals and $z$-congruences on a ring $\mathcal{R}(L)$ and a semiring $\mathcal{R}^+(L)$.  We establish a relationship between $z$-congruence relation on $\mathcal{R}(L)$ and $z$-congruence relation on $\mathcal{R}^+(L)$.  A new characterization of $P$-frames is given via    $z$-congruences on $\mathcal{R}^+(L)$. Also, we show that there is a bijection between the minimal prime ideals  of $\mathcal{R}(L)$ and  coz-ultrafilter on $L$. https://cgasa.sbu.ac.ir/article_105452.html It is well known that the bar resolution can be replaced with any projective resolution of the corresponding algebra when computing the Hochschild (co)homology of that algebra. This is, in fact, a feature of its construction via derived functors. For generalizations and extensions of the Hochschild (co)homology (like the secondary and tertiary Hochschild (co)homology theory, as well as higher order Hochschild (co)homology theory), one uses a bar-like resolution in a simplicial setting within its construction in order to accommodate the changing module structures in every dimension. In this note, we present a method in order to replace these bar-like resolutions. https://cgasa.sbu.ac.ir/article_105322.html In this paper, we present a new way of defining the property of being WRP-Noetherian by making use of principal right poideals. Additionally, we provide a characterization of WRP-Noetherian ordered semigroups through their $S$-posets. Furthermore, we investigate how the property of being WRP-Noetherian behaves under some semigroup-theoretic constructions, like sub ordered semigroups, and quotients. Specifically, we establish necessary and sufficient conditions for the direct product of two ordered semigroups to be WRP-Noetherian. https://cgasa.sbu.ac.ir/article_106675.html https://cgasa.sbu.ac.ir/article_105489.html Let $(\mathcal{R},\otimes)$ be a symmetric monoidal closed Grothendieck category which has enough flat objects.  It is shown that a given object ${\mathcal{G}}$  in $\mathcal{R}$ has finite flat dimension if and only if it is quasi-isomorphic to a bounded complex of objects of finite flat dimension. In the case in which $\mathcal{R}$ has enough projective objects, we prove that finite flat dimension in $\mathcal{R}$ implies finite projective dimension if and only if any object in $\mathcal{R}$ that is quasi-isomorphic to a bounded complex of objects of finite flat dimension has finite projective dimension. This leads to a generalization of  [4, Proposition 2.3] and [15, Theorem]. Moreover, we present a wide class of $n$-perfect rings. https://cgasa.sbu.ac.ir/article_105813.html In [8] Valdis Laan introduced Condition (PW P). Golchin and Mohammadzadeh in [3] introduced Condition (PW P_E), such that Condition (PW P) implies it but the converse is not true in general. In this paper at first we introduce a generalization of Condition (PW P_E), called Condition (PW P_S). Then will give some general properties and a characterization of monoids for which all right acts satisfy this condition. Also, we give a characterization of monoids, by comparing this property of their acts with some others. Finally, we will give a characterization of monoid S, for which S^{I}_{S}, for any non-empty set I and S^{S \times S}_{S}, satisfy Condition(PW P_S). https://cgasa.sbu.ac.ir/article_106169.html Let $R$ be a commutative  ring and $M$ be a finitely generated $R$-module.   Let   I$(M)$ be the first nonzero Fitting ideal of $M$.  In this paper we characterize some modules over Noetherian UFDs, whose first nonzero Fitting ideal is a prime ideal. We show that if $P$ is a prime ideal and $M$ is a finitely generated R-module with I$(M) = P$ and T$(M_P)\neq 0$, then M is isomorphic to $R/P \oplus N$, for some projective R-module $N$ of constant rank. Also,  we investigate some conditions under which  ${M}/$T$(M)$ is free. https://cgasa.sbu.ac.ir/article_106251.html Let $R$ be a ring and  $\mathcal{X} = \mathcal{SH}(R)-\{0\}$ be the set   all  of non-zero strongly hollow ideals (briefly, $sh$-ideals) of   $R$. We first  study the concept   $SH$-topology and investigate some of the basic properties of a topological space with this topology. It is  shown  that, if  $\mathcal X $ is  with $SH$-topology, then  $\mathcal {X}$ is Noetherian if and only if every subset of $\mathcal X$ is quasi-compact if and only if  $R$ has $dcc$ on semi-$sh$-ideals.   Finally,  the relation between the dual-classical Krull dimension of $R$ and the  derived dimension of  $\mathcal {X}$ with a certain topology has been studied. It is proved that,  if $\mathcal {X}$ has derived dimension, then $R$ has the dual-classical Krull dimension and in case $R$ is a $D$-ring (i.e., the lattice of ideals of $R$ is distributive), then the converse is true. Moreover these two dimension differ by at most $1$.  https://cgasa.sbu.ac.ir/article_106869.html Edge coloring of a graph is a function from its edge set to the set of natural numbers (called colours). A path in an edge-colored graph with no two edges sharing the same color is called a rainbow path. An edge-colored graph is said to be rainbow connected if every pair of vertices is connected by at least one rainbow path. Such a coloring is called a rainbow coloring of the graph. The minimum number of colors required to rainbow color a connected graph is called its rainbow connection number, denoted by $rc(G)$. For example, the rainbow connection number of a complete graph is 1, that of a path is its length, and that of a star is its number of leaves. For a basic introduction to the topic, see Chapter 11 in \cite{Ch2} and for a comprehensive treatment of the area see the recent monograph by Li and Sun \cite{Li}. The concept of rainbow coloring was introduced in \cite{Ch1}.