https://cgasa.sbu.ac.ir/ Categories and General Algebraic Structures with Applications en daily 1 Wed, 01 Jul 2026 00:00:00 +0330 Wed, 01 Jul 2026 00:00:00 +0330 https://cgasa.sbu.ac.ir/article_107236.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_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_107062.html Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ are the eigenvalues of $A_{\alpha}(G)$, the $A_{\alpha}$-energy of $G$ is defined as $E_{A_{\alpha}}(G) = \sum_{i=1}^{n} |\rho_i -\frac{2\alpha m}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_\alpha}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_\alpha}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph. https://cgasa.sbu.ac.ir/article_106984.html In this paper, we define an adjunction between two categories: $\sigma$-frames and $\sigma$-spaces, denoted by the pair $(\Sigma^\sigma, \Lambda)$. The functor $\Sigma^\sigma$ is constructed using the concept of $\sigma$-points. We prove that $\sigma$-points are equivalent to $\sigma$-completely prime filters, but unlike in pointfree topology, they do not correspond to prime elements. While every prime element determines a corresponding $\sigma$-point, the converse fails. For $\sigma$-frames, we define the $\sigma$-spatiality condition, which is equivalent to having enough $\sigma$-points. Dually, for $\sigma$-spaces, the $\sigma$-soberness condition is shown to be equivalent to the conjunction of the $\sigma_0$ separation axiom and $\sigma$-pointedness properties. These conditions naturally emerge from careful analysis of the adjunction morphisms. https://cgasa.sbu.ac.ir/article_106968.html In this note, for a certain class of regular continuous frames, we establish conditions that are equivalent to saying that the Freudenthal compactification and the smallest compactification are indistinguishable; in turn, this expands the list of conditions under which the smallest compactification is perfect, which is available in the literature. We define a new class of morphisms between frames, called F-maps, and provide a proof demonstrating that the category of compact regular frames and F-maps forms a coreflective full subcategory of the category of rim-compact frames and F-maps. This coreflection is evidenced by the join map associated with the Freudenthal compactification. Accordingly, this provides an affirmative answer to the question by Herrlich, which inquired whether the Freudenthal compactification can be regarded as a reflection with "sensible" maps. https://cgasa.sbu.ac.ir/article_106967.html In this paper, we define the concept of nucleus map on equality algebras and study related results. Then, using this concept and upsets, a topology on equality algebras is constructed and it is shown that the equality algebra with this topology becomes a topological space. In addition, some properties of topological space such as compactness and connectedness are investigated. Moreover, we study the continuity of all operations with respect to the topology on equality algebras. Finally, the relations between the two topologies in quotient equality algebras are revealed. 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_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_107237.html 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}. https://cgasa.sbu.ac.ir/article_106925.html Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_\ell$ and $(R:T)_r$ denote the largest ideal, left ideal and right ideal of $T$ that are contained in $R$, respectively. It is shown that both $(R:T)_\ell$ and $(R:T)_r$ are prime ideals of $R$, and $|{\rm Min}_R((R:T))|\leq 2$. We prove that if $T_R$ has a maximal submodule, then $(R:T)_\ell$ is a right primitive ideal of $R$. We investigate the conditions under which $(R:T)_r$ is a completely prime (right) ideal of $R$ or of $T$. We show that Char$(R/(R:T)_\ell)={\rm Char}(R/(R:T)_r)$, and if Char$(T)$ is neither zero nor a prime number, then $(R:T)\neq 0$. When $|{\rm Min}(R)|\geq 3$, both $(R:T)$ and $(R:T)_\ell(R:T)_r$ are nonzero ideals. Assuming $R$ is integrally closed in $T$, we prove that $(R:T)_\ell$ and $(R:T)_r$ are prime one-sided ideals of $T$; moreover $(R:T)$ is a semiprime ideal of $T$ and either $(R:T)$ is a prime ideal of $T$ or $(R:T)=(R:T)_\ell\cap (R:T)_r$ is a semiprime ideal of $R$. We observe that if $(R:T)_lT=T$, then $T$ is a finitely generated left $R$-module and $(R:T)_\ell$ is a finitely generated right $R$-module which is also a right primitive ideal of $R$. Finally, we study the transfer of the Noetherian and the Artinian properties between $R$ and $T$. https://cgasa.sbu.ac.ir/article_106945.html In this paper, we first show that the category of measurable spaces is isomorphic to a particular category of structural topological spaces. Next, we define structural measurable space and we introduce a notion of structural outer measure  adapted to a topological structure, along with a corresponding concept of structural measure for objects in the category of structural topological spaces. These concepts are formulated in terms of functions, transformations and natural transformations. Next, we illustrate these concepts with various examples, including several fuzzy topological spaces. Finally, under certain conditions we prove a generalization of Carathéodory's Extension  and Carathéodory's Criterion showing that each notion -structural outer measure and structural measure- induces the other.   https://cgasa.sbu.ac.ir/article_107063.html In this paper, we investigate one-sided (thick) ideals in monoidal categories and explore related concepts, including generating sets, idempotency, radicality and primeness. We establish several structural properties of one-sided ideals, drawing analogies with the ring-theoretic setting. Further results are obtained by specializing to pivotal categories. In particular, one-sided negligible objects are shown to provide examples of one-sided ideals, thanks to the well-established theory of categorical traces in this framework. Moreover, we introduce and study a generalization, called one-sided $\mathcal{U}$-ideals, where we show their nontrivial nature and establish various $\mathcal{U}$-analogues of the preceding results, examining the validity of several fundamental properties. https://cgasa.sbu.ac.ir/article_107064.html This paper aims to redefine the concept of ``continuity of a function at a point'' from a set-theoretic perspective, providing a sufficiently flexible definition that encompasses the various forms of continuity found in the mathematical literature. Let $\mathscr{A}$ and $\mathscr{B}$ denote families of subsets of non-empty sets $X$ and $Y$, respectively. We define an $\mathscr{A}$-$ \mathscr{B}$-continuous map and examine some algebraic properties of structures related to the set $C_{(\mathscr{A}, \mathscr{B})}(X, Y)$, which consists of all $\mathscr{A}$-$ \mathscr{B}$-continuous maps from $X$ to $Y$. Additionally, we show that $C_{_{(\mathscr{A}, \mathcal{O}Y)}}(X, Y)\cong C(X_z, Y)$, where $Y$ is an $f$-ring and $\mathscr{A}$ is closed under finite intersections, and $X_z$ is a topological space induced by $(X, \mathscr{A})$.  https://cgasa.sbu.ac.ir/article_107085.html By a left $PCP$ (or left $P(P)$) monoid we mean a monoid such that all its principal left ideals satisfy Condition $(P)$ (see \cite{14,24}). In this paper, we generalize formalization of left $PCP$ (left $P(P)$) monoids  to right acts and will give characterizations of monoids by this property of their right acts.  https://cgasa.sbu.ac.ir/article_107086.html A Tychonoff space $X$ is said to be a CAP-space if the closure of the interior of every zero-set of $X$ is itself a zero-set of $X$. These spaces were introduced by Golrizkhatami and  Taherifar \cite{GT}. With a view to supplementing the results in the cited paper, we extend this notion to the setting of point-free topology. We thus define a completely regular frame to be capped in case the double pseudocomplement of every cozero element of the frame is also a cozero element. This   makes the comparison of this concept with other disconnectivity notions (such as basic disconnectedness) defined by imposing conditions on pseudocomplements very transparent.    https://cgasa.sbu.ac.ir/article_107089.html We generalize the separation axioms $T_0$, $T_1$ and $T_2$ to LG-spaces and show that the most important results about these concepts can be extended in LG-spaces. https://cgasa.sbu.ac.ir/article_107164.html In this paper, by using the notions of ``quasi" and ``pseudo" in logical algebras,  we introduce two generalizations of equality algebras.  A commutative generalization of  equality algebras, is called quasi-equality algebras and a non-commutative generalization of quasi-equality algebras is called quasi-pseudo equality algebras. Then we investigate some of their properties. In addition, according to \cite{2} and knowing the relation between equality algebras and BCK(C)-meet-semilattices,  we generalize the concepts of BCK-algebras to quasi-BCK-algebras and pseudo BCK-algebras to quasi-pseudo BCK-algebras, too. The related properties and the relation between different kinds of quasi-(pseudo) BCK-algebras are investigated. Moreover, we investigate   the category of quasi-(pseudo) equality algebras and quasi-(pseudo) BCK-algebras and we show that they  are equivalent. https://cgasa.sbu.ac.ir/article_107282.html The present paper addresses the notion of $(u,v)$-absorbing primary hyperideals in commutative multiplicative hyperrings. A proper hyperideal $P$ of $A$ is said to be a $(u,v)$-absorbing primary hyperideal if $x_1 \circ \cdots \circ x_u \subseteq P$ for $x_1,\ldots, x_u \in A \backslash U(A)$ implies either $x_1 \circ \cdots \circ x_v \subseteq P$ or $x_{v+1} \circ \cdots \circ x_u \subseteq rad(P)$. The study investigates the links and characteristics of $(u,v)$-absorbing primary hyperideals, presenting a general framework for realizing their importance in hyperring theory. We give some arguments and examples explaining the relationship between $(u,v)$-absorbing primary hyperideals and other famous classes of hyperideals, such as maximal, prime and primary hyperideals, despitebeing vastly different. Morevoer, we analyze how the $(u,v)$-absorbing primary hyperideals behave concerning images and inverse images of hyperring homomorphisms, quotients and localizations. https://cgasa.sbu.ac.ir/article_107283.html We present a pointfree generalization of outer measures by replacing $\mathcal{P}(X)$ with a bounded distributive lattice $L$, studying functions $\mu: L\to [0,\infty]$ satisfying lattice-theoretic versions of outer measure axioms (null-preserving, monotone, $\sigma$-subadditive). For such an outer measure, we introduce $\mu$-measurable elements via a generalized Carath\'eodory condition and show that they form a sublattice closed under complements. When $L$ is a Boolean $\sigma$-frame, the collection $\mathcal{M}_\mu$ of measurable elements is itself a Boolean $\sigma$-frame; counterexamples demonstrate the necessity of the Boolean hypothesis for closure under countable joins. Moreover, for any $\sigma$-frame $L$ (without requiring the Boolean condition), the restriction of $\mu$ to $\mathcal{M}_\mu$ is $\sigma$-additive. We also examine the categorical behavior of the assignment $\mu \mapsto \mathcal{M}_\mu$, which yields a natural transformation between suitable functors, and pose an open question: whether every Boolean $\sigma$-subframe of $L$ arises as $\mathcal{M}_\mu$ for some outer measure $\mu$. This work extends classical measure theory to pointfree settings, revealing fundamental connections between order-theoretic, measure-theoretic, and categorical structures.