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.
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.
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 $L$.
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.
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).
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.
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.
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.
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.
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.
For a frame L, 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 R(L) and a semiring R+(L). We establish a relationship between z-congruence relation on R(L) and z-congruence relation on R+(L). A new characterization of P-frames is given via z-congruences on R+(L). Also, we show that there is a bijection between the minimal prime ideals of R(L) and coz-ultrafilter on L.
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.
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.
We introduce ,M-spans for a class ,M of morphisms in a category C. Using the equivalence class of ,M-spans under a given equivalence relation, we give the notion of an ,M-relation in C. We first show under what conditions, C-objects together with ,M-relations form a category, called the category of ,M-relations and we construct a quotient of the span category as a byproduct. Then we investigate the connection between ,M-relation categories and quotient span categories. We establish when a category of ,M-relations is isomorphic to a quotient span category. Finally several illustrative examples are given.
In this paper we generalize fibrations by 7-t-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 7-t-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 7-t-fibration has homotopical path lifting property and also prove that a map is an 7-t-fibration if and only if it has a homotopical lifting function.
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 $1$.
In 1997, Golchin and Renshaw introduced Condition (P-E) and showed that this condition implies weak flatness, although the converse is not generally valid. In this paper, we present Condition (P-sc) as a generalization of Condition (P-E). We also see that Condition (P-sc) implies weak flatness, but the converse is not necessarily true. However, for left PSF monoids the converse is holds. Moreover, we discuss some general properties and provide a homological classification of monoids by comparing Condition (P-sc) with some other properties. Furthermore, a new homological classification of monoids is presented by comparing Condition (P-E) with other properties.
In 1997, Golchin and Renshaw introduced Condition $(P_E)$ and showed that this condition implies weak flatness, although the converse is not generally valid.In this paper, we present Condition $(P_{sc})$ as a generalization of Condition $(P_E)$. We also see that Condition $(P_{sc})$ implies weak flatness, but the converse is not necessarily true. However, for left $PSF$ monoids the converse is holds. Moreover, we discuss some general properties and provide a homological classification of monoids by comparing Condition $(P_{sc})$ with some other properties.Furthermore, a new homological classification of monoids is presented by comparing Condition $(P_E)$ with other properties.
In [6] we developed a k-theory for the category of hyperbolic hyperfields (a category that contains a copy of the category of (pre)special groups): this construction extends, simultaneously, Milnor's k-theory ([20]) and Dickmann-Miraglia's k-theory ([13]). An abstract environment that encapsulate all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms is given by the concept of inductive graded rings a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in realm the algebraic theory of quadratic forms for Pythagorean fields. The goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([6]) - equivalent to the category of special groups ([10]) and the categories of inductive graded rings.
We provide an explicit description of the Picard group (the group of isomorphism classes of invertible objects, those that have an inverse under the tensor product) of the dual category of the category of comodules over a supergroup algebra, by using the description of this group for group-theoretical categories. In fact we prove that there is a subgroup relation between these groups. As an interest application of this group in a modular context, it can be used to construct examples of symmetric special Frobenius algebras. They also plays an important role in the theory of braided tensorcategories for the classification of group extensions of fusion categories.
