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Departing from a categorical concept of topogenous orders defined relative to the bifibration of subobjects, we introduce and discuss topogenous orders on forms, i.e., faithful and amnestic functors. These topogenous orders are shown to include both closure and interior operators on forms. We define and study two special morphisms relative to a topogenous order, namely strict and final morphisms. We give a characterization of the two morphisms by the help of their cartesian and cocartesian liftings. Some examples from topology and algebra demonstrating our results are included. [ABSTRACT FROM AUTHOR]

The notions of dense, proper, separated or perfect morphisms and hence of compact, Hausdorff or compact Hausdorff are all consequent to good properties of a family of closed morphisms is well known in literature. Deeper consequences like the Tychonoff product theorem or the Stone Čech compactifications follow from richer properties of the set of closed morphisms. The purpose of this paper is to provide a closure operation on a preneighbourhood space so that the resulting set of closed morphisms possess all the properties mentioned above. [ABSTRACT FROM AUTHOR]

The notions of dense, proper, separated or perfect morphisms and hence of compact, Hausdorff or compact Hausdorff are all consequent to good properties of a family of closed morphisms is well known in literature. Deeper consequences like the Tychonoff product theorem or the Stone Čech compactifications follow from richer properties of the set of closed morphisms. The purpose of this paper is to provide a closure operation on a preneighbourhood space so that the resulting set of closed morphisms possess all the properties mentioned above.

In this article, we have introduced the notion of pairwise compactness in bi-isotonic spaces via finite intersection property and pairwise open cover. Moreover, we have given pairwise compactness of bi-isotonic subspaces with both reduced closure and interior functions. We have characterized pairwise compactness by the neighborhood concept; however, the axioms of bi-isotonic spaces are insufficient to prove the theorem, and we have studied this in bi-closure spaces. For similar reasons, occasionally, bi-closure spaces have been considered with additional explanations, even if some concepts related to compactness have been naturally extended to bi-isotonic spaces. Additionally, interesting results have been obtained considering the pairwise Hausdorffness and compactness relationship. It has also been observed that resembling cross-relationships exist for closed subsets of pairwise compact spaces. Finally, it has been observed that the pairwise compactness of bi-closure spaces is preserved under bi-continuity. [ABSTRACT FROM AUTHOR]

Copyright of Afyon Kocatepe University Journal of Science & Engineering / Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi is the property of Afyon Kocatepe University, Faculty of Science & Literature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We show that the universal X-generated F-inverse monoid F(G), where G is an X-generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion M (G , X ∪ G ¯) of G, with respect to the extended generating set X ∪ G ¯ , where G ¯ is a bijective copy of G which encodes the m-operation in F(G). The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph Cay (G , X ∪ G ¯) and leads to a new and simpler proof of the universal property of F(G). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mahani Mathematical Research Center is the property of Shahid Bahonar University of Kerman, Department of Pure Mathematics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until now, existed in isolation. This paper bridges the gap by showing that greedoids can be defined using a modified violator operator. The established connections not only deepen our understanding of these theories but also provide a new characterization of antimatroids. [ABSTRACT FROM AUTHOR]

The primary aim of this article is to put forward new classes of uni-nullnorms on certain classes of bounded lattices via closure (interior) operators. Due to the new classes of uninorms combining both a t-norm T and a t-conorm S by various kinds of closure operators or interior operators, the relationships and properties among the same class of uninorms on L, we obtain new classes of uni-nullnorms on L via closure (interior) operators. The constructions of uninullnorms on some certain classes of bounded lattices can provide another different perspective of t-norms and the dual of t-norms, uninorms and some other associative aggregation operations on bounded lattices. That is, the constructions seem to be the ordinal like sum constructions, but not limited to the ordinal like sum constructions. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce the structure space of uniformly strongly prime ideals of an ordered semigroup and to study the separation axioms and the compactness property. [ABSTRACT FROM AUTHOR]

Neighborhood relationships play a pivotal role in rough set theory, addressing the limitations of equivalence relations. This article focuses on defining upper and lower approximation operators using neighborhood relationships and exploring their properties in terms of serialization, inverse serialization, reflexivity, symmetry, transitivity, and Euclidean relations. Furthermore, a necessary and sufficient condition for the upper approximation operator to function as a topological closure operator is derived. Overall, this research sheds light on the significance of neighborhood relationships and their implications within rough set theory.

We employ closure operators associated with n-ary relations, n > 1 an integer, to provide the digital space ℤ3 with connectedness structures. We show that each of the six inscribed tetrahedra obtained by canonical tessellation of a digital cube in ℤ3 with edges consisting of 2n − 1 points is connected. This result is used to prove that certain bounding surfaces of the polyhedra in ℤ3 that may be face-to-face tiled with such tetrahedra are digital Jordan surfaces (i.e., separate ℤ3 into exactly two connected components). An advantage of these Jordan surfaces over those with respect to the Khalimsky topology is that they may possess acute dihedral angles π 4 while, in the case of the Khalimsky topology, the dihedral angles may never be less than π 2 . [ABSTRACT FROM AUTHOR]

We show that the category of X-generated E-unitary inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of G. Analogously, we study F-inverse monoids in the extended signature (· , 1 , - 1 , m) , and show that the category of X-generated F-inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of G. As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions. [ABSTRACT FROM AUTHOR]

We introduce the notion of topogenous orders on a poset X to be certain endomaps on X. We build on a Galois connection between endomaps and binary relations on X and study relationships between endomap properties and corresponding relational properties. In particular, we determine the topogenous orders that are in a one-to-one correspondence with (idempotent) closure operators. Extending our considerations to the categorical level, we find a cartesian closed category of topogenous systems. [ABSTRACT FROM AUTHOR]

Several naturally defined lower separation axioms for bitopological spaces obtained by modifying the axioms T0, T1, and R0 appear in the literature. We introduce and study analogous separation axioms for bitopogenous spaces. In particular, we investigate relationships between the axioms and discuss the conditions under which the relationships are similar to those between the corresponding separation axioms for bitopological spaces. [ABSTRACT FROM AUTHOR]

The objective of this article is to characterize each of compact, sober, and Ti for i = 0, 1, 2 constant limit spaces as well as to investigate the relationships between them. Finally, we compare our results in some topological categories. [ABSTRACT FROM AUTHOR]

The work is related to the study of the model-theoretic properties of Jonsson theories, which, generally speaking, are not complete. In the article, on the Boolean of Jonsson subsets of the semantic model of some fixed Jonsson theory, the concept of the Jonsson closure operator Jcl was introduced, defining the J-pregeometry on these subsets, and some results were obtained describing this closure operator.

Uninorms combining t-conorms and t-norms on bounded lattices have lately drawn extensive interest. In this article, we propose two ways for constructing uninorms on a bounded lattice with an identity element. They benefit from the appearance of the t-norm (resp. t-conorm) and the closure operator (resp. interior operator) on a bounded lattice. Additionally, we include some illustrative examples to highlight that our procedures differ from others in the literature. [ABSTRACT FROM AUTHOR]

In a category with a proper ()-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous orders induced by pointed and co-pointed endofunctors. We also lift the topogenous orders along an -fibration. This permits one to obtain the lifting of interior and neighbourhood operators along an -fibration and includes the lifting of closure operators found in the literature. A number of examples presented at the end of the paper demonstrates our results. [ABSTRACT FROM AUTHOR]

Motivated by a problem posed in [10], we investigate the closure operators of the category SLatt of join semilattices and its subcategory SLattO of join semilattices with bottom element. In particular, we show that there are only finitely many closure operators of both categories, and provide a complete classification. We use this result to deduce the known fact that epimorphisms of SLatt and SLattO are surjective. We complement the paper with two different proofs of this result using either generators or Isbell's zigzag theorem. [ABSTRACT FROM AUTHOR]

This note aims to clarify the relations between three ways of constructing complete lattices that appear in three different areas: (1) using ordered structures, as in set-theoretic forcing, or doubly ordered structures, as in a recent semantics for intuitionistic logic; (2) using compatibility relations, as in semantics for quantum logic based on ortholattices; (3) using Birkhoff’s polarities, as in formal concept analysis.

Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in \cite{2020}. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of \emph{continuity} of morphisms with respect to induced closure operators.

In this study, a $ \zeta^*_\Gamma $-local function is defined and its properties are examined. This newly defined local function is compared with the well-known local function and the local closure function according to the relation of being a subset. With the help of this new local function, the $ \Psi_{\zeta^*_\Gamma} $ operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. $ \Psi_{\zeta^*_\Gamma} $-compatibility is defined and its properties are examined. $ \Psi_{\zeta^*_\Gamma} $-compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized.

Uninorms on bounded lattices have been recently a remarkable field of inquiry. In the present study, we introduce two novel construction approaches for uninorms on bounded lattices with a neutral element, where some necessary and sufficient conditions are required. These constructions exploit a t-norm and a closure operator, or a t-conorm and an interior operator on a bounded lattice. Some illustrative examples are also included to help comprehend the newly added classes of uninorms. [ABSTRACT FROM AUTHOR]

In a category C with an (E , M )-factorization structure for morphisms, we prove that any subclass N of M which is closed under pullbacks determines a transitive quasi-uniform structure on C . In addition to providing a categorical characterization of all transitive quasi-uniform structures compatible with a topology, this result also permits us to establish a number of Galois connections related to quasi-uniform structures on C . These Galois connections lead to the description of subcategories of C determined by quasi-uniform structures. Several examples considered at the end of the paper illustrate our results. [ABSTRACT FROM AUTHOR]

In this article, we first introduce a method for obtaining uninorms on a bounded lattice L with the identity element e ∈ L ∖ { 0 L , 1 L } using an interior operator on L and a t-conorm on [ e , 1 L ] 2. Next, based on a closure operator on L and a t-norm on [ 0 L , e ] 2 , a construction method for uninorms on L with the identity element e ∈ L ∖ { 0 L , 1 L } is proposed. Furthermore, we state that these methods are generalizations of those described in [Çaylı, G. D. 2018. "A Characterization of Uninorms on Bounded Lattices by Means of Triangular Norms and Triangular Conorms." International Journal of General Systems 47: 772–793. doi:; Dan, Y., and B. Q. Hu. 2020. "A New Structure for Uninorms on Bounded Lattices." Fuzzy Sets and Systems 386: 77–94. doi:.]. Some corresponding examples are also provided to illustrate that our methods differ from the existing ones. [ABSTRACT FROM AUTHOR]

We extend the bijective correspondence between finite semimodular lattices and Faigle geometries to an analogous correspondence between semimodular lattices of finite lengths and a larger class of geometries. As the main application, we prove that if e is a join-irreducible element of a semimodular lattice L of finite length and h < e in L such that e does not cover h, then e can be "lowered" to a covering of h by taking a length-preserving semimodular extension K of L but not changing the rest of join-irreducible elements. With the help of our "lowering construction", we prove a general theorem on length-preserving semimodular extensions of semimodular lattices. This theorem implies earlier results proved by Grätzer and Kiss (Order 2 351–365, 1986), Wild (Discrete Math. 112, 207–244, 1993), and Czédli and Schmidt (Adv. Math. 225, 2455–2463, 2010) on extensions to geometric lattices, and an unpublished result of E. T. Schmidt. Our approach offers shorter proofs of these results than the original ones. [ABSTRACT FROM AUTHOR]

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system ⟨ X ⟩ P . The closure system ⟨ Y ⟩ P generated by the patch closure Y of X is the patch closure of ⟨ X ⟩ P . If X is contained in the set of nontrivial prime elements of P then ⟨ X ⟩ P is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed. [ABSTRACT FROM AUTHOR]

We study closure operators on complete lattices and define for them the concepts of closed and open bases as well as that of a neighborhood base. Some properties of these concepts are studied including relationships between them. In particular, we find sufficient conditions under which the properties are analogous to those known for topological spaces. [ABSTRACT FROM AUTHOR]

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains with Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, L-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions. [ABSTRACT FROM AUTHOR]

We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.

In this study, a ζ∗Γ-local function is defined and its properties are examined. This newly defined local function is compared with the well-known local function and the local closure function according to the relation of being a subset. With the help of this new local function, the Ψζ∗Γ operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. Ψζ∗Γ-compatibility is defined and its properties are examined. Ψζ∗Γ-compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized. [ABSTRACT FROM AUTHOR]

Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in [49]. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of continuity of morphisms with respect to induced closure operators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notions of hereditarily disconnected and totally disconnected objects in a topological category and examine the relationship as well as interrelationships between them. Moreover, we characterize each of T2, connected, hereditarily disconnected, and totally disconnected objects in some topological categories and compare our results with the ones in the category of topological spaces. [ABSTRACT FROM AUTHOR]

We study a number of categorical quasi-uniform structures induced by functors. We depart from a category C with a proper (E,M)-factorization system, then define the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) on C and use it to describe the quasi-uniformities induced by pointed and copointed endofunctors of C. In particular, we demonstrate that every quasi-uniformity on a reflective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every reflection morphism is continuous. Thinking of categories supplied with quasi-uniformities as large "spaces", we generalize the continuity of C-morphisms (with respect to a quasi-uniformity) to functors. We prove that for an M-fibration or a functor that has a right adjoint, we can obtain a concrete construction of the coarsest quasi-uniformity for which the functor is continuous. The results proved are shown to yield those obtained for categorical closure operators. Various examples considered at the end of the paper illustrate our results. [ABSTRACT FROM AUTHOR]

Recently, topological structures have emerged as one of the most popular rough sets (RS) research topics. It can be stated that it is a fundamental and significant subject in the theory of RS. This study introduces a debate about the structure of rough topological space based on the reflexive relation. To create the rough topological space, we use the representation of RS. We also look at the relationships between approximation operators, closure operators, and interior operators. Also, the relationship between topological space in the universe that is not limited or restricted to be ended, and RS induced by reflexive relations is investigated. Furthermore, we define the relationships between the set of all topologies that satisfy the requirement of compactness C2 and the set of all reflexive relations. Finally, we present a medical application that addresses the issue of dengue fever. The proposed structures are used to determine the impact factors for identifying dengue fever.

The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled. [ABSTRACT FROM AUTHOR]

In this sequel to our previous article (Cordero et al., 2020) on general inference systems for reasoning with if–then dependencies, we study transformations of if–then rules to semantically equivalent collections of if–then rules suitable to solve several problems related to reasoning with data dependencies. We work in a framework of general lattice-based if–then rules whose semantics is parameterized by systems of isotone Galois connections. This framework allows us to obtain theoretical insight as well as algorithms on a general level and observe their special cases by choosing various types of parameterizations. This way, we study methods for automated reasoning with different types of if–then rules in a single framework that covers existing as well as novel types of rules. Our approach supports a large family of if–then rules, including fuzzy if–then rules with various types of semantics. The main results in this article include new observations on the syntactic inference of if–then rules, complete collections of rules, reduced normal forms of collections of rules, and automated reasoning methods. We demonstrate the generality of the framework and the results by examples of their particular cases focusing on fuzzy if–then rules. [ABSTRACT FROM AUTHOR]

We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice based on equality of the topologies induced by uninorms. [ABSTRACT FROM AUTHOR]

We investigate the structure of power-closed ideals of the complex polynomial ring R = C [ x 1 , ... , x d ] and the Laurent polynomial ring R ± = C [ x 1 , ... , x d ] ± = S − 1 C [ x 1 , ... , x d ] , where S is the multiplicatively closed semigroup S = [ x 1 , ... , x d ]. Here, an ideal I is power-closed if f (x 1 , ... , x d) ∈ I implies f (x 1 i , ... , x d i) ∈ I for each natural number i. Important examples of such ideals are provided by the ideals of relations in Minkowski rings of convex polytopes. We investigate related closure and interior operators on the set of ideals of R and R ± and we give a complete description of principal power-closed ideals and of radicals of general power-closed ideals of R and R ±. [ABSTRACT FROM AUTHOR]