Arbitrary Unions Research Articles

Sets Related to Openness and Continuity Decompositions in Primal Topological Spaces

This paper introduces and investigates several new classes of sets called P-α-open sets, Psemiopen sets, P-preopen sets, and P-β-open sets within the framework of primal topological spaces. Their properties and relationships with other open set generalizations are studied through examples. Additionally, the concepts of PR-sets and PRα -sets are defined and their characteristics examined. Also, the notions of P-α-continuous, P-semicontinuous, P-precontinuous and P-β-continuous mappings are initiated and their features and main characterizations determined. A new class of sets called Ψ_P -sets is also introduced in primal topological spaces using the ΨP -operator. Their properties and relationships between Ψ_ P -sets, α-open, semi-open, and pre-open are investigated. Theorems on arbitrary unions and finite intersections of Ψ_P are discussed.

μ-L-Closed Subsets of Noetherian Generalized Topological Spaces

In the final years of the 20th century, the notion of generalized topological spaces was introduced, marking a significant shift in the field of topology. This paper focuses on a subset of on a non-empty set that is closed under arbitrary unions, defining a generalized topology and subsequently a generalized topological space (GTS) denoted by . Within this framework, we explore the concept of Noetherian generalized topological spaces and delve into the properties of closed subsets within the Noetherian GTS. The investigation reveals that subspaces of a Noetherian GTS , with the induced topology, inherit the Noetherian property and exhibit finitely many non-empty irreducible components. Furthermore, the study extends to the analysis of hereditary properties, regular , , irreducible closed subsets, and the product properties of closed subsets under continuous functions. We also establish the closure property of finite unions in Noetherian GTS and clarify the homeomorphic nature of Noetherian GTS to itself.

Algebraic properties of periodic points to the additive cellular automata

This paper characterizes the periodic points of additive cellular automata such as, arbitrary union and arbitrary intersection, relationship between periodic sets and existence of additive cellular automata with cardinality of periodic points one and two and existence for higher cardinality more than two

$\Psi_{\Gamma}-C$ Sets in Ideal Topological Spaces

In this paper, we present a new type of set called $\Psi_{\Gamma}-C$ set by using the operator $\Psi_{\Gamma}$. We investigate the relationships of these sets with some special sets which were studied in the literature. For instance $\theta$-open set, semi $\theta$-open set, $\theta$-semiopen set, regular $\theta$-closed set. In particular, we show that $\Psi_{\Gamma}-C$ set is weaker than $\theta$-open set. Furthermore, we prove that the collection of $\Psi_{\Gamma}-C$ set is closed under arbitrary union. Finally, we obtain the conclusion that the collection of $\Psi_{\Gamma}-C$ set forms a supratopology.

Subspace graph topological space of graphs

A graph topology defined on a graph G is a collection 𝒯 of subgraphs of G which satisfies the properties such as K0, G ∈ 𝒯 and 𝒯 is closed under arbitrary union and finite intersection. Let (X, T) be a topological space and Y ⊆ X then, TY = {U ∩ Y : U ∈ T} is a topological space called a subspace topology or relative topology defined by T on Y. In this P1 we discusses the subspace or the relative graph topology defined by the graph topology 𝒯 on a subgraph H of G. We also study the properties of subspace graph topologies, open graphs, d-closed graphs and nbd-closed graphs of subspace graph topologies.

On Grill Sβ-Open Set in Grill Topological Spaces

In this article we originate a new class of Grill Set, namely GSβ-Open Set, which is parallel to the β Open Set in Grill Topological Space (X, θ, G). In addition, we entitle GSβ-continuous and GSβ-open functions by applying a GSβ-Open Set and we review some of its important properties. Many examples are given to explain the concept lucidly. The properties of GSβ open sets are investigated and studied. The theorems based on the arbitrary union and finite intersections are discussed with counter examples. Moreover, some operators like GSβ−closure and GSβ−interior are introduced and investigated. The concept of GSβ−continuous functions are compared with the idea of G−Semi Continuous function. The theorems based on GSβ−continunity have been proved.

Notes on “On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices”

Jiang and Hu characterized (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices with L-fuzzy negations, where L denotes a complete lattice. However, there are some faults in the study of (O,G)-fuzzy rough sets. As there exist complete lattices without strict L-fuzzy negations or L-fuzzy involutive negations, the properties of G-lower L-fuzzy rough approximation operators need be further discussed. Meanwhile, it is wrong that the G-lower L-fuzzy rough approximation operators preserve the arbitrary union of L-fuzzy sets and O-upper L-fuzzy rough approximation operators preserve the arbitrary intersection of L-fuzzy sets. Hence, the characterizations of (O,G)-fuzzy rough sets and multigranulation (O,G)-fuzzy rough sets need be rectifies. For the convenience of readers, a table is provided to show the correspondences among the conclusions.

On Hyper Ideals of Γ – hypernear-ring

In 2010, hyperideals of Γ- hypernear-ring introduced by B. Davvaz [8]. In this paper, arbitrary union and intersection of a family of hyperideals of Γ- hypernear-ring is hyperideal of Γ- hypernear-ring are proved. This study showed that, the quotient Γ-hypernear-ring ( M/I, ⊕, ⊙ ) is Γ-hypernear-ring.
 Key Words: near-ring, subnear-ring, hyperoperation, hypergroupoid, hypernear-rings, Γ – hypernear-ring, hyperideal of Γ – hypernear-ring, and Γ – hypernear-ring homomorphism.

A Generalized Topology from the Edge Set of Maximal Paths of Directed Graphs

This study intends to provide a fundamental step towards studying the properties of directed graphs with their corresponding generalized topological spaces. A generalized topology (GT) \(\mu\) on a nonempty set X is defined as a family of subsets of X such that \(\Theta\) and an arbitrary union of sets in \(\mu\) is in \(\mu\) . In this study, we introduce a new generalized topology generated by the set of edges of maximal paths of the directed graph D called the maximal path edge generalized topology (MPE\(\Gamma\)), denoted by \(\Gamma\)MP (D). The basic topological properties and connectedness in the context of this new structure are explored and illustrated. In particular, this paper established that (E(D); \(\Gamma\)MP (D)) is a strong generalized topological space and characterized the open and closed sets in this space. Moreover, it was seen that the MPE\(\Gamma\) space of every disconnected digraph is \(\Gamma\)MPMP -disconnected and the MPE\(\Gamma\) space of every connected digraph is also characterized.

A Novel Fuzzy Structure: Infra-Fuzzy Topological Spaces

Obtaining a weaker condition that preserves some inspired topological properties is always desirable. As a result, we introduce the concept of infra-fuzzy topology, which is a subset family that degrades the concept of fuzzy topology by omitting the condition of closedness under arbitrary unions. Fundamental properties of infra-fuzzy topological spaces are investigated, including infra-fuzzy open and infra-fuzzy closed sets, infra-fuzzy interior and infra-fuzzy closure operators, and the infra-fuzzy boundary of a fuzzy set. It is not possible to expect the latter concepts to have properties identical to those in ordinary fuzzy topological spaces. More precisely, the infra-fuzzy interior of a set need not be infra-fuzzy open, and the infra-fuzzy closure and boundary of a set may not be infra-fuzzy closed. Then, employing infra-fuzzy neighborhood systems, infra-fuzzy Q-neighborhood systems, the basis of infra-fuzzy topology, and infra-fuzzy relative topology, we propose several approaches for generating infra-fuzzy topologies. Finally, we define the notions of continuity, openness, closedness, and homeomorphism of mappings in the context of infra fuzziness and investigate some of their properties and characterizations. We show that the usual characterization of earlier notions in the infra-fuzzy structure is incorrect. We demonstrate that the family of all infra-fuzzy homeomorphisms on an infra-fuzzy topological space forms a group under mappings composition. We finish this work by proving that each infra-fuzzy homeomorphism between two infra-fuzzy topological spaces produces an isomorphism on groups of infra-fuzzy homeomorphisms of the corresponding spaces.

Total Irregularity Strengths of an Arbitrary Disjoint Union of (3,6)- Fullerenes.

A fullerene graph is a mathematical model of a fullerene molecule. A fullerene molecule, or simply a fullerene, is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Chemical graph theory is a combination of chemistry and graph theory, where theoretical graph concepts are used to study the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has application not only within mathematics but also in computer science, coding theory, medicine, communication networking, chemistry, among other fields. For example, in chemistry, vertex labeling is being used in the constitution of valence isomers and transition labeling to study chemical reaction networks Method: In terms of graphs, vertices represent atoms while edges stand for bonds between the atoms. By tvs (tes) we mean the least positive integer for which a graph has a vertex (edge) irregular total labeling such that no two vertices (edges) have the same weights. A (3,6)-fullerene graph is a non-classical fullerene whose faces are triangles and hexagons Results: Here, we study the total vertex (edge) irregularity strength of an arbitrary disjoint union of (3,6)-fullerene graphs and provide their exact values. The lower bound for tvs (tes) depends on the number of vertices. Minimum and maximum degree of a graph exist in literature, while to get different weights, one can use sufficiently large numbers, but it is of no interest. Here, by proving that the lower bound is the upper bound, we close the case for (3,6)-fullerene graphs.

New Soft Structure: Infra Soft Topological Spaces

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.

Sets with constant normal in Carnot groups: properties and examples

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a metric cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower subRiemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.

On psi _{{mathcal{H}}}( . )-operator in weak structure spaces with hereditary classes

Weak structure space (briefly, wss) has master looks, when the whole space is not open, and these classes of subsets are not closed under arbitrary unions and finite intersections, which classify it from typical topology. Our main target of this article is to introduce psi _{{mathcal {H}}}(.)-operator in hereditary class weak structure space (briefly, {mathcal {H}}wss) (X, w, {mathcal {H}}) and examine a number of its characteristics. Additionally, we clarify some relations that are credible in topological spaces but cannot be realized in generalized ones. As a generalization of w-open sets and w-semiopen sets, certain new kind of sets in a weak structure space via psi _{{mathcal {H}}}(.)-operator called psi _{{mathcal {H}}}-semiopen sets are introduced. We prove that the family of psi _{{mathcal {H}}}-semiopen sets composes a supra-topology on X. In view of hereditary class {mathcal {H}}_{0}, w T_{1}-axiom is formulated and also some of their features are investigated.

Self-injectivity of versus modulo its socle

Let be a field of subsets of a set X and be the ring of all real valued -measurable functions on X. If is closed under arbitrary unions, the self-injectivity of are given by Azadi-Henriksen-Momtahan in 2009. In this article, we improve the latter result and show that is self-injective if and only if is a complete and - additive field of sets. Finally, it is observed that if is a σ-field, modulo its socle is self-injective if and only if is a complete and - additive field of sets with a finite number of atoms.

English

Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R, then R contains one of them under various conditions.

ON EFFICIENCY GAINS FROM MULTIPLE INCOMPLETE SUBSAMPLES

Cost-effective survey methods such as multi(R)-phase sampling typically generate samples that are collections of monotonic subsamples, i.e., the variables observed for the units in subsample r are also observed for the units in subsample r + 1 for r = 1,…,R – 1. These subsamples represent subpopulations that can be systematically different if the selection of a unit in each phase of sampling depends on the observed variables for that unit from past phases. Our article is about optimally combining all the subsamples for the efficient estimation of a finite dimensional parameter defined by moment restrictions on a generic target population that is an arbitrary union of these subpopulations. Only the R-th subsample is assumed to contain all the variables that are arguments of the moment function. Semiparametric efficiency bounds for estimation are obtained under a unified framework, allowing for full generality of the selection on observables in the sampling design. Contribution of each subsample toward efficient estimation is analyzed; and this turns out to differ fundamentally from that in setups where the same collection of subsamples is instead generated unplanned by unknown sampling. Uniquely, our setup enables all the subsamples to contribute to the efficient estimation for all the target populations, which we show is not possible in other setups. Efficient estimation is standard. Simulation evidence of substantive efficiency gains from using all the subsamples is provided for all the targets.

Attribute reducts of multi-granulation information system

In recent years, more and more methods and theories of multi-granulation information systems have been explored. However, there is very limited investigation on the attribute reducts of multi-granulation rough sets. Therefore, the main objective of this paper is to draw attention to the attribute reducts of multi-granulation information system. For any subset of information system, we usually characterize it by its upper and lower approximations. In order to calculate the upper and lower approximations faster, we must reduce the redundant information of the information system. According to the preceding analysis, we first introduce three types of attribute reduct, which are called arbitrary union reduct, neighborhood union reduct and neighborhood intersection reduct, respectively. Then many basic and important results of these reducts are deeply explored. In order to apply the theories of attribute reducts to deal with practical issues, we develop three algorithms so as to compute multi-granulation upper and lower approximations. Next, we further study the interrelationships among these attribute reducts. Finally, we present a multi-granulation information system with respect to thirty students’ exam scores and calculate the corresponding attribute reducts by using the algorithms listed in the paper.

A duality between LM-fuzzy possibility computations and their logical semantics

Let X be a dcpo and let L be a complete lattice. The family σL(X) of all Scott continuous mappings from X to L is a complete lattice under pointwise order, we call it the L-fuzzy Scott structure on X. Let E be a dcpo. A mapping g : σL(E) −> M is called an LM-fuzzy possibility valuation of E if it preserves arbitrary unions. Denote by πLM(E) the set of all LM-fuzzy possibility valuations of E. The denotational semantics assigning to an LM-fuzzy possibility computation from a dcpo D to another one E is a Scott continuous mapping from D to πLM(E), which is a model of non-determinism computation in Domain Theory. A healthy LM-fuzzy predicate transformer from D to E is a sup-preserving mapping from σL(E) to σM(D), which is always interpreted as the logical semantics from D to E. In this paper, we establish a duality between an LM-fuzzy possibility computation and its LM-fuzzy logical semantics.

A New Supervisor Synthesis Approach for Discrete-Event Systems With Infinite Behavior

w-closure (together with w-controllability) plays an important role in the supervisor synthesis for liveness specifications imposed on discrete-event systems (DESs) with infinite behavior. However, it is not closed under arbitrary unions, and thus there does not exist the supremal w-closed (and w-controllable) sublanguage. In this paper, we first develop an algorithm to compute a subautomaton of the automaton representing an arbitrarily given w-language by deleting all bad loops (which lead to un-closedness of the given language) in the automaton, and show that the obtained subautomaton represents an w-closed sublanguage and is maximal from the perspective of an automaton transition graph. With this algorithm, we propose a new approach to construct a supervisor synthesizing an w-language even if it is not w-closed. Furthermore, we prove that, compared with the supervisor computed by the Thistle's supervisor synthesis approach, the supervisor synthesized by our approach is often more permissive. Examples are presented for illustration.