Prime Filters Research Articles

\(\mathcal{L}\)−weakly 1−Absorbing Prime Ideals and Filters

In this manuscript, we have presented the concept of \(\mathcal{L}\)-weakly 1-absorbing prime ideals and \(\mathcal{L}\)-weakly 1-absorbing prime filters within an ADL. Mainly, we illustrate the connections between \(\mathcal{L}\)-weakly prime ideals (filters) and \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters), as well as between \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) and \(\mathcal{L}\)-weakly 2-absorbing ideals (filters). Lastly, we have shown that both the image and inverse image of \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters) result in \(\mathcal{L}\)-weakly 1-absorbing prime ideals (filters).

Fuzzy prime filter theorem in multilattices

This paper mainly focuses on building the fuzzy prime filter theorem for multilattices. Firstly, we introduce the notion of a fuzzy filter generated by a fuzzy subset of a multilattice and we give a characterization. Also, we define four types of fuzzy prime filters and establish some relationships between them. Finally, we state and prove the fuzzy prime filter theorem in distributive multilattices.

The L-Fuzzy Prime Filter Degrees on Lattices and Its Induced L-Fuzzy Convex Structure

The aim of this paper is to examine the L-fuzzy prime filter degrees on lattices and their induced L-fuzzy convex structure. Firstly, the notion of L-fuzzy prime filter degrees on lattices is established using the implication operator when L is a completely distributive lattice. Secondly, an equivalent characterization of L-fuzzy prime filter degrees on lattices is provided. The equivalence relation, through the definitions of reflexivity, symmetry, and transitivity, provides a method for partitioning subsets within a lattice that possesses the prime filter property. Finally, the L-fuzzy convex structure induced by the L-fuzzy prime filter degrees on lattices is examined. Simultaneously, the properties of L-fuzzy prime filter degrees on lattices in relation to images and preimages under homomorphic mappings are discussed.

Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the Gödel-Dummett axiom (p→q)∨(q→p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-GD) of extensions of bi-GD.We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of bi-GD. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of bi-GD. We introduce a sequence of co-trees, called the finite combs, and show that a logic in Λ(bi-GD) is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of bi-GD and consequently, a unique pre-locally tabular extension of bi-GD. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.

Normal Paradistributive Latticoids

For any filter $\LomP$ of a paradistributive latticoid, $\LomO(\LomP)$ is defined and it is proved that $\LomO(\LomP)$ is a filter if $\LomP$ is prime. It is also proved that each minimal prime filter belonging to $\LomO(\LomP)$ is contained in $\LomP$, and $\LomO(\LomP)$ is the intersection of all the minimal prime filters contained in $\LomP$. The concept of a normal paradistributive latticoid is introduced and characterized in terms of the prime filters and minimal prime filters. We proved that every relatively complemented paradistributive latticoid is normal.

The Space of Fuzzy Minimal Prime Filters In an Almost Distributive Lattice

In this paper, we establish and describe the notions of fuzzy minimal prime filters of an Almost Distributive Lattice (ADL). We prove that a fuzzy filter is the point wise infimum of all minimal prime fuzzy filters (all fuzzy minimal prime filters) of an ADL. Mainly, we introduce the topological space on the set of all fuzzy minimal prime filters of an ADL (denoted by M(R)). For an ADL R, we prove that an open set M(R) is a base for the topology and it constitutes a base for the subspace topology on a closed set N and they are the only compact open sets.

Mp-Residuated Lattices

This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime filter contains a unique minimal prime filter. A combination of algebraic and topological methods is applied to obtain new and structural results on mp-residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Especially, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, equipped with the dual hull-kernel topology, is normal. The class of mp-residuated lattices is characterized by means of pure filters. It is shown that a residuated lattice is mp if and only if its pure filters are precisely its minimal prime filters, if and only if its pure spectrum is homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel topology.

Study hoop algebras by fuzzy (n-fold) obstinate filters

Abstract In this article, we introduce the concept of fuzzy (n-fold) obstinate filter on hoop algebras and study some of the properties. We define and study fuzzy prime filter and fuzzy n-fold implicative filter on hoop algebras. Also, the relationship between fuzzy obstinate filter and some other fuzzy filters likeness fuzzy prime and fuzzy positive implicative filters are investigated. Then we show that the quotient of this structure is a Boolean algebra and obtained some condition equivalent with fuzzy n-fold implicative filter. Finally we show that every fuzzy n-fold obstinate filter with some conditions is a fuzzy n-fold implicative filter.

Quasihyperarchimedean BL-algebras

In this paper, we introduce a new class of BL-algebras called quasihyperarchimedean (q-hyperarchimedean) BL-algebras and investigate some related properties. In addition, we consider and study filters in BL-algebras whose quotients are q-hyperarchimedean. Finally, we explore the relations between the introduced classes of filters, especially their connections to prime filters, radical filters and maximal filters.

The spectra of quasi-Boolean algebras

Abstract In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal filters of quasi-Boolean algebras, showing that the prime filters of a quasi-Boolean algebra are precisely the maximal filters and the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space. Finally, we define and study the reticulation of a quasi-Boolean algebra.

P3FA: Unified Unicast/Multicast Forwarding Algorithm for High-Performance Router/Switch

High-performance multicast packet switching technologies are evolving to meet the growing demand for scalability on the Internet and datacenters, etc. Implementing a high-performance switch/router relies on a polynomial-time group membership query algorithm within the Packet Forwarding Engines (PFEs) to determine whether a packet is forwarded through an egress. Among these, Bloom filter (BF)-based and Residue Number System (RNS)-based are being considered as two representatives of the membership query algorithms. However, both approaches suffer from some fatal weaknesses such as false-positive probability and time inefficiencies, especially for a carrier-grade PFE with high port-density features. According to similar properties of the RNS, we propose a simplified forwarding algorithm in this paper, named Per-Port Prime Filter Array (P3FA). The simulation results indicate that the P3FA can significantly improve space efficiencies under specific lower egress-diversities conditions. Under the same space constraints, P3FA improves multicast and unicast time efficiency by 1 to 4 orders of magnitude in the port-density 16–1024 range compared to previous works. Although it comes at the expense of hardware cost, it is still acceptable compared to recently improved previous work.

Spectra and reticulation of semihoops

Abstract In this article, we further study the filter theory of semihoops. Moreover, we use the prime (maximal) filters to construct the prime (maximal) spectrum on semihoops, and prove that the prime spectrum is a compact T 0 {T}_{0} topological space and that the maximal spectrum is a compact T 2 {T}_{2} topological space. As an application, in order to study the relationship between the spectrum of semihoops and the spectrum of lattices, we introduce the reticulations of semihoops and obtain some related results.

ℒ -Fuzzy Cosets of ℒ -Fuzzy Filters of Residuated Multilattices

This paper mainly focuses on building the ℒ -fuzzy filter theory of residuated multilattices. Firstly, we introduce the concepts of ℒ -fuzzy filter and ℒ -fuzzy deductive system of residuated multilattices. Then, we highlight their properties and show how they are linked. Secondly, we introduce the concept of prime ℒ -fuzzy filter and propose some illustrative examples. Then, we bring out their properties and show how they are related to the concept of ℒ -fuzzy prime filter. Thirdly, we characterize ℒ -fuzzy maximal filter and maximal ℒ -fuzzy filter by atoms and coatoms. In the case where ℒ is a distributive lattice, we prove that maximal ℒ -fuzzy filters are prime. Finally, we are interested in ℒ -fuzzy cosets of an ℒ -fuzzy filter, and we prove that the set of all ℒ -fuzzy cosets of any ℒ -fuzzy filter of a residuated multilattice is a residuated multilattice.

Three types of prime filters in strong quasi-ordered residuated system

The concept of quasi-ordered residuated system was introduced in 2018 by Bonzio and Chajda. The author introduced and analyzed the concept of filters as well as some types of filters in such an algebraic system. Additionally, the author also dealt with a strong quasi-ordered residuated system in which he determined prime and irreducible filters. In this paper, the author introduce three types of prime filters in a strong quasi-ordered residuated system and analyze their interconnections.

The Prime Filter Theorem for Multilattices

The aim of this paper is to establish the prime filter theorem for multilattices.

Study strong Sheffer stroke non-associative MV-algebras by fuzzy filters

In this paper, some types of fuzzy filters of a strong Sheffer stroke non-associative MV-algebra (for short, strong Sheffer stroke NMV-algebra) are introduced. By presenting new properties of filters, we define a prime filter in this algebraic structure. Then (prime) fuzzy filters of a strong Sheffer stroke NMV-algebra are determined and some features are proved. Finally, we built quotient strong Sheffer stroke NMV-algebra by a fuzzy filter.

An extension of Jónsson‐Tarski representation and model existence in predicate non‐normal modal logics

AbstractWe give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas.

Stone Duality for Kolmogorov Locally Small Spaces

In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.

The hull-kernel topology on prime filters in residuated lattices

The hull-kernel topology on prime filters in residuated lattices

Two Types of Prime UP-Filters of Meet-Commutative UP-Algebras

The concept of meet-commutative UP-algebra was introduced by Sawika et al. in 2016 and the concept of prime filters in such algebras was introduced and analyzed by Muhiuddin, Romano and Jun. In this article, as a continuation of the previous one, we introduce one more type of prime UP-filters in meet-commutative UP-algebras and we analyze the interrelationships of these two types of prime UP-filters.