Implication Algebra Research Articles

On join-complete implication algebras

In this paper, first, we consider an algebra that has a binary operation and a join of arbitrary nonempty subset. A lattice implication algebra is a lattice with a binary operation, which has a join and a meet of finite nonempty subsets. In this work, the notion of join-complete implication algebras L is defined as a join-complete lattice with a binary operation, and some properties of this algebra L are searched. Moreover, we prove that the interval [a, 1] in L is a lattice implication algebra and show that L satisfies the completely distributive law when it has the smallest element 0. Finally, we state the concept of filter and multipliers of L and provide finite and infinite examples of them. In addition, we research some properties of these concepts in detail.

A conceptual clustering method for large-scale group decision-making with linguistic truth-valued lattice implication algebra

The increasing complexity of decision-making environments has led to a rise in the involvement of decision-makers (DMs) in group decision-making problems. Clustering is widely used in large-scale group decision-making (LSGDM) to categorize DMs into smaller groups. Ensuring reasonable decision-making results requires providing explanations for the generated groups during the clustering process. To address the clustering problem in LSGDM within uncertain linguistic environments, this paper proposes a conceptual clustering method based on the linguistic concept lattice. The method efficiently manages comparable and incomparable linguistic information. To achieve interpretable clustering results for DMs, attribute and expert induction matrices are first introduced. Cluster stability analysis is then employed to automatically determine the optimal number of clusters. Second, linguistic truth-valued aggregation operators are proposed to aggregate the linguistic evaluation information of DMs in each cluster. In addition, a consensus reaching process is conducted within each cluster, and a feedback mechanism is established to iteratively update clusters when consensus cannot be reached. Finally, numerical examples and comparative analyses are presented that verify the effectiveness of the proposed approach in effectively addressing the LSGDM problem within uncertain linguistic environments.

Varieties of unary-determined distributive $\ell$-magmas and bunched implication algebras

A distributive lattice-ordered magma ($d\ell$-magma) $(A,\wedge,\vee,\cdot)$ is a distributive lattice with a binary operation $\cdot$ that preserves joins in both arguments, and when $\cdot$ is associative then $(A,\vee,\cdot)$ is an idempotent semiring. A $d\ell$-magma with a top $\top$ is unary-determined if $x{\cdot} y=(x{\cdot}\!\top\wedge y)$ $\vee(x\wedge \top\!{\cdot}y)$. These algebras are term-equivalent to a subvariety of distributive lattices with $\top$ and two join-preserving unary operations $\mathsf p,\mathsf q$. We obtain simple conditions on $\mathsf p,\mathsf q$ such that $x{\cdot} y=(\mathsf px\wedge y)\vee(x\wedge \mathsf qy)$ is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models. We find all subdirectly irreducible algebras up to cardinality eight in which $\mathsf p=\mathsf q$ is a closure operator, as well as all finite unary-determined bunched implication chains and map out the poset of join-irreducible varieties generated by them.

On categorical structures arising from implicative algebras: From topology to assemblies

Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.

On Implicative and Positive Implicative GE Algebras

GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebrasverifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented.

A concept lattice-based expert opinion aggregation method for multi-attribute group decision-making with linguistic information

During the multi-attribute group decision-making (MAGDM) processing, the individuals often hold different opinions about the alternatives. It is necessary to aggregate the different individual opinions into a unified group opinion. In the real world, experts sometimes use linguistic expressions to evaluate attributes in uncertain environments. To address the problem of reducing the information loss of expert opinion aggregation in MAGDM, this paper proposes a MAGDM approach based on linguistic concept lattices in the context of uncertain linguistic expression. A linguistic concept lattice for multi-expert linguistic formal context is first constructed based on linguistic truth-valued lattice implication algebra, which can express both comparable and incomparable linguistic information in the decision-making process. Different expert opinions are aggregated via the extent of fuzzy linguistic concepts, which can reduce information loss in the aggregation process. Second, meet-irreducible elements in the linguistic concept lattice are introduced to reduce the computational complexity of obtaining all fuzzy linguistic concepts in the decision-making process. the distance between the intents of different fuzzy linguistic concepts is considered to enhance the rationality of linguistic decision results. In addition, the expert’s decision-making process for each alternative is visualized via linguistic concept lattices. Finally, the case study and comparative analysis illustrate the validity and rationality of the proposed approach in MAGDM with linguistic information.

괴델과 우카시에비치의 다치논리

Gödel and Łukasiewicz proposed the three-valued logic by adding the third logical situation, which includes uncertainty and ambiguity, to the classical two logical values, true or false. These logical systems were generalized to the many types of many valued logics, and especially, Gödel’s many valued logic was developed to Heyting algebra and Łukasiewicz’s one to lattice implication algebra. In this paper, we introduce the many valued logics of Gödel and Łukasiewicz, and Heyting’s algebra and lattice implication algebra that are generalizations of Gödel’s and Łukasiewicz’s logic, respectively. Also, we research the properties and relationship of Heyting algebras and lattice implication algebras, especially by defining another implication on a finite lattice implication algebra, we prove finite implication algebra is a special case of Heying algebras.

On Boolean elements and derivations in 2-dimension linguistic lattice implication algebras

A 2-dimension linguistic lattice implication algebra (2DL-LIA) can build a bridge between logical algebra and 2-dimension fuzzy linguistic information. In this paper, the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed. Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated. Moreover, it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.

Application of Neutrosophic Filters in Lattice Implication Algebra

Neutrosophic set theory is applied to lattice implication algebra, and the concept of neutrosophic filters and neutrosophic lattice filters in lattice implication algebra are introduced. Several properties are investigated. Characterizations of a neutrosophic filter are discussed. Finally, we proved that every neutrosophic filter is a neutrosophic lattice filter, but the converse is invalid.

Application of Neutrosophic implicative filters and Neutrosophic positive implicative filters in Lattice implication algebra

We introduced Neutrosophic implicative filters and Neutrosophic positive implicative filters in Lattice implication algebra. We proved some properties and equivalent conditions of both the filters. Finally we proved that “Every Neutrosophic positive implicative filter is a Neutrosophic implicative filter” and “Every Neutrosophic positive implicative filter is a Neutrosophic filter”.

Properties of Homomorphism and Quotient Implication Algebra on Implication Algebras

The concept of homomorphisms on implication algebra is introduced. The notion of sub algebras, normal subalgebras in an implication algebra are investigated. Quotient implication algebras and kernels in an implication algebra ,and Homomorphisms and isomorphism theorems are elaborated.

MℬJ-Filters on Lattice Implication Algebras

We explored the some equivalent conditions and properties of MℬJ-filters of Lattice implication algebras. As well we define MℬJ-lattice filters of Lattice implication algebras. Finally, it was established that MℬJ-Lattice filter is an MℬJ-filter, but not the other way around.

A variety of algebras closely related to subordination algebras

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

Chinese Research on Mathematical Logic and the Foundations of Mathematics

This paper outlines the Chinese research on mathematical logic and the foundations of mathematics. Firstly, it presents the introduction and spread of mathematical logic in China, especially the teaching and translation of mathematical logic initiated by Bertrand Russell’s lectures in the country. Secondly, it outlines the Chinese research on mathematical logic after the founding of the People’s Republic of China. The research in this period experienced a short revival under the criticism of the Soviet Union, explorations under the heavy influence of the Cultural Revolution, and the vigorous development of mathematical logic teaching and research after the period of “Reform and Opening Up” that started in the late 1970s, and the full integration of Chinese mathematical logic research into the international academic circle in the new century after 2000. In the third part, it focuses on the unique and original results of the Chinese mathematical logic research teams from the following three aspects: medium logic, lattice implication algebras and their lattice-valued systems of logic, and Chinese notation of logical constants, which can be used as a substantive supplement to the relevant literature on the history of mathematical logic in China. The last part is a reflection on the shortcomings of contemporary Chinese research on mathematical logic and the foundations of mathematics.

Positive implicative equality algebras and equality algebras with some types

The notion of a positive implicative equality algebras are defined, and related properties are studied. Characterizations of a positive implicative equality algebra is investigated. Conditions for an equality algebra to be positive implicative are provided. Equality algebra with some types is considered, and several properties are investigated. Using equality algebra with some types, we characterize a commutative equality algebra and a positive implicative algebra.

Implicative equality algebras and annihilators in equality algebras

In this paper, the concept of an implicative equality algebra is defined and related properties are investigated. Characterizations of an implicative equality algebra and the relation among implicative and positive implicative equality algebras are discussed. Also, we define the notion of annihilator of equality algebras and investigate some traits of it and prove that annihilator of any nonempty set of equality algebras is a deductive system. Moreover, by using this notion, we define the operation and prove that for any commutative equality algebra is a bounded implicative BCK-algebra.

Weak LI-ideals in implicative almost distributive lattices

In the field of many valued logic, lattice valued logic (especially ideals) plays an important role. Nowadays, lattice valued logic is becoming a research area. Researchers introduced weak LI-ideals of lattice implication algebra. Furthermore, other scholars researched LI-ideals of implicative almost distributive lattice. Therefore, the target of this paper was to investigate new development on the extension of LI-ideal theories and properties in implicative almost distributive lattice. So, in this paper, the notion of weak LI-ideals and maximal weak LI- ideals of implicative almost distributive lattice are defined. The properties of weak LI- ideals in implicative almost distributive lattice are studied and several characterizations of weak LI-ideals are given. Relationship between weak LI-ideals and weak filters are explored. Hence, the extension properties of weak LI-ideal of lattice implication algebra to that of weak LI-ideal of implicative almost distributive lattice were shown.

Hilbert Implication Algebra and Some Properties

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.

Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.

Research on the Disease Intelligent Diagnosis Model Based on Linguistic Truth-Valued Concept Lattice

Uncertainty natural language processing has always been a research focus in the artificial intelligence field. In this paper, we continue to study the linguistic truth-valued concept lattice and apply it to the disease intelligent diagnosis by building an intelligent model to directly handle natural language. The theoretical bases of this model are the classical concept lattice and the lattice implication algebra with natural language. The model includes the case library formed by patients, attributes matching, and the matching degree calculation about the new patient. According to the characteristics of the patients, the disease attributes are firstly divided into intrinsic invariant attributes and extrinsic variable attributes. The calculation algorithm of the linguistic truth-valued formal concepts and the constructing algorithm of the linguistic truth-valued concept lattice based on the extrinsic attributes are proposed. And the disease bases of the different treatments for different patients with the same disease are established. Secondly, the matching algorithms of intrinsic attributes and extrinsic attributes are given, and all the linguistic truth-valued formal concepts that match the new patient’s extrinsic attributes are found. Lastly, by comparing the similarity between the new patients and the matching formal concepts, we calculate the best treatment options to realize the intelligent diagnosis of the disease.