Nondistributive Lattices Research Articles

Characterizations for convolution lattices based on non-distributive lattices

Recently, De Miguel, Bustince and De Baets have conducted a systematic study on convolution lattices based on distributive lattices. There have been few reports on applying non-distributive lattices to a domain of functions. As a complement to their work, in this paper, we carry out an in-depth investigation of convolution operations of the functions between a non-distributive lattice (domain) and a frame (co-domain). We first present an equivalence characterization between non-distributive lattices and idempotent functions and further show that a subset of the set of idempotent functions is closed under convolution operations. We demonstrate that this subset also is a bisemilattice and satisfies the Birkhoff equation under join- and meet-convolution operations. Finally, we analyze and study the lattice structure related to the obtained algebraic structure.

Remarks on Sugeno Integrals on Bounded Lattices

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study the mentioned unique extension property concerning the possible extension of a Sugeno integral to non-distributive lattices. We show that this property is equivalent to the distributivity of the underlying bounded lattice. As a byproduct, an alternative proof of Iseki’s result, stating that a lattice having prime ideal separation property for every pair of distinct elements is distributive, is provided.

On the Common Logical Structure of Classical and Quantum Mechanics

At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this question by showing that the original arguments promoting quantum logic contain serious flaws, and that quantum theory does satisfy the classical distributivity law once the full meaning of quantum propositions is properly taken into account. Moreover, we show that quantum mechanics can generate a distributive lattice of propositions, which, unlike the one of quantum logic, includes statements about expectation values which are of undoubtable physical interest. Lastly, we show that the lattice of statistical propositions in classical mechanics follows the same structure, yielding an analogue non-commutative sublattice of classical propositions. This fact entails that the purported difference between classical and quantum logic stems from a misconstructed parallel between the two theories.

Implication in finite posets with pseudocomplemented sections

It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced so-called sectionally pseudocomplemented lattices, i.e. lattices with top element 1 where for every element y the interval [y, 1], the so called section, is pseudocomplemented. We extend this concept to posets with top element. Our goal is to show that such a poset can be considered as an algebraic semantics for a certain kind of more general intuitionistic logic provided an implication is introduced as shown in the paper. We prove some properties of such an implication. This implication is “unsharp” in the sense that the value for given entries need not be a unique element, but may be a subset of the poset in question. Using this implication we show that we can even recover the order of the original poset. Further, a new “unsharp” operator odot of conjunction can be introduced which is adjoint to “unsharp” implication and hence we obtain an “unsharp” residuated poset.

New Examples of Radical Join-Meet Ideals

For non-distributive lattice L, it is not yet known classes of L with the property that the join-meet ideal of L is radical. In this paper, we give a partial answer to this problem.

Kleene posets and pseudo-Kleene posets

The concept of a Kleene algebra (sometimes also called Kleene lattice) was already generalized by the first author for non-distributive lattices under the name pseudo-Kleene algebra. We extend these concepts to posets and show how (pseudo-)Kleene posets can be characterized by identities and implications of assigned commutative meet-directoids. Moreover, we prove that the Dedekind-MacNeille completion of a pseudo-Kleene poset is a pseudo-Kleene algebra and that the Dedekind-MacNeille completion of a finite Kleene poset is a Kleene algebra. Further, we introduce the concept of a strict (pseudo-)Kleene poset and show that under an additional assumption a strict Kleene poset can be organized into a residuated structure. Finally, we prove by using the so-called twist construction that every poset can be embedded into a pseudo-Kleene poset in some natural way.

Non-distributive Relatives of ETL and NFL

In this paper we devise non-distributive relatives of Exactly true logic (ETL) by Pietz and Riveccio and its dual Non-falsity logic (NFL) by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL’s and NFL’s entailment relations on the de Morgan lattice 4. We generalise these pre-orders and determine which distributive properties that hold on 4 are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics we also devise a truth table semantics which uses non-distributive lattice M3 as their lattice of truth values. We also provide analytic tableaux systems which work with sequents of the form $$\phi \vdash \chi $$. We then prove correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices.

Relatively residuated lattices and posets

Abstract It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.

Duality Results for (Co)Residuated Lattices

We present dualities (discrete duality, duality via truth and Stone duality) for implicative and (co)residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orlowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as well as full Stone duality for (co)residuated lattices. Our results have immediate applications to the semantics of related substructural (resource consious) logical calculi.

New inclusion relation of neutrosophic sets with applications and related lattice structure

The main purpose of this paper is to study the inclusion relations of neutrosophic sets and some applications in multiple attribute decision making. At first, the existing two definitions of inclusion relation (called type-1 and type-2 inclusion relation, respectively) of neutrosophic sets are analyzed, and the deficiencies of type-1 and type-2 inclusion relations are illustrated by examples (in fact, they are actually two extreme cases). Second, a new definition of inclusion relation of neutrosophic sets (call it type-3 inclusion relation) is introduced, and a new method of ranking of neutrosophic sets is given. The effectiveness of the ranking method is presented by some application examples in multiple attribute decision making. Finally, type-3 inclusion relation of neutrosophic sets and related lattice structure are investigated in a systematic way, the definitions of type-3 union and type-3 intersection operations are proposed, and the following important result is proved: all of neutrosophic sets based on a certainty domain constitute a generalized De Morgan algebra (non-distributive lattice) with respect to type-3 union, type-3 intersection and complement operations. From this, the essential difference between neutrosophic set and fuzzy set (and intuitionistic fuzzy set) is clarified theoretically.

A partial solution to an open problem of Amadio and Curien

Amadio raised the question of whether the category of stable bifinite domains of Amadio–Droste is the largest Cartesian closed full subcategory of the category of ω-algebraic meet-cpos with stable functions. Zhang and Jiang showed that, for any ω-algebraic meet-cpo D, if the stable function space [D→sD] (with stable order) satisfies property M, then D is finitary (i.e. each compact element dominates only finitely many elements). In this paper, we show that, for any ω-algebraic meet-cpo D and the diamond lattice M (the classical non-distributive lattice), if [D→sM] and [[D→sM]→s[D→sM]] are ω-algebraic, then(1)D is finitary;(2)if [[D→sM]→s[D→sM]] is finitary, then D is stable bifinite. So the category of stable bifinite domains is a maximal Cartesian closed full subcategory of the category of ω-algebraic meet-cpos with stable functions. This gives a partial solution to the problem.

Some Remarks on Quantum Logics and Quantum Mechanics

Some remarks on the use of quantum logic and corresponding nondistributive lattices in macroscopic situations—in the theory of automata and macroscopic quantum games—are discussed. Comparison with microscopic quantum mechanics is made.

A Characterization of a Semimodular Lattice

A geometric lattice is the lattice of closed subsets of a closure operator on a set which is zero-closure, algebraic, atomistic and which has the so-called exchange property. There are many profound results about this type of lattices, the most recent one of which, due to Czedli and Schimdt (Adv Math 225:2455–2463, 2010), says that a lattice L of finite length is semimodular if and only if L has a cover-preserving embedding into a geometric lattice G of the same length. The goal of our paper is to offer the following result: a lattice of finite length is semimodular if and only if every cell in L is a 4-element Boolean lattice and the 7-element non-distributive atomistic lattice having 3 atoms is not a cover-preserving sublattice of L.

Effective categoricity for distributive lattices and Heyting algebras

We study complexity of isomorphisms between computable copies of lattices and Heyting algebras. For a computable ordinal α, the Δ α 0 dimension of a computable structure S is the number of computable copies of S, up to Δ α 0 computable isomorphism. The results of Goncharov, Harizanov, Knight, McCoy, Miller, Solomon, and Hirschfeldt, Khoussainov, Shore, Slinko imply that for every computable successor ordinal α and every non-zero natural number n, there exists a computable non-distributive lattice with Δ α 0 dimension n. In this paper, we prove that for every computable successor ordinal α ≥ 4 and every natural number n > 0, there is a computable distributive lattice with Δ α 0 dimension n. For a computable successor ordinal α ≥ 2, we build a computable distributive lattice M such that the categoricity spectrum of M is equal to the set of all PA degrees over O(α). We also obtain similar results for Heyting algebras.

Homogeneous Modular Lattices are Distributive

We prove the claim in the title based on failure of amalgamation for classes of non-distributive modular lattices.

Abstract conflict driven learning

Modern satisfiability solvers implement an algorithm, called Conflict Driven Clause Learning, which combines search for a model with analysis of conflicts. We show that this algorithm can be generalised to solve the lattice-theoretic problem of determining if an additive transformer on a Boolean lattice is always bottom. Our generalised procedure combines overapproximations of greatest fixed points with underapproximation of least fixed points to obtain more precise results than computing fixed points in isolation. We generalise implication graphs used in satisfiability solvers to derive underapproximate transformers from overapproximate ones. Our generalisation provides a new method for static analysers that operate over non-distributive lattices to reason about properties that require disjunction.

CONGRUENCE LIFTING OF SEMILATTICE DIAGRAMS

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted Mn). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

Intuitionistic Quantum Logic of an n-level System

A decade ago, Isham and Butterfield proposed a topos theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M_n(C) of complex n x n matrices. This leads to an explicit expression for the pointfree quantum phase space and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen-Specker Theorem. In our approach, the nondistributive lattice P(M_n(C)) of projections in M_n(C)(which forms the basis of the traditional quantum logic of Birkhoff and von Neumann)is replaced by a specific distributive lattice of functions from the poset of all unital commutative C*-subalgebras of M_n(C) to P(M_n(C)). The latter lattice is essentially the (pointfree) topology of the quantum phase space mentioned above, and as such defines a Heyting algebra. Each element of the lattice corresponds to a ``Bohrified'' proposition, in the sense that to each classical context it associates a yes-no question pertinent to this context, rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.

Standard Logics Are Valuation-Nonmonotonic

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of non-orthomodular and thus non-distributive lattices that properly contain standard orthomodular and Boolean classes, respectively. In this article we prove that these logics are complete even for those classes of the former lattices from which the standard orthomodular lattices and Boolean algebras are excluded. We also show that neither quantum nor classical computers can be founded on the latter models. It follows that logics are valuation-nonmonotonic in the sense that their possible models (corresponding to their possible hardware implementations) and the valuations for them drastically change when we add new conditions to their defining conditions. These valuations can even be completely separated by putting them into disjoint lattice classes by a technique presented in the paper.

Nondiversity in substructures

AbstractFor a model of Peano Arithmetic, let Lt() be the lattice of its elementary substructures, and let Lt+ () be the equivalenced lattice (Lt(),≅), where ≅ is the equivalence relation of isomorphism on Lt(). It is known that Lt+() is always a reasonable equivalenced lattice.Theorem. Let L be a finite distributive lattice and let (L, E) be reasonable. If 0 is a nonstandard prime model of PA, then 0 has a cofinal extension such that Lt+() ≅ (L,E).A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.