Lattice-theoretic Approach Research Articles

Strategic complementarity in games

The lattice-theoretic approach has had a significant impact in all fields of economics, being progressively incorporated into the standard toolbox. This paper presents a selective survey with an emphasis on basic tools, some important results, and applications in industrial organization, dynamic games, games of incomplete information, and mechanism design. Frontier theoretical research employing lattice-theoretic methods continues to be developed in areas such as mean-field games and information design.

On Fuzzy Negations and Laws of Contraposition. Lattice of Fuzzy Negations

Summary This the next article in the series formalizing the book of Baczyński and Jayaram “Fuzzy Implications”. We define the laws of contraposition connected with various fuzzy negations, and in order to make the cluster registration mechanism fully working, we construct some more non-classical examples of fuzzy implications. Finally, as the testbed of the reuse of lattice-theoretical approach, we introduce the lattice of fuzzy negations and show its basic properties.

Some further applications of a lattice theoretic method in the study of singular LCM matrices

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S={x1,x2,…,xn} be a set of positive integers with x1<x2<⋯<xn. The LCM matrix [S] on the set S is the n×n matrix with lcm(xi,xj) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set S (which means that gcd⁡(xi,xj)∈S for all i,j∈{1,2,…,n}), suffices to guarantee the invertibility of the matrix [S]. However, a few years later this conjecture was proven false first by Haukkanen et al. and then by Hong. It turned out that the conjecture holds only on GCD closed sets with at most 7 elements but not in general for larger sets. However, the given counterexamples did not give much insight on why does the conjecture fail exactly in the case when n=8. This situation was later improved in a couple of articles, where a new lattice theoretic approach was introduced (the method is based on the fact that because the set S is assumed to be GCD closed, the structure (S,|) actually forms a meet semilattice). For example, it has been shown that in the case when the set S has 8 elements and the matrix [S] is singular, there is only one option for the semilattice structure of (S,|), namely the cube structure.Since the cases n≤8 have been thoroughly studied in various articles, the next natural step is to apply the methods to the case n=9. This was done by Altınışık and Altıntaş as they consider the different lattice structures of (S,|) with nine elements that can result in a singular LCM matrix [S]. However, their investigation leaves two open questions, and the main purpose of this presentation is to provide solutions to them. We shall also give a new lattice theoretic proof for a result referred to as Sun's conjecture, which was originally proven by Hong via number theoretic approach.

A Lattice Theoretic Approach on Prime Topological Spaces

A Lattice Theoretic Approach on Prime Topological Spaces

A lattice-theoretical approach to extensions of filters in algebras of substructural logic

Commutative bounded integral residuated lattices (residutaed lattices, in short) form a large class of algebras containing algebras which are algebraic counterparts of certain propositional fuzzy logics. The paper deals with the so-called extended filters of filters of residuated lattices. It is used the fact that the extended filters of filters associated with subsets coincide with those associated ones with corresponding filters. This makes it possible to investigate the set of all extended filters of residuated lattices within the Heyting algebras of their filters by means of the structural methods of the theory of such algebras.

Brane involutions on irreducible holomorphic symplectic manifolds

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a K3 or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a K3 surface and K3[2]-type manifolds.

Lattice theoretical approach in strongly correlated electron systems

The effective lattice models in strongly correlated electron systems are \emph{derived} in particular for the cuprate superconductors, that incorporate the quantum fluctuations of the spin Berry's phase and the antiferromagnetic fluctuation. Consistent with the field-theoretical approach, the density modulation, the weak ferromagnetism, and the superconductivity are reproduced. We discussed the pros and cons of the effective-model approach and demonstrate that both positive and negative Hubbard model are the effective models subject to the occurrence of the quantum fluctuations of the correlation degrees of freedom.

A lattice-theoretic approach to the Bourque–Ligh conjecture

ABSTRACTThe Bourque–Ligh conjecture states that if is a gcd-closed set of positive integers with distinct elements, then the LCM matrix is invertible. It is known that this conjecture holds for but does not generally hold for . In this paper, we study the invertibility of matrices in a more general matrix class, join matrices. At the same time, we provide a lattice-theoretic explanation for this solution of the Bourque–Ligh conjecture. In fact, let be a lattice, let be a subset of P and let be a function. We study under which conditions the join matrix on S with respect to f is invertible on a meet closed set S (i.e. .

Efficient Rough Set Theory Merging

Theory exploration is a term describing the development of a formal approach to selected topic, usually within mathematics or computer science, with the help of an automated proof-assistant. This activity however usually doesn't reflect the view of science considered as a whole, not as separated islands of knowledge. Merging theories essentially has its primary aim of bridging these gaps between specific disciplines. As we provided formal apparatus for basic notions within rough set theory (as e.g. approximation operators and membership functions), we try to reuse the knowledge which is already contained in available repositories of computer-checked mathematical knowledge, or which can be obtained in a relatively easy way. We can point out at least three topics here: topological aspects of rough sets – as approximation operators have properties of the topological interior and closure; possible connections with formal concept analysis; lattice-theoretic approach giving the algebraic viewpoint (e.g. Stone algebras). In the first case, we discovered semiautomatically some connections with Isomichi's classification of subsets of a topological space and with the problem of fourteen Kuratowski sets. This paper is also a brief description of the computer source code which is a feasible illustration of our approach – nearly two thousand lines containing all the formal proofs (essentially we omit them in the paper). In such a way we can give the formal characterization of rough sets in terms of topologies or orders. Although fully formal, still the approach can be revised to keep the uniformity all the time.

A lattice-theoretic approach to multigranulation approximation space.

In this paper, we mainly investigate the equivalence between multigranulation approximation space and single-granulation approximation space from the lattice-theoretic viewpoint. It is proved that multigranulation approximation space is equivalent to single-granulation approximation space if and only if the pair of multigranulation rough approximation operators forms an order-preserving Galois connection, if and only if the collection of lower (resp., upper) definable sets forms an (resp., union) intersection structure, if and only if the collection of multigranulation upper (lower) definable sets forms a distributive lattice when n = 2, and if and only if . The obtained results help us gain more insights into the mathematical structure of multigranulation approximation spaces.

On the conditional independence implication problem: A lattice-theoretic approach

Conditional independence is a crucial notion in the development of probabilistic systems which are successfully employed in areas such as computer vision, computational biology, and natural language processing. We introduce a lattice-theoretic framework that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI statements and a finite, sound and complete inference system relative to semi-lattice inclusions is presented. This system is shown to be (1) sound and complete for inferring general from saturated CI statements and (2) complete for inferring general from general CI statements. We also show that the general probabilistic CI implication problem can be reduced to that for elementary CI statements. The completeness of the inference system together with its lattice-theoretic characterization yields a criterion we can use to falsify instances of the probabilistic CI implication problem as well as several heuristics that approximate this falsification criterion in polynomial time. We also propose a validation criterion based on representing constraints and sets of constraints as sparse 0–1 vectors which encode their semi-lattices. The validation algorithm works by finding solutions to a linear programming problem involving these vectors and matrices. We provide experimental results for this algorithm and show that it is more efficient than related approaches.

ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

Fleury's spanning dimension and chain conditions on non-essential elements in modular lattices

Based on a lattice theoretical approach, we give a complete characterization of modules with Fleury’s spanning dimension. An example of a non-Artinian, non-hollow module satisfying this finiteness condition is constructed. Furthermore we introduce and characterize the dual notion of Fleury’s spanning dimension.

When does an AB5*module have finite hollow dimension?

Using a lattice-theoretical approach we find characterizations of modules with finite uniform dimension and of modules with finite hollow dimension.

Lattice Theory and the Consumer's Problem

This paper explores and explains the application of the lattice theoretic approach to classic comparative statics in consumer theory. Through examples of preferences that are not quasiconcave, or not differentiable, or not continuous, the approach is shown to characterize income effects more powerfully than the standard approach. The underlying partial order is key to applying the method. Therefore, several adapted partial orders are introduced and discussed.

Lattices of Radicals of Involution Rings

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.

Lattices of Radicals of Involution Rings

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We show that certain sublattices are isomorphic to sublattices of the lattice of radicals of rings. We characterize the atoms of certain lattices of radicals of involution rings.

Confluence and Koszulity

A lattice-theoretic approach to Koszulity is well known. This paper gives a lattice-theoretic approach to polynomiality, i.e., existence of a P.B.W. basis in Priddy's sense. This provides a straightforward proof for Priddy's theorem. Polynomiality is interpreted in terms of confluence in noncommutative computational algebra. The basic notion of a reduction operator is investigated in some detail. The reduction algebras are introduced to study the confluence of two reduction operators in terms of representation theory. The meet and the join of two reduction operators are constructed in an algorithmic way. A geometric characterization of the confluence is obtained.

Lattice theory of crystal edges and bars

In this paper, a lattice theoretical approach to the structure and dynamics of crystal edges is presented. The relaxation of infinite quadratic bars is investigated in detail using various lattice-dynamical models. The phonon spectra of the bars have been calculated and phonon modes have been found which are strongly localized at the tips of the four edges of the bar. Our results obtained in the framework of lattice dynamics are compared with investigations based on continuum theory in the long-wavelength limit.

A treatment of plurals and plural quantifications based on a theory of collections

Collective entities and collective relations play an important role in natural language. In order to capture the full meaning of sentences like “The Beatles sing ‘Yesterday’”, a knowledge representation language should be able to express and reason about plural entities — like “the Beatles” — and their relationships — like “sing” — with any possible reading (cumulative, distributive or collective). In this paper a way of including collections and collective relations within a concept language, chosen as the formalism for representing the semantics of sentences, is presented. A twofold extension of theA−C concept language is investigated: (1) special relations introduce collective entities either out of their components or out of other collective entities, (2) plural quantifiers on collective relations specify their possible reading. The formal syntax and semantics of the concept language is given, together with a sound and complete algorithm to compute satisfiability and subsumption of concepts, and to compute recognition of individuals. An advantage of this formalism is the possibility of reasoning and stepwise refining in the presence of scoping ambiguities. Moreover, many phenomena covered by the Generalized Quantifiers Theory are easily captured within this framework. In the final part a way to include a theory of parts (mereology) is suggested, allowing for a lattice-theoretical approach to the treatment of plurals.