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Concept lattices and closed set lattices are graphs with the lattice property. They have been increasingly used this last decade in various domains of computer science, such as data mining, knowledge representation, databases or information retrieval. A fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table. A consequence is the existence of a bijective link between lattices, contexts (via the table) and a set of implicational rules (via the canonical (direct) basis). The possible transformations between these objects give rise to relevant tools for data analysis. In this paper, we present a survey of lattice theory, from the algebraic definition of a lattice, to that of a concept lattice, through closure systems and implicational rules; including the exploration of fundamental bijective links between lattices, reduced contexts and bases of implicational rules; and concluding with the presentation of the main generation algorithms of these objects.

In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furthermore, we apply our result to Lennard--Jones type interacting potentials that appear to be accurate in many physical and biological models. The goal of this investigation is to understand how the minimum of the Lennard--Jones lattice energy varies with respect to the density of the points. Considering the lattices of fixed area A, we find the maximal open set to which A must belong so that the triangular lattice is a minimizer (resp. a maximizer) among lattices of area A. Similarly, we find the maximal open set to which A must belong so that the square lattice is a minimizer (resp. a saddle point). Finally, we present a complete conjecture, based on numerical investigations and rigorous results among rhombic and rectangular lattices, for the minimality of the classical Lennard--Jones energy per point with respect to its area. In particular, we prove that the minimizer is a rectangular lattice if the area is sufficiently large., Comment: 30 pages. 18 Figures. Published in Nonlinearity, Volume 31, Issue 9, p. 3973-4005. DOI:10.1088/1361-6544/aac75a

© 2016 Elsevier B.V. In this paper the answer to a problem posed in [19] is given using known and some newly introduced types of lattices and lattice constructions. The main result is that the collection of cuts of an interval-valued fuzzy set forms a complete finitely spatial meet-between planar lattice.

We show that an Urysohn lattice pre-homomorphism defined on a normal sublattice D of a vector lattice E can be extended to the whole space E and the extended operator is an Urysohn lattice homomorphism. We introduce a new class of nonlinear operators which called φ -operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice E and taking value in Dedekind complete vector lattice F .

We consider a lattice generated by three elements, one of which is completely modular. The free lattice with this property is proved to be finite. It is not modular and contains exactly 39 elements. We have also found a finite set of defining relations for the generating elements of this lattice.

We prove extension theorems for group-valued modular functions defined on orthomodular lattices or modular complemented lattices and for modular measures defined on (pseudo-)D-lattices generalizing results of Riecan (Mat Casopis Sloven Akad Vied 19:44–49, 1969; 20:239–244, 1970; Comment Math Univ Carolin 20:309–316, 1979), Avallone and De Simone (Ital J Pure Appl Math 9:109–122, 2001) and Avallone et al. (Bollettino UMI 9-B(8):423–444, 2006; Math Slovaca 66:421–438, 2016). As basic tool, we first prove an extension theorem for lattice uniformities. This also yields as immediate consequence a result on the extension of modular functions on arbitrary lattices similar to that of Fox and Morales (Fundam Math 78:99–106, 1973) and Kranz (Fundam Math 91:171–178, 1976).

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.

It is well known that continuous lattices and algebraic lattices can be respectively represented by the family of all fixed points of the projection operator and the closure operator preserving sups of directed sets on the power set of a set X . Similar to the algebraic ⋂-structure as the concrete representation of algebraic lattices, can we have a the concrete representation of continuous lattices by families of sets? We give a positive answer to this in this paper. Also, as a special type of continuous lattice, a concrete representation of completely distributive complete lattices by families of sets is obtained.

This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.

Many studies have investigated lattices of fuzzy algebraic systems. One of them belongs to Borzooei et al. (Soft Comput 12:739–749, 2008) who found some properties of lattices of fuzzy algebraic structures. In this study, we solve the problem of finding necessary and sufficient conditions for distributivity and modularity of lattice of fuzzy hyperideals of a hyperring which was one of the open problems in Borzooei’s paper.

In this article, we generalize a bubble sort map to a map on the lattice, and characterize the modular law and the distributive law of lattices from the viewpoint of the Yang–Baxter maps.

In the logico-algebraic approach to the foundation of quantum mechanics we sometimes identify the set of events of the quantum experiment with an orthomodular lattice (“quantum logic”). The states are then usually associated with (normalized) finitely additive measures (“states”). The conditions imposed on states then define classes of orthomodular lattices that are sometimes found to be universal-algebraic varieties. In this paper we adopt a conceptually different approach, we relax orthomodular to orthocomplemented and we replace the states with certain subadditive mappings that range in the Łukasiewicz groupoid. We then show that when we require a type of “fulness” of these mappings, we obtain varieties of orthocomplemented lattices. Some of these varieties contain the projection lattice in a Hilbert space so there is a link to quantum logic theories. Besides, on the purely algebraic side, we present a characterization of orthomodular lattices among the orthocomplemented ones. - The intention of our approach is twofold. First, we recover some of the Mayet varieties in a principally different way (indeed, we also obtain many other new varieties). Second, by introducing an interplay of the lattice, measure-theoretic and fuzzy-set notions we intend to add to the concepts of quantum axiomatics.

We consider a lattice generated by three elements, two of which are semi-normal. We have proved it is infinite and have also found an additional condition for the lattice to be modular. As a result, it is proved that the sublattice generated in the subgroup lattice by two normal subgroups and any subgroup is modular.

Orthomodular lattices with a two-valued Jauch–Piron state split into a generalized orthomodular lattice (GOML) and its dual. GOMLs are characterized as a class of L-algebras, a quantum structure which arises in the theory of Garside groups, algebraic logic, and in connections with solutions of the quantum Yang–Baxter equation. It is proved that every GOML X embeds into a group G(X) with a lattice structure such that the right multiplications in G(X) are lattice automorphisms. Up to isomorphism, X is uniquely determined by G(X), and the embedding $$X\hookrightarrow G(X)$$ is a universal group-valued measure on X.

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.

We present a mathematical model of sets with incomplete information about identity of elements and the membership relation. The model is based on L-valued sets where L is a complete atomic Boolean algebra. We present general foundations of the theory with emphasis on partially ordered sets and complete lattices. As an application, we show a method for constructing structures that represent the concept lattice of an incomplete binary dataset.

Summary In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].

We study the lattice-theoretic analogue of rings and σ-rings, which are well-known concepts from Algebra and are also used in the development of classical Measure Theory. The elements of the lattice rings and lattice σ-rings under consideration are elements of the set of sequences of non-negative real numbers, which form a lattice with an appropriate partial order.

In this paper, we show that the set An(L) of all n-ary aggregation functions on a complete lattice L is a complete lattice and we study some properties of this lattice. If L is a bounded lattice and n is a natural number such that n≠0, we obtain that Lm for m∈[n] can be embedded into the lattice An(L). We generate aggregation functions from monotone functions. We introduce the concept of internal product of aggregation functions. We give some examples of aggregation functions on bounded lattices.

Using the Hasse diagrams of concept lattices, we investigate therelations between matroids and geometric contexts, followed byjudging a mathematical construction to be a matroid. We provide anidea to find out the dual of a matroid from the ways of conceptlattice drawing. In addition, we utilize the Hasse diagrams ofconcept lattices to discuss the minors of matroids, direct sum ofmatroids and the connectivity of a matroid. All the consequences demonstrate that the theory of concept lattice drawing can be usedinto matroids. This generalizes the applied fields of conceptlattices.DOI : http://dx.doi.org/10.22342/jims.22.2.267.183-190

In this paper, we study S-implication operator and QL-operation on a lattice L . We tabulate properties of S-implication and QL-operation with respect to different triangular norms and negations. We study properties of S-implication and QL-operation on uniquely complemented and orthocomplemented lattice.

Continuum partial differential equations are obtained from a set of discrete stochastic evolution equations of both non-Markovian and Markovian processes and applied to the diffusion within the context of the lattice gas model. A procedure allowing to construct one-dimensional lattices that are topologically equivalent to two-dimensional lattices is described in detail in the case of a hexagonal lattice which has the particular feature that need four types of dynamical variables. This example shows additional features to the general procedure and some extensions are also suggested in order to provide a wider insight in the present approach.

It was shown previously (Shan, 2010) that the lattice weights of the lattice Boltzmann method are the solutions to an optimization problem subjecting to a set of linear equality and inequality constraints. Here, a solution strategy is developed and complete solutions for such optimization problem are obtained in one-, two-, and three-dimensions up to 9th order. In addition to recovering the well-known lattices, a number of high-order lattices are also discovered for the first time. These lattices can be used to construct high-order lattice Boltzmann schemes for non-equilibrium flow simulation.

Continuum partial differential equations are obtained from a set of discrete stochastic evolution equations of both non-Markovian and Markovian processes and applied to the diffusion within the context of the lattice gas model. A procedure allowing to construct one-dimensional lattices that are topologically equivalent to two-dimensional lattices is described in detail in the case of a rectangular lattice. This example shows the general features that possess the procedure and extensions are also suggested in order to provide a wider insight in the present approach.

We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.

We deal with size reduction of concept lattices by means of factorization by block relations defined on corresponding formal context. We show that all block relations with a multiplication defined by means of relational composition form a (non-commutative) residuated lattice. Such residuated lattice can be then thought of as a scale of truth degrees using which we evaluate formulas of predicate logic specifying the desired parameters of the factorization. We also introduce efficient algorithms computing operations on a residuated lattice of block relations. The naive way how to design such algorithms is to compute all the formal concepts of a given context in advance, and then apply some well-known properties of residuated lattices. Our algorithms get rid of the time-consuming precomputation of all concepts.

The Gratzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that1.the set of indices of computable lattices that are complete is $$\Pi ^1_1$$ź11-complete;2.the set of indices of computable lattices that are algebraic is $$\Pi ^1_1$$ź11-complete;3.the set of compact elements of a computable lattice is $$\Pi ^{1}_{1}$$ź11 and can be $$\Pi ^1_1$$ź11-complete; and4.the set of compact elements of a distributive computable lattice is $$\Pi ^{0}_{3}$$ź30, and there is an algebraic distributive computable lattice such that the set of its compact elements is $$\Pi ^0_3$$ź30-complete.

This paper introduces a concept of lattice-theoretic contexts as well as their concept lattices. A lattice-theoretic context is a triple (G, M, I) with two complete lattices G, M and their Galois ideal I. A lattice-theoretic context and its concept lattice are a common generalization of classical FCA, Pocs’s formal fuzzy context, one-sided concept lattices, generalized concept lattices and L-fuzzy concept lattices (with hedges). When the lattices G, M are completely distributive, a reduction of the relation I in the lattice-theoretic context (G, M, I) can be obtained. Related algorithms to construct concept lattices of L-fuzzy contexts considered as lattice-theoretic contexts are presented. In the case of L being a completely distributive lattice, we can reduce the number of elements (objects or/and attributes) before computing the whole concept lattice. Then the related algorithm has lower complexity.

Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

In a pseudocomplemented de Morgan algebra, it is shown that the set of kernel ideals is a complete Heyting lattice, and a necessary and sufficient condition that the set of kernel ideals is boolean (resp. Stone) is derived. In particular, a characterization of a de Morgan Heyting algebra whose congruence lattice is boolean (resp. Stone) is given.

We define the notion of a residuated lattice valued function on a set as Jun and Song have done in BCK-algebras. We also investigate related properties of residuated lattice valued function. We establish the codes generated by residuated lattice valued function and conversely we give residuated lattice valued function and residuated lattice obtained by the giving binary block-code.

This paper is devoted to the structure that describes the construction of finite distributive lattices. From the viewpoint of application, we consider algorithms of construction and enumeration of distributive lattices and partially ordered sets for finite distributive lattices: A formula for finding the maximum anti-chain with respect to nonintersection is given, it is shown that elements of the lattice can be split into pairs according to comparison, the point of the maximum number of elements in the lattices is considered, and the structure of lattice congruence is described.

In this paper, we are mainly concerned with semicontinuity of complete lattices and their distributive reflections, introduced by Rav in 1989. We prove that for a complete lattice L, the distributive reflection Ld is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. It is also proved that if a complete lattice L is semicontinuous and every semiprime element x ∈ L is the largest in d(x), then Ld is continuous whenever the distributive reflector d is Scott continuous. We construct counterexamples to confirm a conjecture and solve two open problems posed by Zhao in 1997.

Slim rectangular lattices are special planar semimodular lattices introduced by G. Gratzer and E. Knapp in 2009. They are finite semimodular lattices L such that the ordered set Ji L of join-irreducible elements of L is the cardinal sum of two nontrivial chains. After describing these lattices of a given length n by permutations, we determine their number, |SRectL(n)|. Besides giving recursive formulas, which are effective up to about n = 1000, we also prove that |SRectL(n)| is asymptotically (n - 2)! · \({e^{2}/2}\). Similar results for patch lattices, which are special rectangular lattices introduced by G. Czedli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given.

A number of research articles have established the significant role of lattice-ordered groups (l-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as l-groups with a modal operator. These connections are just the tip of the iceberg. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of l-groups, which analyzes the structure of individual or classes of l-groups by primarily using strictly lattice theoretic properties of their lattices of convex l-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x \ e ≈ e / x . An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

International audience; The natural join and the inner union operations combine relations of a database. Tropashko and Spight realized that these two operations are the meet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to the theory of databases alternative to the relational algebra. Litak et al. proposed an axiomatization of relational lattices over the signature that extends the pure lattice signature with a constant and argued that the quasiequational theory of relational lattices over this extended signature is undecidable. We prove in this paper that embeddability is undecidable for relational lattices. More precisely, it is undecidable whether a finite subdirectly-irreducible lattice can be embedded into a relational lattice. Our proof is a reduction from the coverability problem of a multimodal frame by a universal product frame and, indirectly, from the representability problem for relation algebras. As corollaries we obtain the following results: the quasiequational theory of relational lattices over the pure lattice signature is undecidable and has no finite base; there is a quasiequation over the pure lattice signature which holds in all the finite relational lattices but fails in an infinite relational lattice.

International audience; It is well-known that the weak Bruhat order on the symmetric group on a finite number n of letters is a lattice, denoted by P(n) and often called the permutohedron on n letters, of which the Tamari lattice A(n) is a lattice retract. The equational theory of a class of lattices is the set of all lattice identities satisfied by all members of that class. We know from earlier work that the equational theory of all P(n) is properly contained in the one of all A(n). We prove the following results. Theorem I. The equational theory of all P(n) and the one of all A(n) are both decidable. Theorem II. There exists a lattice identity that holds in all P(n), but that fails in a certain 3338-element lattice. Theorem III. The equational theory of all extended permutohedra, on arbitrary (possibly infinite) posets, is trivial. In order to prove Theorems I and II, we reduce the satisfaction of a given lattice identity in a Cambrian lattice of type A to a certain tiling problem on a finite chain. Theorem I then follows from Büchi's decidability theorem for the monadic second-order theory MSO of the successor function on the natural numbers. It can be extended to any class of Cambrian lattices of type A with MSO-definable set of orientations.

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a unimodular sequence in $\mathbb{Z}^2$. This formula implies the generalized twelve-point theorem in [12]. We then introduce the notion of lattice multi-polygons which is a generalization of lattice polygons, state the generalized Pick's formula and discuss the classification of Ehrhart polynomials of lattice multi-polygons and also of several natural subfamilies of lattice multi-polygons., 21 pages, 7 figures, Kyoto J. Math. to appear

Graphical abstractDisplay Omitted HighlightsConcept of lattice ordered soft sets is introduced.This type of soft sets can help to represent linguistic terms having certain order.Properties of lattice order soft sets have been studied.Some algebraic structures associated with this type of soft sets have been given.In certain decision making problems these soft sets and operations on them can be very helpful. Certain type of linguistic terms such as satisfactory, good, very good and excellent have an order among them. In this paper we introduce a new concept of soft sets with some order among the parameters. Some properties of lattice ordered soft sets are given. Lattice ordered soft sets are very useful in particular type of decision making problems where some order exists among the elements of parameters set.

This paper investigates the class of ordered sets that are embeddable into a distributive lattice in such a way that all existing finite meets and joins are preserved. The main result is that the following decision problem is NP-complete: Given a finite ordered set, is it embeddable into a distributive lattice with preservation of existing meets and joins? The NP-hardness of the problem is proved by polynomial reduction of the classical 3SAT decision problem into it, and the NP-completeness by presenting a suitable NP-algorithm.

The amalgamation property (AP) is of particular interest in the study of residuated lattices due to its relationship with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the AP. The variety SemRL of semilinear residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. However, we prove that this is not the case, and in the process we establish that the same is true for the variety SemCanRL of semilinear cancellative residuated lattices. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ∧ z)w ≈ xyw ∧ xzw also fails the AP.

Palfy and Pudlak (Algebra Universalis 11, 22–27, 1980) posed the question: is every finite lattice isomorphic to an interval sublattice of the lattice of subgroups of a finite group? in this paper we will look at examples of lattices that can be realized as subloop lattices but not as subgroup lattices. This is a first step in answering a new question: is every finite lattice isomorphic to an interval sublattice of the subloop lattice of a finite loop?

The lattice of closed right ideals is an important invariant of a C*-algebra and naturally generalizes the spectrum of a commutative C*-algebra. As the C*-algebra is a union of its commutative sub-C*-algebras, the lattice can be considered as a “piecewise frame”. It is discussed here that there is no right residuated total operation extending the meet operation on compatible elements, no “Sasakian” product, and no active lattice structure.

In this paper, we introduce the concept of a Semi Heyting Almost Distributive Lattice (SHADL) as a generalization of a Semi Heyting Algebra in the class of ADLs. We characterize an SHADL in terms of its principal ideals. We also give different sets of equations that define an SHADL.

We axiomatize the smallest variety that contains both the variety of MV-algebras and the variety (term equivalent to the variety) of orthomodular lattices.

We introduce the notion of α-filters in BL-algebras and then state and prove some theorems which determine the relationships of these filters and other filters in BL-algebras. Some properties of the lattice of all α-filters are investigated too.

In this paper, we characterize the congruence lattice of a symmetric extended De Morgan algebra $$L$$ . We show that the congruence lattice of the algebra $$L$$ is a pseudocomplemented lattice, and that such a congruence lattice is a Stone lattice if and only if the lattice of the compact congruences on $$L$$ forms a complete Boolean lattice. In particular, we prove that the congruence lattice of $$L$$ is a Boolean lattice if and only if, it is a relative Stone lattice, which is the case, if and only if $$L$$ is finite.