Search

Isomorphic representations of finite distributive lattices by lattices of functions and fuzzy sets are constructed. These class es the case of finite lattices are the generalization of the representation of distributive lattices by set lattices whose existence is proved by Birkhoff’s theorem. Homomorphic image and subdirect representations of finite distributive lattices are characterized, and figure of two-dimensional distributive lattices are geometrically characterized. This theoretical basis underlies the general description of systems of linear and partially ordered parallel discrete processes with mutual constraints on step order acting between them. The application of the developed tools is demonstrated on the examples from the graph theory, parallel calculations, artificial intelligence, and power engineering.

In this paper, as a generalization of derivation on a lattice,the notion of f -derivation for a lattice is introduced and some relatedproperties are investigated. 1. Introduction and preliminariesLattices play an important role in many fields such as information theory,information retrieval, information access controls and cryptanalysis [2, 6, 13, 8].Recently the properties of lattices were widely researched [1, 2, 5, 7, 8, 9, 13, 14].In the theory of rings and near rings, the properties of derivations are animportant topic to study [3, 4, 11, 12]. Y. B. Jun and X. L. Xin [10] appliedthe notion of derivation in ring and near ring theory to BCI-algebras. In[15], J. Zhan and Y. L. Liu introduced the notion of left-right (or right-left) f -derivation of a BCI algebra and investigated some properties. In [14], X. L. Xin,T. Y. Li, and J. H. Lu introduced the notion of derivation on a lattice anddiscussed some related properties.In this paper, as a generalization of derivation on a lattice, the notion of

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices.

We prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.

We present two examples of distributive algebraic lattices which are not isomorphic to the congruence lattice of any lattice. The first such example was discovered by F. Wehrung in 2005. One of our examples is defined topologically, the other one involves majority algebras. In particular, we prove that the congruence lattice of the free majority algebra on (at least) ℵ2 generators is not isomorphic to the congruence lattice of any lattice. Our method is a generalization of Wehrung’s approach, so that we are able to apply it to a larger class of distributive semilattices.

We prove modularity of the lattice Lat (T) of closed invariant subspaces for a C 0-operator T with property (P).

Pseudo *-autonomous lattices are non-commutative generalizations of *-autonomous lattices. It is proved that the class of pseudo *-autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo *-autonomous lattices can be described as their normal ideals.

In this paper, we study further the filter theory of residuated lattices. First, we discuss the concepts of filters and normal filters of residuated lattices and propose some equivalent conditions for them. Then we introduce and investigate the notions of v-filter and normal v-filter of a residuated lattice with a weak vt-operator and lay bare the formulas for calculating the v-filters and the normal v-filters generated by subsets. Finally we show that the lattices of v-filters and normal v-filters of a residuated lattice with a vt-operator are both complete Brouwerian lattices.

A class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ1 and ℱ2 the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ1 and ℱ2 are partially ordered by the class-theoretical inclusion. We prove that ℱ1 satisfies the infinite distributivity law and that ℱ2 is isomorphic to a principal ideal of ℱ1.

We show that for any infinite cardinal κ, every complete lattice where each element has at most one complement can be regularly embedded into a uniquely complemented κ-complete lattice. This regular embedding preserves all joins and meets, in particular it preserves the bounds of the original lattice. As a corollary, we obtain that every lattice where each element has at most one complement can be embedded into a uniquely complemented κ-complete lattice via an embedding that preserves the bounds of the original lattice.

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank d contains a complemented rank 3 interval, then the lattice is ( d + 1 ) -connected. In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain ( d ∈ N 0 ). Then the comparability graph of L is ( d + 1 ) -connected if and only if L has no simplicial elements, where z ∈ L is simplicial if the elements comparable to z form a chain.

The paper presents results on factorization by similarity of fuzzy concept lattices with hedges. A fuzzy concept lattice is a hierarchically ordered collection of clusters extracted from tabular data. The basic idea of factorization by similarity is to have, instead of a possibly large original fuzzy concept lattice, its factor lattice. The factor lattice contains less clusters than the original concept lattice but, at the same time, represents a reasonable approximation of the original concept lattice and provides us with a granular view on the original concept lattice. The factor lattice results by factorization of the original fuzzy concept lattice by a similarity relation. The similarity relation is specified by a user by means of a single parameter, called a similarity threshold. Smaller similarity thresholds lead to smaller factor lattices, i.e. to more comprehensible but less accurate approximations of the original concept lattice. Therefore, factorization by similarity provides a trade-off between comprehensibility and precision. We first describe the notion of factorization. Second, we present a way to compute the factor lattice directly from input data, i.e. without the need to compute the possibly large original concept lattice. Third, we provide an illustrative example to demonstrate our method.

In the present paper we deal with generalized MV-algebras (GMV-algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, GMV-algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of GMV-algebras. The relations between GMV-algebras and lattice ordered groups are essential for this investigation.

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics--classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.

A new type of lattices, lattice of cubes, is defined and described. It is proved that the number of all subcubes of the cube of the dimension m is 3 m . It is shown that the set of such subcubes with an appropriate choice of the operations of union and intersection is a lattice, called the lattice of cubes. An algorithm of constructing this lattice is proposed, and the problem of minimization and maximization of supermodular functions is considered on it. Particular examples of such functions are given. Optimizations algorithms, as well as the possibility of setting and solving a new class of problems on the cubes lattices are discussed.

Relationship between automorphisms and residuated bounded mappings in atomistic lattices is studied.

In mathematical morphology for sets (binary images), the geodesic reconstruction associates to a set called mask and a subset of it called marker, the union of all connected components of the mask intersected by the marker. It is obtained by iteration of a geodesic dilation applied to the marker inside the mask. This operation extends naturally to numerical functions (grey-level images), where it allows us to reconstruct flat zones. Considering that the family of images constitutes a complete lattice under some partial order relation, a general theory of geodesic dilations and reconstructions in a complete lattice was given in Ronse and Serra [Fundam Inf 46(4):349–395, 2001]. It relies on the assumption that the lattice is infinitely supremum distributive, and it fails for pictorial objects forming a non-distributive lattice. In this paper we give a more general theory of geodesic operations, that can be applied to non-distributive lattices; it is compatible with the previous theory when the lattice is infinitely supremum distributive. We study one particular form of geodesic operator, the impulsive geodesic dilation, which gives good results for images with values in a bundle lattice (e.g., the lattice of labels and the reference lattice). We also briefly discuss geodesy on the lattice of partitions.

From the viewpoint of semantics, lattice implication algebras provide a basis to establish lattice-valued logic with truth value in a relatively general lattice. In this paper, we first introduce two notions of lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, which induced by lattice implication algebras. Secondly, we study some of their basic properties and prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice ordered semigroup. We also define the homomorphism mapping between lattice implication n-ordered semigroups. Finally, we discuss some properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups.

In this paper, we introduce the notion of derivation for a lattice and discuss some related properties. We give some equivalent conditions under which a derivation is isotone for lattices with a greatest element, modular lattices, and distributive lattices. We characterize modular lattices and distributive lattices by isotone derivation. Moreover, we prove that if d is an isotone derivation of a lattice L, the fixed set Fix"d(L) is an ideal of L. Finally, we prove that D(L) is isomorphic to L in a distributive lattice L.

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.

This paper proves that a geometric lattice of rank n is a modular lattice if its every maximal chain contains a modular element of rank greater than 1 and less than n. This result is generalized to a more general lattices of finite rank.

We show that, for a large class of countable order types α, a modular algebraic lattice L contains no chain of type α if and only if K(L), the join-semilattice of compact elements of L, contains neither a chain of type α nor a join-subsemilattice isomorphic to [ω] < ω , the set of finite subsets of ω ordered by inclusion. We give a description of the indivisible members of this class. It includes the order types ω * of the chain of negative integers and η of the chain of rational numbers.

Pseudo-BCK-algebras arise as the { \ , / , 1 } -subreducts of integral residuated lattices. In this note we characterize pseudo-BCK-algebras that are subdirect products of linearly ordered pseudo-BCK-algebras.

In this paper, the definition of a variable threshold concept lattice is introduced. Based on a Galois connection, three kinds of variable threshold concept lattices, in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold, are defined. The number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice. The three kinds of variable threshold concept lattices are constructed between two crisp sets, between a crisp set and a fuzzy set, and between a fuzzy set and a crisp set. Their properties are analogous to that of the classical concept lattices, and can be induced by the fuzzy concept lattice.

We consider the problem of counting the number of lattice vectors of a given length. We show that problem is @?P-complete resolving an open problem. Furthermore, we show that the problem is at least as hard as integer factorization even for lattices of bounded rank or lattices generated by vectors of bounded norm. Next, we discuss a deterministic algorithm for counting the number of lattice vectors of length d in time 2^O^(^r^s^+^l^o^g^d^), where r is the rank of the lattice, s is the number of bits that encode the basis of the lattice. The algorithm is based on the theory of modular forms.

Lattice algebras replace + and - with max and min, but then can create many of the operations of normal algebra. In particular lattice algebras can be used in the study of neural networks and extraction of images in the presence of noise. This paper is not a full introduction but a start with some of the simple algorithms that can be used.

In this paper we deal with weakly homogeneous direct factors of lattice ordered groups. The main result concerns the case when the lattice ordered groups under consideration are archimedean, projectable and conditionally orthogonally complete.

In this paper, we study the lattice structure of some fuzzy algebraic systems such as (G-)fuzzy groups, some fuzzy ordered algebras and fuzzy hyperstructures. We prove that under suitable conditions, these structures form a distributive or modular lattice.

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary.

We show that the only compatible lattice order on a matrix ring over the integers for which the identity matrix is positive is (up to isomorphism) the usual, entrywise, lattice order. We also find a condition that guarantees that the only compatible lattice order on a matrix ring over the integers is formed by multiplying the positive cone of the usual, entrywise, lattice order by a matrix with positive entries. Using this condition, we show that such orders are the only compatible ones in the two-by-two case.

It is shown that the variety generated by a finite modular lattice has only finitely many locally finite covers.

SK-partitions were introduced by Sharma and Kaushik, who defined distances (metrics) between vectors in terms of partitions of the alphabet set Fq , the set of the ring of integers modulo q. These distances were applied in Coding Theory. The research examined algebraic and topological aspects of SK-partitions and related sets of measures. A lattice of SK-partitions was introduced and shown to have distributive sub lattices. Generator sets of the lattice were obtained and the structure of its ideals and filters examined. Corresponding results for lattices of measures were presented.

A lattice ordered group valued subadditive measure is extended from an algebra of subsets of a set to a σ-algebra.

The purpose of this paper is to introduce the lattice of convex partitions for a lattice L. Then we will show some properties of this lattice. Finally, we will show that if the convex partition lattice of L is finite and modular if and only if L is a finite chain.

In this paper we prove that an atomistic lattice L of finite length is geometric if it has the nontrivial modular cutset condition, that is, every maximal chain of L contains a modular element which is different from the minimum element and the maximum element of L.

S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices dened in (6). If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only nitely many s-extremal strongly modular lattices of even minimum.

Original algorithmic approaches in integer programming rely on the availability of different representations for all the lattice points in the feasible region. We present three results that are applicable to fairly general sets of lattice points and characterize a nonnegative integer linear representation.

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that ensures that no non-Shannon inequalities exist. It is demonstrated that 3-dimensional distributive lattices cannot have non-Shannon inequalities and planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group., Ten pages. Submitted to ISIT 2015. The appendix will not appear in the proceedings

We claim and justify that the future of a fuzzy logic is in the interconnection of various well-developed theories. We are focused on a lattice valued analysis that unifies the treatments of atomic elements, sets of atomic elements, functions between sets of atomic elements and their properties. We clarify the relationship between a fuzzy function and its ordinary core. We discuss the property of continuity of a fuzzy function in a lattice valued topology.

In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Other than distributive lattices, the main examples of trim lattices are the Tamari lattices and various generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim, and we conjecture that all Cambrian lattices are trim., Comment: 19 pages, 4 figures. Version 2 includes small improvements to exposition, corrections of typos, and a new section showing that if a group G acts on a trim lattice by lattice automorphisms, then the sublattice of L consisting of elements fixed by G is trim

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems., Comment: 29 pages

In the paper [2] we introduced and investigated complete orthomodular lattices generated by graphs of continuous functions. A natural question arises: can such a lattice be represented by the lattice of projectors in a Hilbert space (the standard quantum logic)? The answer is no, because the covering law is not satisfied in this case.

Given the primitive vectors of an arbitrary Bravais lattice and the Miller indices of a set of lattice planes in it, it is shown how to construct an alternative set of primitive vectors that are adapted to the lattice planes in the following sense: all but one of these alternative vectors serve as a basis for the points in one of the lattice planes, and the remaining vector serves to shift any of these lattice planes into a neighboring one. This construction is described for a three-dimensional Bravais lattice and then generalized to arbitrary d-dimensional lattices. This construction can be used to generate computer images of lattice points in a succession of crystal lattice planes, which could be useful for instructional purposes.

Let L be a residuated lattice. Then, given a monoid M , we define a Galois connection from the lattice of the compatible L -preorders in M and the lattice of L -submonoids of M . Given a set S we define a Galois connection between the lattice of the L -preorders in S and the lattice of L -submonoids of the monoid ( S S , ∘ , i ) . A link with the notion of quasi-metric is also established.

Representing and reasoning with an agent's preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice.

Repnitskii proved that any lattice embeds in a subsemigroup lattice of some commutative, cancellative, idempotent free semigroup with unique roots. In that proof, use is made of a result by Bredikhin and Schein stating that any lattice embeds in a suborder lattice of suitable partial order. Here, we present a direct proof of Repnitskii’s result which is independent of Bredikhin-Schein’s, thus giving the answer to the question posed by Shevrin and Ovsyannikov.

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.