A maximal chain approach to topology and order

On ordered sets (posets, lattices) we regard topologies (or, more general convergence structures) which on any maximal chain of the ordered set induce its own interval topology. This construction generalizes several well-known intrinsic structures, and still contains enough to produce interesting results on for instance compactness and connectedness. The maximal chain compatibility between topology (convergence structure) and order is preserved by formation of arbitrary products, at least in case the involved order structures are conditionally complete lattices

Lattice Type Fuzzy Order and Closure Operators in Fuzzy Ordered Sets

Complete lattices and closure operators in ordered sets are considered from the point of view of fuzzy logic. A typical example of a fuzzy order is the graded subsethood of fuzzy sets. Graded subsethood makes the set of all fuzzy sets in a given universe into a completely lattice fuzzy ordered set (i.e. a complete lattice in fuzzy setting). Another example of a completely lattice fuzzy ordered set is the set of all so-called fuzzy concepts in a given fuzzy context; the respective fuzzy order is the graded subconcept/superconcept relation. Conversely, each completely lattice fuzzy ordered set is isomorphic to some fuzzy ordered set of fuzzy concepts of a given fuzzy context. These natural examples motivate us to investigate some general properties of complete lattice-type fuzzy order. Particularly, in this paper we focus mainly on closure operators in fuzzy ordered sets. Preliminaries The notion of a (partial) order plays a central role in mathematics and its applications. It goes back to 19-th century investigations in logi

A generalization of monotone comparative statics

In this paper, we generalize the lattice theoretical comparative statics by Li Calzi and Veinott, and Milgrom and Shannon. While their theorem is constructed on lattices, particularly on partially ordered sets, we do not require the antisymmetry on a binary relation defined on the set. On the basis of this result, we can deal with the comparative statics of constrained optimization problems, including the cases with nonlinear constraints, in a very intuitive, but considerably general fashion. Specifically, we can extend the Âgvalue order methodsÂh proposed by Antoniadou and Mirman and Ruble in the context of consumer problems with linear constraints. It is also worth noting that our results on the value order can be applicable for any comparative criterion as long as it is a complete preorder on the domain of the objective function.

Spectral Lattices of reducible matrices over completed idempotent semifields

Proceedings of: 10th International Conference on Concept Lattices and Their Applications. (CLA 2013). La Rochelle, France, October 15-18, 2013.Previous work has shown a relation between L-valued extensions of FCA and the spectra of some matrices related to L-valued contexts. We investigate the spectra of reducible matrices over completed idempotent semifields in the framework of dioids, naturally-ordered semirings, that encompass several of those extensions. Considering special sets of eigenvectors also brings out complete lattices in the picture and we argue that such structure may be more important than standard eigenspace structure for matrices over completed idempotent semifields.FJVA is supported by EU FP7 project LiMoSINe (contract 288024). CPM has been partially supported by the Spanish Government-Comisión Interministerial de Ciencia y Tecnología project TEC2011-26807 for this paper.Publicad

On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity. In previous work, it has been proven that, for any mapping f : L → [special characters omitted] between two complete lattices L and [special characters omitted], there exists a largest residuated mapping ρf dominated by f, and the notion of the shadow σ f of f is introduced. A complete lattice [special characters omitted] is completely distributive if, and only if, the shadow of any mapping f : L → [special characters omitted] from any complete lattice L to [special characters omitted] is residuated. Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice. We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → [special characters omitted] between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = {[special characters omitted][x] | x ∈ L} and fo = [special characters omitted]. The umbral number for any isotone mapping f is equal to the umbral number for fo, and [special characters omitted] for any ordinal number α. Let [special characters omitted] be the maximal umbral number for all isotone mappings f : L → [special characters omitted] between two complete lattices. If L is a ∼ finite lattice, then [special characters omitted]. The computation of [special characters omitted] is less than or equal to that of [special characters omitted], we have a more efficient method to calculate the umbral number [special characters omitted]. The previous results indicate that the umbral number [special characters omitted] determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width