A Fresh Perspective on Canonical Extensions for Bounded Lattices (original) (raw)

M.: Reconciliation of approaches to the construction of canonical extensions of bounded lattices

2014

ABSTRACT. We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps. c©2014

Reconciliation of approaches to the construction of canonical extensions of bounded lattices

Mathematica Slovaca, 2014

We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps.

Bounded Lattice Expansions

Journal of Algebra, 2001

The notion of a canonical extension of a lattice with additional operations is introduced. Both a concrete description and an abstract characterization of this extension are given. It is shown that this extension is functorial when applied to lattices whose additional operations are either order preserving or reversing, in each coordinate, and various results involving the preservation of identities under canonical extensions are established.

Some nonstandard methods applied to distributive lattices

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1990

In memory of Abraham Robinson, on the occasion of his 70th birthday Distributive lattices are locally finite algebras, i.e. their finitely generated sublattices are finite. The general theory of finite distributive lattices is remarkably transparent, due to one or both of the following principles: (i) Any filter (or ideal) is principal. (ii) The lattice is join-generated by its join-irreducible elements, namely by those elements z for which x v y = z implies x = z or y = z .

The extension of measures on D-lattices

Fuzzy Sets and Systems, 2014

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues.

Key words and phrases. Bounded distributive lattice, Priestley duality

2012

Abstract. The ordering relation of a bounded distributive lattice L is a (distributive) (0, 1)-sublattice of L × L. This construction gives rise to a functor Φ from the category of bounded distributive lattices to itself. We examine the interaction of Φ with Priestley duality and characterise those bounded distributive lattices L such that there is K with Φ(K) ∼ = L. 1 1. Some conventions and definitions For any poset P we say that A ⊆ P is a lower set or down-set if a ∈ A, x ∈ P,x ≤ a imply x ∈ A. The dual notion is that of an up-set. We assume all lattices to be distributive and bounded by 0,1 such that 0 ̸ = 1. A nonempty down-set I of a bounded distributive lattice L is said to be an ideal if a,b ∈ I implies a ∨ b ∈ I. An up-set with the dual property is called a filter. Moreover, I is a prime ideal if I ̸ = L and if a,b ∈ L \ I implies a ∧ b ∈ L \ I. Note that a down-set of L is a prime ideal if and only if its complement is a filter. Let L be a bounded distributive lattice. Th...

Aggregation Functionals on Complete Lattices

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. Two different types of aggregation functionals are introduced and investigated. We consider a target-based approach that has been studied in Decision Theory and we focus on the equivalence between a utility-based approach and target-based approach. Moreover we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices.