Bounded Lattice Expansions

Abstract

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.

Coalgebraic completeness-via-canonicity for distributive substructural logics

“…The dual topology will be denoted by γ ↓ , and their join by γ. It is not too hard to show (see [GH01,Ven06]) that γ ↑ ⊆ σ ↑ , γ ↓ ⊆ σ ↓ , and γ ⊆ σ. We denote the product of topologies by ×, and the n-fold product of a topology τ by τ n .…”

Relational semantics for full linear logic

“…In [GH01] it is analysed in which case it does. The equalities (1) and (3) rely on special instances of this general theory, (2) follows from the assumption for !…”

“…Canonical extensions were introduced in the 1950s by Jónsson and Tarski exactly for BAOs [JT51,JT52]. Thereafter their ideas have been developed further, which has led to a smooth theory of canonical extensions applicable in a broad setting [GH01,GJ94]. In [DGP05] canonical extensions of partially ordered algebras are defined to obtain relational semantics for the implication-fusion fragment of various substructural logics.…”

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

“…This paper was prompted by results obtained in [4], where we presented a new approach to the canonical extension of a bounded lattice, based on a lesser-known dual representation for the variety L of bounded lattices due to Ploščica [15]. The canonical extension L δ of a bounded lattice L ∈ L was introduced by Gehrke and Harding [8]; their L δ arises as the complete lattice of Galois-closed sets associated with the Galois connection between ℘ (Filt(L)) and ℘ (Idl(L))…”

“…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.…”