nLab
DistLat

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

(0,1)(0,1)-Category theory

Contents

Definition

DistLatDist Lat is the category whose objects are distributive lattices and whose morphisms are lattice homomorphisms, that is functions which preserve finitary meets and joins (equivalently, binary meets and joins and the top and bottom elements). DistLatDist Lat is a subcategory of Pos and a replete subcategory of Lat.

Properties

DistLatDist Lat is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories. The free distributive lattice on a set XX is the set of irredundant finite subsets of the finite power set 𝒫 fin\mathcal{P}_{fin} of XX; a collection 𝒞\mathcal{C} of subsets of XX is irredundant if, whenever ABA \subseteq B for A,B𝒞A, B \in \mathcal{C}, we have A=BA = B (that is, no two distinct elements of 𝒞\mathcal{C} may be comparable). Note that 𝒫 finX\mathcal{P}_{fin}X is the free semilattice on XX; we may interpret an element of it as a finitary join of elements of XX. Then we interpret an element of 𝒫 fin𝒫 finX\mathcal{P}_{fin}\mathcal{P}_{fin}X as a finitary meet of elements of 𝒫 finX\mathcal{P}_{fin}X. However, if two of the elements of 𝒫 finX\mathcal{P}_{fin}X that appear in this meet are comparable, then we need only the smaller of them, so we take only the irredundant elements of 𝒫 fin𝒫 finX\mathcal{P}_{fin}\mathcal{P}_{fin}X. (We could equally well take joins of meets instead of meets of joins; then we keep only the larger of two meets that appear in a given join.)

category: category

Revised on March 15, 2012 16:35:01 by Urs Schreiber (82.169.65.155)