nLab DistLat

Definition

$Dist Lat$ 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). $Dist Lat$ is a replete subcategory of Pos and a full and replete subcategory of Lat.

Properties

$Dist Lat$ 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 $X$ is the set of finite antichains of the finite power set $\mathcal{P}_{fin}$ of $X$; a collection $\mathcal{C}$ of subsets of $X$ is an antichain if, whenever $A \subseteq B$ for $A, B \in \mathcal{C}$, we have $A = B$ (that is, no two distinct elements of $\mathcal{C}$ may be comparable). Note that $\mathcal{P}_{fin}X$ is the free semilattice on $X$; we may interpret an element of it as a finitary join of elements of $X$. Then we interpret an element of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$ as a finitary meet of elements of $\mathcal{P}_{fin}X$. However, if two of the elements of $\mathcal{P}_{fin}X$ that appear in this meet are comparable, then we need only the smaller of them, so we take only the irredundant elements of $\mathcal{P}_{fin}\mathcal{P}_{fin}X$. (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.)

 Related concepts

• coherent category

• coherent hyperdoctrine

• Pos, Frm

category: category