An extension of Birkhoff's representation theorem to locally-finite distributive lattices

Abstract

Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. We present a simplified version of Stone's extension of this theorem to general distributive lattices. We then apply this formulation to locally finite distributive lattices to produce a novel representation theorem: The lattice is isomorphic to the order ideals of the poset of prime filters of the lattice whose symmetric difference from a particular ideal is finite.

Figures (6)

• Figure 1: The lattice $\mathbb{Z} \times \mathbb{Z}$
• Figure 2: The lattice $\mathbb{Z} \times \mathbb{Z}$ with some filters outlined
• Figure 3: The lattice $t$-SkewStrip (right) and its poset of join-irreducible elements $t$-Plait (left). (Taken from Fom1994a*Fig. 7.)
• Figure 4: The lattice of filters $\mathcal{F}$ for $\mathcal{L} = \mathbb{Z} \times \mathbb{Z}$ with its prime elements outlined, all of which are sur-primes
• Figure 5: The poset of prime filters $\mathcal{P}$ for $\mathcal{L} = \mathbb{Z}$. $\mathcal{P} = A \sqcup B$

Theorems & Definitions (65)

• Theorem 1: Birkhoff Birk1937aBirk1967a*§ III.3 p. 58
• Theorem 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Lemma 8
• proof
• Lemma 9