A Note on the Set-Theoretic Representation of Arbitrary Lattices

Every lattice is isomorphic to a lattice whose elements are sets of sets and whose operations are intersection and the operation ∨∗ defined by A ∨∗ B = A ∪ B ∪ {Z : (∃X ∈ A)(∃Y ∈ B)X ∩ Y ⊆ Z}. This representation spells out precisely Birkhoff’s and Frink’s representation of arbitrary lattices, which is related to Stone’s set-theoretic representation of distributive lattices. (AMS Subject Classification, 1991: 06B15) As a generalization of his representation theory for Boolean algebras, Stone has developed in [4] a representation theory for distributive lattices. This representation theory has set-theoretic and topological aspects. Set-theoretically, every distributive lattice L is isomorphic to a set lattice L∗, i.e. a lattice whose elements are sets and whose operations are intersection and union. In Stone’s representation, the elements of L∗ are certain subsets of the set F (L) of prime filters of L. Topologically, F (L) can be viewed as a T0-space with the elements of L∗ constituting a subbasis. Following ideas of Priestley’s [3], Urquhart has developed in [5] the topological aspects of this representation theory to cover arbitrary bounded lattices. However, Birkhoff and Frink had already in [1] (section 6) a simple set-theoretic representation for arbitrary lattices, also inspired by Stone, but different from Urquhart’s representation. In the Birkhoff-Frink representation, every lattice L is isomorphic to a lattice L∗ whose elements are sets of sets, whose meet operation is intersection and whose join operation is a set-theoretic operation ∨∗ unspecified by Birkhoff and Frink. The elements of L∗ are certain subsets of a set F (L), which may be either the set of all filters of L, or the set of all principal filters of L, or any set of filters of L that for every pair of distinct elements of L has a filter containing one element of the pair but not the other. Stone’s set-theoretic representation for distributive lattices may be viewed as a special case of the Birkhoff-Frink representation: if for a distributive lattice L we take F (L) to be the set of all prime filters of L, then ∨∗ collapses into set-theoretic union. The aim of this note is to make precise some details of the Birkhoff-Frink representation, which doesn’t seem to be very well known. We shall explicitly characterize the operation ∨∗ when F (L) is the set of all filters of L, or of all