A note on representable posets

Abstract

The problem of characterizing those posets which arise as posets of prime ideals (or prime filters) of distributive lattices was raised in Chen and Gr~itzer [-3] - see also problem 33 of Gr~itzer [-4]. The first solution to this problem was given in Speed [,7], although it was also considered in Balbes [,1]. Speed's characterization has the disadvantage that it cannot be easily applied to construct representable posets. In this note we give a new characterization of representable posers which facilitates the construction of examples. Let 2 = {0, 1 } with the natural order 0 < 1. Then the poset of prime filters of a distributive lattice A, ordered by inclusion, is order isomorphic to the poser of nonconstant 2-valued homomorphisms r ~2, ordered pointwise. This is a dual order isomorphism for the poset of prime ideals. For this reason we work with the poser of prime filters. DEFINITION 1. A poset X is representable if it is isomorphic to the poser of prime filters of a distributive lattice Ax. The lattice A x is then said to represent the poset X. The characterization of representable posets is in terms of locally closed families of 2-valued functions. DEFINITION 2. Let I be any set and let A and B be subsets of 2 x with A___B. A map ff in B is locally a member of A if for each finite subset J of I there exist ~0 in A (depending upon ~k and J) such that ~k I J= q~ ] J. A is locally closed in B if A contains every ~k in B which is locally a member of A'. The set 2 is endowed with the discrete topology and for any set I we endow 2 t with the product topology and order. A subset of 2 ~ is given the relative topology. Observe that 2 x is a compact totally order disconnected space in the sense of Priestley ['5], [6]. The constant maps into 2 will be denoted by 0 and 1.