Partial order embeddings with convex range

Abstract

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for example, we prove that every product order embedding sigma between the irrationals (i.e. the family of functions from N into N) with convex range is of the form sigma(x)(n) = ((x o g) + y)(n) if n in K, and sigma(x)(n) = y(n) otherwise, for all irrationals x, where K is a subset of N, g:K -> N is a bijection and y is an irrational. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form N^I for some index set I, modulo equality on a comeager subset of the domain, with its `natural' ordering.