Decidability and ℵ0-categoricity of theories of partially ordered sets
James H. Schmerl · Journal of Symbolic Logic · 1980
Abstract This paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ0-categorical there is a linear order of A such that ( , <) is ℵ0-categorical. Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory. Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory. There is a section dealing just with decidability of partially ordered sets, the main result of this section being Theorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.