Journal of Symbolic Logic

Decidability and $\aleph_0$-Categoricity of Theories of Partially Ordered Sets

James H. Schmerl

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This paper is primarily concerned with $\aleph_0$-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on $\aleph_0$-categoricity. Among the latter are the following. Corollary 3.3. For every countable $\aleph_0$-categorical $\mathfrak{U}$ there is a linear order of $A$ such that $(\mathfrak{U}, <)$ is $\aleph_0$-categorical. Corollary 6.7. Every $\aleph_0$-categorical theory of a partially ordered set of finite width has a decidable theory. Theorem 7.7. Every $\aleph_0$-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 $K_P$ is the class of partially ordered sets which do not embed $(P, <)$, then $\mathrm{Th}(K_P)$ is decidable iff $K_P$ contains only reticles.

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J. Symbolic Logic, Volume 45, Issue 3 (1980), 585-611.

First available in Project Euclid: 6 July 2007

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Schmerl, James H. Decidability and $\aleph_0$-Categoricity of Theories of Partially Ordered Sets. J. Symbolic Logic 45 (1980), no. 3, 585--611.

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