Partially ordered sets and the independence property. (English) Zbl 0699.03020

Summary: No theory of a parially ordered set of finite width has the independence property, generalizing Poizat’s corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.


03C65 Models of other mathematical theories
06A06 Partial orders, general
06D99 Distributive lattices
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