Non-elementary classes of representable posets

Abstract

A poset is $(\omega ,C)$-representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of $(\omega ,C)$-representable posets cannot be axiomatized in first order logic using the standard language of posets. We generalize this result to $(\alpha ,\beta )$-representable posets for certain values of $\alpha$ and $\beta$.

This revised version edits and expands some of the explanatory passages, primarily in the introduction but also elsewhere, and also lightly edits the references. Some proofs have been shortened, though the key ingredients remain the same. The main results are unchanged. This paper is to appear in Proceedings of the AMS. The latest revision corrects some typos

References in corpus (1)

• Representable posets

Cited by in corpus (3)

• No finite axiomatizations for posets embeddable into distributive lattices
• Recursive axiomatizations for representable posets
• Recursive axiomatisations from separation properties