
TL;DR
This paper investigates the conditions under which posets can be embedded into fields of sets while preserving finite meets and joins, establishing elementary class properties and axiomatization limitations.
Contribution
It characterizes when classes of $(eta,eta)$-representable posets are elementary and shows the non-finite axiomatizability for certain cases, also exploring pseudoelementary classes.
Findings
The class of $(eta,eta)$-representable posets is elementary for $2\,\leq\,\beta\leq\omega$.
No finite axiomatization exists when either $eta$ or $eta=\,\omega$.
Classes with countable or all-meets-and-joins-preserving representations are pseudoelementary.
Abstract
A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals and a poset is said to be -representable if an embedding into a field of sets exists that preserves meets of sets smaller than and joins of sets smaller than . We show using an ultraproduct/ultraroot argument that when the class of -representable posets is elementary, but does not have a finite axiomatization in the case where either or . We also show that the classes of posets with representations preserving either countable or all meets and joins are pseudoelementary.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
