Some semilattices of definable sets in continuous logic

TL;DR

This paper explores which semilattices of definable sets can occur in continuous logic, showing that any finite semilattice can be realized in a superstable theory and extending results to certain infinite semilattices.

Contribution

It demonstrates that all finite semilattices can be represented as the collection of definable sets in superstable theories, and extends this to specific infinite semilattices.

Findings

Any finite semilattice arises as definable sets in a superstable theory.

Certain infinite semilattices, including those containing an exact pair above ω, are realizable.

Semilattices of filters in countable meet-semilattices can be represented in continuous theories.

Abstract

In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend this construction to some infinite semilattices. In particular, we show that the following semilattices arise in continuous theories: $\alpha+1$ and $(\alpha+1)^\ast$ for any ordinal $\alpha$, a semilattice containing an exact pair above $\omega$, and the lattice of filters in $L$ for any countable meet-semilattice $L$. By previous work of the author, this establishes that these semilattices arise in stable theories. The first two are done in languages of cardinality $\aleph_0 + |\alpha|$, and the latter two are done in countable languages.