Some semilattices of definable sets in continuous logic

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 -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: α+1 and (α+1)^ for any ordinal α, a semilattice containing an exact pair above ω, 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 ₀ + |α|, and the latter two in countable languages.