Name for logical predicates which act like homomorphisms from set union to logical conjunction? Is there a name for predicates defined over sets with the property that:
$$P(S\cup Q)\iff P(S)\land P(Q)$$
For example the predicate $P(Q)=``Q\text{ is empty"}$ would be one such predicate because:
$$S\cup Q=\emptyset\iff (S=\emptyset)\land (Q=\emptyset)$$
It looks sort of like the definition for a homomorphism between algebraic structures.

These are semilattice homomorphisms. A semilattice is a set with a binary operation that is associative, commutative, and idempotent (i.e., $x \vee x = x$). As in algebra, a semilattice homomorphism is a function that preserves the operation. Examples of semilattices include:


*

*subsets of a given set, where the operation is union

*subsets of a given set, where the operation is intersection

*propositions, where the operation is conjunction

*propositions, where the operation is disjunction.


So, if you specify what your semilattices are, you get the notion of homomorphism you're looking for.

Something to file away for later, since I'm guessing you haven't seen category theory yet: it is very natural, also, to view these maps as functors between certain categories. You're then asking questions related to whether the functors are "left exact" or "right exact."
