Name for logical predicates which act like homomorphisms from set union to logical conjunction?

Question

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.

Answer 1

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.

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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."