present a problem in that the argument of a predicate can be an entity comprised of multiple basic individuals (those we ordinarily quantify over, and ascribe properties to). Most approaches to this problem employ a plural operator, say, plur, allowing us to map a singular predicate P into a plural predicate plur(P), applicable to collective entities. These collective entities are usually assumed to form a join semilattice with atomic elements (singular entities) that are ordinary individuals (e.g., Scha 1981; Link 1983; Landman 1989, 2000). When an overlap relation is assumed, and when all elements of the semilattice are assumed to have a supremum (completeness), the result is a complete Boolean algebra except for lack of a bottom element (because there is no null entity that is a part of all others). One theoretical issue is the relationship of the semilattice of plural entities to the semilattice of material parts of which entities are constituted. Though there are differences in theoretical details (e.g., Link 1983; Bunt 1985), it is agreed that these semilattices should be aligned in this sense: When we take the join of material parts of which several singular or plural entities are constituted, we should obtain the material parts of the join of those singular or plural entities. Note that while some verbal predicates, such as (intransitive) gather, are applicable only to collections, others, such as ate a pizza, are variously applicable to individuals or collections. Consequently, a sentence such as