(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
Given any object $X$ in any category $C$, the subobjects of $X$ form a poset (though in general it may be a large one, i.e. a partially ordered proper class; cf. well-powered category). This is called, naturally enough, the poset of subobjects of $X$, or the subobject poset of $X$.
Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).
If $C$ is finitely complete, then the subobjects form a meet-semilattice, so we may speak of the semilattice of subobjects.
In any coherent category (such as a pretopos), the subobjects form a distributive lattice, so we may speak of the lattice of subobjects.
In any Heyting category (such as a topos), the subobjects of $X$ form a Heyting algebra, so we may speak of the algebra of subobjects.
The reader can probably think of other variations on this theme.
Discussions of this can be found in A.1.3 of Johnstone’s Elephant, and also this MO discussion. ↩