(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
Could not include topos theory - contents
Given any object $X$ in any category $C$, the subobjects of $X$ form a poset, 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.