A map between non-empty partially ordered sets is said to be order-preserving if entails . The collection of order-preserving maps can be endowed with an ordering relation in a natural way by setting
in iff for all .
When considering any partially ordered set, or poset, for short, a natural question arising is that of the existence of (least) upper bounds, and dually, (greatest) lower bounds. A subset of a poset is said to have an upper bound if there exists such that for all . The collection of upper bounds of is denoted by .
The basic question of this post is: When do have a (least) upper bound?
When you start tinkering with and its elements, you notice that the existence of bounds in the codomain is more important than in the domain . This can be illustrated by two examples:
(1) If and have no upper bound in for some , then are not bounded in .
(2) If is a lattice, then so is .
As for the proof of statement (2), let us denote the supremum of by and the infimum by . Let . Then it is easy to verify that the map given by
is the supremum of ; the infimum is constructed in a similar manner.
Let’s have a look at the converse statements of (1) and (2).
The converse of (1) would say that if for all , then have an upper bound in . We can construct a finite counterexample to this statement by letting ordered by and ordered by for . So if we draw we get a v-shaped figure, and looks like a butterfly.
Let be the maps given by and and and . Then for all , but it is easily seen that there is no order-preserving map that is an upper bound of .
The converse of (2), on the other hand, is true. In order to tackle this, we need the concept of a retract. We say that a poset is a retract of a poset if there are order-preserving maps and such that is the identity map.
Lemma 1. is a retract of .
Proof. Let be defined by where is the constant map that maps everything from to . For defining , fix and for let . A routine check shows that are order-preserving and .
Lemma 2. Any retract of a lattice is a lattice.
Proof. Let be a lattice and a poset and and such that is the identity map. For it suffices to check that is the least upper bound of . It clearly is an upper bound. Suppose . Then since is order-preserving, therefore . Using the fact that is order-preserving we get that , so is the least upper bound.
In summary, we get that is a lattice if and only if is a lattice. (Note that this equivalence only works for ).
We gave partial answers, but are still lacking a full one, to the main question of this post: When do have a (least) upper bound?