Editing Order theory (section)

== Functions between orders ==
It is reasonable to consider functions between partially ordered sets having certain additional properties that are related to the ordering relations of the two sets. The most fundamental condition that occurs in this context is [[monotonic function|monotonicity]]. A function ''f'' from a poset ''P'' to a poset ''Q'' is '''monotone''', or '''order-preserving''', if ''a'' ≤ ''b'' in ''P'' implies ''f''(''a'') ≤ ''f''(''b'') in ''Q'' (Noting that, strictly, the two relations here are different since they apply to different sets.). The converse of this implication leads to functions that are '''order-reflecting''', i.e. functions ''f'' as above for which ''f''(''a'') ≤ ''f''(''b'') implies ''a'' ≤ ''b''. On the other hand, a function may also be '''order-reversing''' or '''antitone''', if ''a'' ≤ ''b'' implies ''f''(''a'') ≥ ''f''(''b'').

An '''[[order-embedding]]''' is a function ''f'' between orders that is both order-preserving and order-reflecting. Examples for these definitions are found easily. For instance, the function that maps a natural number to its successor is clearly monotone with respect to the natural order. Any function from a discrete order, i.e. from a set ordered by the identity order "=", is also monotone. Mapping each natural number to the corresponding real number gives an example for an order embedding. The [[complement (set theory)|set complement]] on a [[powerset]] is an example of an antitone function.

An important question is when two orders are "essentially equal", i.e. when they are the same up to renaming of elements. '''[[Order isomorphism]]s''' are functions that define such a renaming. An order-isomorphism is a monotone [[bijective]] function that has a monotone inverse. This is equivalent to being a [[surjective]] order-embedding. Hence, the image ''f''(''P'') of an order-embedding is always isomorphic to ''P'', which justifies the term "embedding".

A more elaborate type of functions is given by so-called '''[[Galois connection]]s'''. Monotone Galois connections can be viewed as a generalization of order-isomorphisms, since they constitute of a pair of two functions in converse directions, which are "not quite" inverse to each other, but that still have close relationships.

Another special type of self-maps on a poset are '''[[closure operator#Closure operators on partially ordered sets|closure operator]]s''', which are not only monotonic, but also [[idempotent]], i.e. ''f''(''x'') = ''f''(''f''(''x'')), and '''[[Closure operator|extensive]]''' (or ''inflationary''), i.e. ''x'' ≤ ''f''(''x''). These have many applications in all kinds of "closures" that appear in mathematics.

Besides being compatible with the mere order relations, functions between posets may also behave well with respect to special elements and constructions. For example, when talking about posets with least element, it may seem reasonable to consider only monotonic functions that preserve this element, i.e. which map least elements to least elements. If binary infima ∧ exist, then a reasonable property might be to require that ''f''(''x'' ∧ ''y'') = ''f''(''x'') ∧ ''f''(''y''), for all ''x'' and ''y''. All of these properties, and indeed many more, may be compiled under the label of limit-preserving functions.

Finally, one can invert the view, switching from ''functions of orders'' to ''orders of functions''. Indeed, the functions between two posets ''P'' and ''Q'' can be ordered via the [[pointwise order]]. For two functions ''f'' and ''g'', we have ''f'' ≤ ''g'' if ''f''(''x'') ≤ ''g''(''x'') for all elements ''x'' of ''P''. This occurs for example in [[domain theory]], where [[function space]]s play an important role.