Editing Adjoint functors (section)

====Posets====
Every [[partially ordered set]] can be viewed as a category (where the elements of the poset become the category's objects and we have a single morphism from ''x'' to ''y'' if and only if ''x'' ≤ ''y''). A pair of adjoint functors between two partially ordered sets is called a [[Galois connection]] (or, if it is contravariant, an ''antitone'' Galois connection). See that article for a number of examples: the case of [[Galois theory]] of course is a leading one. Any Galois connection gives rise to [[closure operator]]s and to inverse order-preserving bijections between the corresponding closed elements.

As is the case for [[Galois group]]s, the real interest lies often in refining a correspondence to a [[duality (mathematics)|duality]] (i.e. ''antitone'' order isomorphism). A treatment of Galois theory along these lines by [[Irving Kaplansky|Kaplansky]] was influential in the recognition of the general structure here.

The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:
* adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status
* closure operators may indicate the presence of adjunctions, as corresponding [[monad (category theory)|monads]] (cf. the [[Kuratowski closure axioms]])
* a very general comment of [[William Lawvere]]<ref>[[William Lawvere|Lawvere, F. William]], "[http://www.tac.mta.ca/tac/reprints/articles/16/tr16abs.html Adjointness in foundations]", ''Dialectica'', 1969. The notation is different nowadays; an easier introduction by Peter Smith [http://www.logicmatters.net/resources/pdfs/Galois.pdf in these lecture notes], which also attribute the concept to the article cited.</ref> is that ''syntax and semantics'' are adjoint: take ''C'' to be the set of all logical theories (axiomatizations), and ''D'' the power set of the set of all mathematical structures. For a theory ''T'' in ''C'', let ''G''(''T'') be the set of all structures that satisfy the axioms ''T''; for a set of mathematical structures ''S'', let ''F''(''S'') be the minimal axiomatization of ''S''. We can then say that ''S'' is a subset of ''G''(''T'') if and only if ''F''(''S'') logically implies ''T'': the "semantics functor" ''G'' is right adjoint to the "syntax functor" ''F''.
* [[division (mathematics)|division]] is (in general) the attempt to ''invert'' multiplication, but in situations where this is not possible, we often attempt to construct an ''adjoint'' instead: the [[ideal quotient]] is adjoint to the multiplication by [[ring ideal]]s, and the [[material conditional|implication]] in [[propositional calculus|propositional logic]] is adjoint to [[logical conjunction]].