in mathematics, especially in order theory, a galois connection is a particular correspondence (typically) between two partially ordered sets (posets). the same notion can also be defined on preordered sets or classes; this article presents the common case of posets. galois connections generalize the correspondence between subgroups and subfields investigated in galois theory (named after the french mathematician Ãvariste galois). they find applications in various mathematical theories. a galois connection is rather weak compared to an order isomorphism between the involved posets, but every galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. the literature contains two closely related notions of "galois connection". in this article, we will distinguish between the two by referring to the first as (monotone) galois connection and to the second as antitone galois connection. the term galois correspondence is sometimes used to mean bijective galois connection, just is simply an order isomorphism (or dual order isomorphism, dependently on whether we take monotone or antitone galois connections). == definitions == === (monotone) galois connection === let  and  be two partially ordered sets. a monotone galois connection between these posets consists of two monotonemonotonicity follows from the following condition. see the discussion of the properties. it is only explicit in the definition to distinguish it from the alternative antitone definition. one can also define galois connections as a pair of monotone functions that satisfy the laxer condition that for all  in ,  and for all y in b, f(g(y)) â¤ y. functions:  and , such that for all  in  and  in , we have : if and only if . in this situation,  is called the lower adjoint of  and  is called the upper adjoint of f. mnemonically, the upper/lower terminology refers to where the function application appears relative to â¤;gierz, p. 23 the term "adjoint" refers to the fact that monotone galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. other terminology encountered here is coadjoint (resp. adjoint) for the lower (resp. upper) adjoint. an essential property of a galois connection is that an upper/lower adjoint of a galois connection uniquely determines the other: : is the least element {{math|
Abstract_interpretation Closure_operators Galois_theory Order_theory