Galois connection

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 Évariste 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; this is simply an order isomorphism (or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).

Let $(A, \leq)$ and $(B, \leq)$ be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions: $F : A \to B$ and $G : B \to A$ , such that for all $a$ in $A$ and $b$ in $B$ , we have

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