# 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. 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.

Like Galois theory, Galois connections are named after the French mathematician Évariste Galois.

### Other articles related to "galois connection, galois connections, galois":

Galois Connection - Applications in The Theory of Programming
... Galois connections may be used to describe many forms of abstraction in the theory of abstract interpretation of programming languages ...
Consequence Operator - Closure Operators On Partially Ordered Sets
... Every Galois connection (or residuated mapping) gives rise to a closure operator (as is explained in that article) ... operator arises in this way from a suitable Galois connection ... The Galois connection is not uniquely determined by the closure operator ...
Residuated Mapping - Consequences
... A residuated function and its residual form a Galois connection under the (more recent) monotone definition of that concept, and for every (monotone) Galois connection ... Therefore, the notions of monotone Galois connection and residuated mapping essentially coincide ... mapping if and only if f A → B° and f * B° → A form a Galois connection under the original antitone definition of this notion ...
Adjoint Functors - History - Posets
... two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection) ... for a number of examples the case of Galois theory of course is a leading one ... Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements ...
Closure Operators On Partially Ordered Sets
... Every Galois connection (or residuated mapping) gives rise to a closure operator (as is explained in that article) ... The Galois connection is not uniquely determined by the closure operator ... One Galois connection that gives rise to the closure operator cl can be described as follows if A is the set of closed elements with respect to cl, then cl P → A is the lower adjoint of a Galois connection ...

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