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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections.
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 term Galois correspondence is sometimes used to mean a 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, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions, F : A → B and G : B → A, such that for all a in A and b in B, we have
In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤. 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 left adjoint (respectively right adjoint) for the lower (respectively upper) adjoint.
An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection uniquely determines the other:
A consequence of this is that if F or G is bijective then each is the inverse of the other, i.e. F = G −1.
Given a Galois connection with lower adjoint F and upper adjoint G, we can consider the compositions GF : A → A, known as the associated closure operator, and FG : B → B, known as the associated kernel operator. Both are monotone and idempotent, and we have a ≤ GF(a) for all a in A and FG(b) ≤ b for all b in B.
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Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections.
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 term Galois correspondence is sometimes used to mean a 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, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone functions, F : A → B and G : B → A, such that for all a in A and b in B, we have
In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤. 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 left adjoint (respectively right adjoint) for the lower (respectively upper) adjoint.
An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection uniquely determines the other:
A consequence of this is that if F or G is bijective then each is the inverse of the other, i.e. F = G −1.
Given a Galois connection with lower adjoint F and upper adjoint G, we can consider the compositions GF : A → A, known as the associated closure operator, and FG : B → B, known as the associated kernel operator. Both are monotone and idempotent, and we have a ≤ GF(a) for all a in A and FG(b) ≤ b for all b in B.
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