Galois connections
Jul. 14th, 2006 01:14 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
anhinga_draftsA remark on (monotone) Galois connections (соответствия Галуа).
Consider two partially ordered sets, (A, ≤A), and (B, ≤B), and two monotone functions, F : A → B, G : B → A.
F and G form a Galois connection, if for all a in A, b in B, F(a) ≤B b ⇔ a ≤A G(b).
In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F.
Consider a Galois connection (F,G) and true statements G(b) ≤A G(b), F(a) ≤B F(a). Then we obtain: for all b∈B, F(G(b)) ≤B b, and for all a∈A, a ≤A G(F(a)).
Consider two partially ordered sets, (A, ≤A), and (B, ≤B), and two monotone functions, F : A → B, G : B → A.
F and G form a Galois connection, if for all a in A, b in B, F(a) ≤B b ⇔ a ≤A G(b).
In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F.
Consider a Galois connection (F,G) and true statements G(b) ≤A G(b), F(a) ≤B F(a). Then we obtain: for all b∈B, F(G(b)) ≤B b, and for all a∈A, a ≤A G(F(a)).
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Date: 2006-07-14 06:34 am (UTC)