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Galois connection

Defines: 
upper adjoint, lower adjoint
Synonym: 
Galois correspondence, Galois connexion
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

06A15 no label found

Comments

I'm having a problem with the following sentence in this article:
"The existence of a Galois connection between P and Q is the same as the existence of a Galois connection between Q and P so the definition is unambiguous."

Is this right? Does it use the same maps? f^* and f_*, just interchanged?
I can't seem to prove this.
Perhaps I am just misundertanding.

-Mike Zabrocki

Yes, use the duals of the orderings (you can define $a {\le}^{\prime} b$ if and only if $b {\le} a$) on each of the posets, and switch the maps $f^*$ and $f_*$.

Does this answer your question?

Chi

I guess. But what this says to me is
"The existence of a Galois connection between P and Q is the same as the existence of a Galois connection between Q^D and P^D."
where P^D means the dual posets. The way I am reading the sentence in
the article says that there is a Galois connection between P and Q iff there
is a Galois connection between Q and P. Is this right? Thanks.

-Mike Zabrocki

I clarified the wording in the entry a little. Does this help?

Chi

I think that it is clear now.
BTW, you are missing an open parenthesis at "Q, \leq_Q')".

The confusion for me came up because I identify P with
the poset (P, \le_P) which you were not doing explictly. So I interpret
"a Galois connection between P and Q" to mean
"a Galois connection between (P, \le_P) and (Q, \le_Q)"

When you say there is a
"We denote a Galois connection between P and Q" you mean,
"there exists orders \le_P and \le_Q such that there
is a Galois connection between (P, \le_P) and (Q, \le_Q)."
right?

-Mike

That's right. And thanks for the comment.

Chi

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