# Questions tagged [galois-connections]

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### "function monotonicity" condition is required in definition of Galois connection?

In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf definition of Galois connection is: A Galois connection between preorders P and Q is a pair of monotone maps $f:P→Q$ and $g:Q→P$ ...
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### Understanding infimum in a complete lattice

For any two formal concepts, $(A_1,B_1)$ and $(A_2,B_2)$ of a formal context, the standard definition for the supremum and infimum in a complete lattice are as ...
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### Action of $\operatorname{Gal}(K(X))$ on the normalization of $X$ in $K(X)^\text{sep}$

I'm reading this part of The Stacks Project regarding ramification theory. Right before defining decomposition and inertia groups for schemes there is the following passage: Let $X$ be a normal ...
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### Trivial covering of a topological space

We have a connected topological space $X$ and $p:E\to X$ a Galois covering projection with $E$ not necessarily connected, let us call $\text{Aut}(E/X)=G$. Suppose it is given that every homomorphism ...
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### A faster way to find the quadratic extensions of a field extension?

The usual method I have for finding Galois correspondence goes like this : Say we have the Splitting field of $x^4-3$, i.e. $\Bbb Q(i,\sqrt[4]{3})$. Then it's generating automorphisms are those ...
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### A reference for an explicit statement of the Galois correspondence in a Galois category

The definition of a Galois category was cooked up intentionally to create the general setting where Galois correspondences appear. There are plenty of the resources (e.g. here and here) that go into ...
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### Equivalence between Category of Covers and $\pi_1(X)$ Sets

I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at page 38): In order to show the category equivalence claimed in Thm 2....
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### What makes a good mathematical theory?

I recently read about Galois connections, and that they show up in a lot of different places in mathematics. Given there apparent ubiquity, I thought they might have a rich theory. However, when ...
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### Infimum and Supremum (of sets) - Formal Concept Analysis

I am taking a course of Introduction to Formal Concept Analysis and I have an uncertainty about the definition of supremum (least comum superconcept) and infimum (greatest comum subconcept) of formal ...
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### Adjoint to multiplication in a GCD lattice

Consider the lattice on the nonzero natural numbers where the meet $a \wedge b$ is defined to be the greatest common divisor of $a$ and $b$, and the join $a \vee b$ is the least common multiple. ...
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### Image and Preimage - Proof of Galois Connection

Here is a problem from my Graduate Abstract Algebra course. I'm not quite sure how to go about part d at all, though the rest of the parts were easily proved using some basic machinery I already knew. ...
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### Does $f[A] \cap B\subseteq f[A\cap f^{-1}[B]]$ generalize beyond sets?

If $f:X\to Y$, and $A\subseteq X$, $B\subseteq Y$, then the equation $f[A] \cap B\subseteq f[A\cap f^{-1}[B]]$ holds. Indeed, let $y\in f[A]\cap B$, then $y=f(x)$ for some $x\in A$; since $f(x)\in B$ ...
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### Conditions for embedding to be part of Galois connection?

I am working though 7 sketches in compositionality and have almost reached the end of chapter 1, which is very much concerned with Galois Connections. One of the questions on the subject that is not ...
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### Under what conditions $c = \gamma(\alpha(c))$ for a Galois connection?

I have the basic definition of Galois connection. Let $(C,\leq)$ and $(A,\sqsubseteq)$ be partial orders and $\alpha: C \rightarrow A$, $\gamma: A \rightarrow C$ monotonic functions. They form a ...
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### Reference request: the category of adjunctions between posets as categories that induce a partiuclar monad

I am interested in the category $A$ of adjunctions that induce a monad $c : C \to C$ where $C$ is a poset. (The description of $A$ is in a previous math.se post.) For a general $C$, of course, $A$ ...
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### Show $F(U) = K((x^q -x)^{q-1})$.

Let $K$ be a finite field with $q$ elements. Show that if $U$ is the subgroup of $Aut(K(x)/K)$ which consists of all mappings $\sigma$ of the form $(\sigma \theta)(x) = \theta(ax+b)$ with $a \neq 0$ ...
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### Let $K$ be a finite field with $q$ elements then $Aut(K(x)/K)$ has $q^3 -q$ elements. [closed]

Let $K$ be a finite field with $q$ elements. $(a)$ Show that $Aut(K(x)/K)$ has $q^3 -q$ elements. $(b)$ Show that $F(Aut(K(x)/K)) = K(\phi)$ where \phi(x) = \frac{(x^{q^2} -x)^{q+1}}{(x^q -x)^{q^...
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### Is this always a Galois connection?

In continuation of this question: Suppose that we know that (for every elements $a$, $b$ of some posets) both: $F(a) = \inf \{ c \mid a \leq G(c) \}$; $G(b) = \sup \{ c \mid b \geq F(c) \}$. Does ...
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### How Galois connections between powersets correspond to binary relations?

How to show the well-known bijective correspondence between Galois connections (or rather polarities) between two powersets on some (fixed) sets with binary relations between these sets? You can also ...
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I have the next doubt about this problem: In a Galois Connection between posets, show that the subset $\{p\mid p=RLp\}$ of $P$ is equal $\{p\mid p=Rq \; for\; some \; q\}$ and give a bijection from ...