Questions tagged [galois-connections]

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Generalisation of "discrete" and "indiscrete" left/right adjoint to the forgetful functor for general "ordered" Categories with monotone. morphisms.

The following is taken from "An introduction to Category Theory" by Harold Simmons $\color{Green}{Background:}$ $\textbf{Example (Galois connection):}$ We modify the category of $\textbf{...
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Galois connection arising from discussion of flat module and pure exact sequence.

There is somewhat of symmetry in the definition of flat module and pure short exact sequence which can be made precise as follows. Let $\mathcal{R}$ be the class of all right $R$-modules, $\mathcal{S}$...
Zhenhui Ding's user avatar
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Is there a way to state the fundamental theorem of Galois theory in a categorical sense?

I will state the fundamental theorem of Galois theory to make things clear. Let $F/K$ be a finite dimensional Galois extension. Let $A$ be the set of all intermediate fields of $F/K$, and let $B$ be ...
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2 answers
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Galois theory for arbitrary field extension

I know that for an arbitrary field extension $E/F$ there may be distinct subgroups $G$ and $H$ of $Aut(E/F)$ with the same fixed field: $E^{G} = E^{H}$. When can we conclude that for subextensions $E/...
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If $(f, g)$ is a Galois connection between two bounded lattices, then if $T$ is an ideal we have $f^{-1}(T)$ is an ideal

Let $\mathcal{I}(L)\:$ and $\mathcal{I}(N)\:$ be the ideal lattices of the bounded lattices $L$ and $N$ and let $(f, g)$ be a Galois connection between $L$ and $N$, then show that $\:\forall \:\: T \...
AleVanDerBauch's user avatar
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1 answer
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Prove that if $N, M \leq\operatorname{Aut}(K/F) $ in a galois extension $K/F $, the fixed field of $\langle N, M \rangle $ is $F^N \cap F^M $

The Fundamental Theorem of Galois Theory states that in a Galois extension $K/F$, there is a bijection of subfields $E$ of $K$ containing $F$ and the subgroupsn $H$ of $G$. For any subgroup $N \leq\...
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$I(A)$ and $I(B)$ ideal lattices, then $F(J) = \downarrow \psi(J)$ and $G(U)=\downarrow \phi(U)$ is a connection of Galois between $I(A)$ and $I(B)$.

Let $A$ and $B$ be bounded lattices, $\mathcal{I}(A)$ and $\mathcal{I}(B)$ the ideal lattices of $A$ and $B$ and let $(\phi, \psi)$ be a Galois connection between $A$ and $B$: show that $\forall J \in ...
F.inc's user avatar
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Galois Connection from $fgf=f$ and $gfg=g$.

Let $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be a $\bigvee$-morphism and a $\bigwedge$-morphism of complete lattices, respectively. If $f\circ g\circ f=f$ and $g\circ f\circ g=g$, then it is true ...
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left adjoint preserve join applicable on empty set?

In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2.87(b) & answer of 2.104(1) assume that left adjoint preserve join applicable on empty set, i.e. $ \left( v \otimes \bigvee_\...
Shaq Xu's user avatar
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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 $ ...
Shaq Xu's user avatar
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Charecterization of Topologies via Galois Connections

Let $X$ and $Y$ be two sets and let $F: \mathcal{P}(X) \to \mathcal{P}(Y)$ and $G: \mathcal{P}(Y) \to \mathcal{P}(X)$ be two set functions that satisfy $$F(A) \subseteq B \iff A \subseteq G(B). \tag{1}...
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3 answers
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Image of sup/inf under Galois connection between lattices

This is Lemma 2.4 in Pete Clark's notes on commutative algebra. Given two posets $(X,\le)$ and $(Y,\le)$, an (antitone) Galois connection between $X$ and $Y$ is a pair of maps $F:X\to Y$ and $G:Y\to X$...
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Constraint satisfaction problem with a semilattice polymorphism is polynomial-soluble

Let $\mathbb{A}=(A;R)$ be a relation structure, where $A$ is a finite set, $\varnothing\neq R\subseteq A^2$ is a relation preserved by a semilattice operation $f$. This means that $f:A^2\to A$ is ...
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0 answers
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Galois Theory of Ramified Coverings & Classical Galois Theory

The question adresses the answer in this thread: Algebraic closure of $k((t))$ In the answer reuns used a theory relating classical Galois theory with Galois theory of ramified coverings. I'm an ...
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1 answer
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How to prove that image and inverse image form a Galois connection

Let $f'\colon X \to Y$ and $P(X)$ and $P(Y)$ be the powersets of $X$ and $Y$. Let $f \colon P(X) \to P(Y)$, $f(U) = \{ y \in Y | \exists x \in U (f'(x) = y) \},$ and $f^{-1} \colon P(Y) \to P(X)$, ...
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An exercise on Galois Connections

From "An invitation to General Algebra and Universal Constructions" of George M. Bergman. (i) Let $X$ be a set, $S = T = P(X)$, the set of all subsets of $X$, and let $R$ be the relation of ...
Giovanni Barbarani's user avatar
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Does $\mathcal C=\mathrm{Pol}(\mathrm{Inv}(\mathcal C))$ hold for clones on an infinite set?

I would like to know, whether the following theorem holds also for clones on an infinite set? Theorem (Geiger; Bodnarcuk, Kaluznin, Kotov, Romov). Let $A$ be a finite set, then $$\mathrm{Clo}(A)=...
user771160's user avatar
1 vote
1 answer
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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 ...
Amrith Krishna's user avatar
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1 answer
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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 ...
Pippo's user avatar
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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 ...
shadow10's user avatar
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2 votes
2 answers
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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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1 answer
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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 ...
Santana Afton's user avatar
2 votes
1 answer
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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....
user267839's user avatar
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2 votes
2 answers
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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 ...
user38770's user avatar
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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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5 votes
1 answer
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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. ...
Apocalisp's user avatar
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1 answer
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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. ...
WhatsDUI's user avatar
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1 answer
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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 $f(x)\in B$ ...
Mario Carneiro's user avatar
3 votes
1 answer
123 views

The relation between the left- and right- adjoints participating in Galois connections w.r.t. a common functor

Consider a monotone Galois connection $(F,G_r)$. Suppose that $(G_l,F)$ is another monotone Galois connection, where $F$ is the same functor in both connections. Has the relation between $G_l$ and $...
Evan Aad's user avatar
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1 answer
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Meaning of dots in Lattice / Galois Connection

Wondering what the dots mean (and what the whole element it's a part of means) in the slide below (above pos and neg). I have ...
Lance's user avatar
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2 votes
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An interior/closure Galois connection

In A Primer on Galois Connections, the authors define a Galois connection thus (definition 1, p. 104). Consider posets $\mathcal{P} = \langle P, \leq\rangle$ and $\mathcal{Q} = \langle Q,\...
Evan Aad's user avatar
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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 ...
LudvigH's user avatar
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Condition for Being Galois

I am thinking about the following excerpt of Patrick Morandi's "Field and Galois Theory" from chapter 1 section 2, page 21: Here, $K$ is a field, and $\mathcal{F}(G)$ denotes the subfield $\{ x \in K ...
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1 answer
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Galois connection, basic property, inclusion

Let $A,B$ be sets and $\mu:{\cal P}(A)\to {\cal P}(B),$and $\iota:{\cal P}(B) \to {\cal P}(A),$where $\cal P$ denotes the power set. Let $X,X'\subseteq A$ and $Y,Y'\subseteq B.$ Suppose that we have a ...
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3 votes
2 answers
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Galois connections between quasi orders?

A binary relation over a set $A$ is a quasi order if it is reflexive and transitive. It would be a partial order if it also were anti-symmetric. I've always seen Galois connections be defined over ...
Nicola Gigante's user avatar
2 votes
0 answers
98 views

Can the twin prime conjecture be stated in terms of ideals in a useful way?

Let $\mathcal{I}^{\bullet}(\Bbb{Z})$ be the set of nonzero ideals of $\Bbb{Z}$. It is a cancellative multiplicative monoid since $\Bbb{Z}$ is a PID. Define $\mathcal{I}^{\bullet} \xrightarrow{\phi} \...
Daniel Donnelly's user avatar
3 votes
1 answer
229 views

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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1 vote
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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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2 votes
1 answer
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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 ...
porton's user avatar
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1 vote
1 answer
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Longer chains of adjoints :-)

The first question: Is in true that there exist chains of Galois connections (let's limit to Galois connections between posets) of arbitrary lengths $n$? $F_0(a) \leq b$ if and only if $a \leq F_1(b)...
porton's user avatar
  • 5,063
5 votes
1 answer
273 views

Galois connection and order-isomorphisms

Assume that we have a Galois connection formed by two monotone maps $f\colon X\to Y$ and $g\colon Y\to X$. I want to know whether the following statement is true: if $f$ is bijective, then $f$ is an ...
J. Karen's user avatar
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$f\in K[t]$ is irreducible if and only if $G$ $(\Gamma(L:K))$ acts transitively on the roots of $f$. [duplicate]

Let $K$ be a field, $f\in K[t]$ separable over $K$, $L$ the splitting field of $f$ over $K$, and $G=\Gamma(L:K)$. I need to prove that $f$ is irreducible if and only if $G$ acts transitively on the ...
Ldog327's user avatar
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-1 votes
1 answer
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Two nonisomorphic tensor products - where is the error?

Where is the error in the following? In the category of all posets with increasing maps: With product order it becomes a monoidal category. With "product" which maps a pair posets into the set of ...
porton's user avatar
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1 vote
0 answers
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If a Galois connection does exists, how is it called?

Let $\phi$ be a function from a poset $B$ to a poset $A$. $f \mapsto \min \{ g\in B \mid \phi(g) \geq f \}$ is called the lower adjoint of $\phi$ and $\phi$ is called an upper adjoint. These two ...
porton's user avatar
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1 vote
1 answer
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Greatest Galois connection

Let $\mathfrak{A}$, $\mathfrak{B}$ be bounded posets. The main question: Explicitly describe (and prove that it exists) the greatest Galois connection between $\mathfrak{A}$ and $\mathfrak{B}$ (as ...
porton's user avatar
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2 votes
1 answer
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Understanding the Basic Theorem on Concept Lattices

In Ganter and Wille's Applied Lattice Theory: Formal Concept Analysis, one can find the following definition: Basic Theorem on Concept Lattices. Let $K := (G, M, I)$ be a formal context. Then $\...
Bartek Dzieńkowski's user avatar
1 vote
0 answers
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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 ...
porton's user avatar
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Galois Connection Between Posets

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 ...
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