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Questions tagged [galois-connections]

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4
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1answer
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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 ...
2
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1answer
57 views

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....
0
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1answer
48 views

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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1answer
56 views

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 ...
3
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1answer
54 views

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. ...
0
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1answer
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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. ...
0
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1answer
51 views

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 $f(x)\in B$ ...
3
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1answer
53 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 $...
0
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1answer
34 views

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 ...
2
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0answers
72 views

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,\...
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0answers
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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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0answers
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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 ...
0
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1answer
44 views

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 ...
2
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2answers
40 views

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 ...
2
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0answers
90 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} \...
3
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1answer
71 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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0answers
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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, $A$ ...
2
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1answer
103 views

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 $a \neq 0$ ...
0
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1answer
155 views

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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1answer
50 views

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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1answer
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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)...
3
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1answer
114 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 ...
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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 ...
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1answer
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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 ...
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0answers
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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 ...
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1answer
96 views

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 ...
2
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1answer
127 views

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 $\...
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0answers
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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 ...
0
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1answer
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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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4answers
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The Galois connection between topological closure and topological interior

[Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.] Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and ...
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0answers
57 views

Classification of Galois covering maps over a bouquet of 2 circles

The b-sheeted Galois covering maps over $C^*$ are equivalent to $z\mapsto z^b$. I wonder if there is an analogous statement for such Galois covers over C except two points $0,1$. Is that true that ...
4
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1answer
81 views

Galois comodules

I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not? I am really interested in Comodule theory, and I am very curious to know the answer for ...
6
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2answers
271 views

Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
4
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1answer
94 views

Do any familiar adjunctions arise from this construction?

I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide. For any set ...
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1answer
134 views

Question on Galois Connections

I have been learning about Galois Connections, and came across Steve Roman's book "Lattices and Ordered Sets." Two questions: There is a theorem which says that if $(\Pi,\Omega)$ is a Galois ...
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2answers
518 views

A function on binary relations

Let $\rho$ is a function mapping every binary relation $f$ (on some set $U$) into a function which maps binary relations into binary relations by the formula $$(\rho(f))(g) = f\circ g.$$ Is $\rho$: ...