# Questions tagged [galois-connections]

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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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 $\rho$: ...