Questions tagged [automorphisms]
An automorphism is an isomorphism from an object to itself, and which also preserves the objects structure .
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Automorphism on the hyperreals
$\DeclareMathOperator\hal{hal}$A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered ...
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Is Acyclic ZF consistent with downshifting automorphisms?
Recall the criterion of acyclic comprehension. This is shown to be equivalent to stratified comprehension for language $\sf FOL(=, \in)$, given minimal assumptions. [See here, and here].
Let Acyclic ...
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Can MLU prove symmetric comprehension?
Working in $\sf ML$$\sf U$:
Define: $x \in^f y \iff f(x) \in y$
by $\varphi^f$ we mean the formula obtained by merely replacing each "$\in$" symbol in formula $\varphi$ by the symbol "...
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Can we have an inverted iterative hierarchy?
Let stratified $\sf Z$ have all axioms of $\sf Z$-$\sf Reg.$ with Infinity stipulated by existence of a Dedekind infinite set, and with Separation restricted to use only stratified formulas, where ...
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Can we have external automorphisms over intersectional models?
Is the following inconsistent:
By "intersectional" set I mean a set having the intersectional set of every nonempty subset of it, being an element of it.
$\forall S \subset M: S\neq \...
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Extension of automorphism of shift of finite type
$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two subshifts of finite type and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any homeomorphism $\...
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What is a cogroup and what are coactions?
What is a cogroup and what are coactions?
A very nice way to think about a group action on an object $X$ is as a group homomorphism from $G$ to $\operatorname{Aut}(X)$. Is there something similar for ...
user525442
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Combinatorial classes where not almost all objects are asymmetric
Let $\mathcal{C} = \bigcup_{n=0}^{\infty}\mathcal{C}_n$ be a class of finite (labeled) combinatorial objects, where $\mathcal{C}_n$ is the set of objects on $[n] = \{1,2,\dotsc,n\}$. For example, $\...
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Can automorphism equivalence in a free group be detected in a nilpotent quotient?
If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...
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Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
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Subfields of division rings of degree $2$ which are not invariant
Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...
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Twisting a graded algebra by an automorphism (Transitivity)
Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...
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The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings
I am looking for further proofs, preferably in the literature, of the following result:
Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...
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Composition of correspondences pulled back to $\mathrm{CH}_0$
Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition,
$$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...
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What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?
A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...
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