An n-mean (also called a “topological social choice rule”) on a topological space X is a continuous function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p:X^n\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>:</mo> <msup> <mi>X</mi> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p(x,\dots , x)=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p(x_1,\dots , x_n)=p(x_{\sigma (1)},\dots x_{\sigma (n)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <msub> <mi>x</mi> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any permutation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{1,\dots , n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. If, in addition, X is a G-space and p is equivariant with respect to the diagonal action of G on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, we say that p is an equivariant n-mean. In this paper, we continue the work initiated by Juárez-Anguiano (Topol Appl 279, 2020. <a href="https://doi.org/10.1016/j.topol.2020.107246">https://doi.org/10.1016/j.topol.2020.107246</a>), examining conditions on a G-space X under which the existence of an equivariant n-mean guarantees that X is a G-absolute retract. We also explore this problem when we remove the symmetry condition on the definition of an n-mean.

Equivariant means

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2025年9月15日

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Natalia Jonard-Pérez,
Ananda López-Poo

摘要

An n-mean (also called a “topological social choice rule”) on a topological space X is a continuous function $p:X^n\rightarrow X$ $p : X^{n} \to X$ satisfying $p(x,\dots , x)=x$ $p (x, \dots, x) = x$ for every $x\in X$ $x \in X$ and $p(x_1,\dots , x_n)=p(x_{\sigma (1)},\dots x_{\sigma (n)})$ $p (x_{1}, \dots, x_{n}) = p (x_{σ (1)}, \dots x_{σ (n)})$ for any permutation $\sigma $ $σ$ of $\{1,\dots , n\}$ ${1, \dots, n}$ . If, in addition, X is a G-space and p is equivariant with respect to the diagonal action of G on $X^n$ $X^{n}$ , we say that p is an equivariant n-mean. In this paper, we continue the work initiated by Juárez-Anguiano (Topol Appl 279, 2020. https://doi.org/10.1016/j.topol.2020.107246), examining conditions on a G-space X under which the existence of an equivariant n-mean guarantees that X is a G-absolute retract. We also explore this problem when we remove the symmetry condition on the definition of an n-mean.