Suppose <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$({\mathbb {R}}^n,\oplus )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mo>⊕</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a commutative semigroup such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A(a\oplus b)=A(a)\oplus A(b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⊕</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a,b \in {\mathbb {R}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A\in SO_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi>S</mi> <msub> <mi>O</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, then S has a maximal subgroup G. The group G is invariant under <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$SO_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and either <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G=\{ 0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> or G is isomorphic to the Abelian group <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$({\mathbb {R}}^n,+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The latter case holds if and only if there is an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a\in {\mathbb {R}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a\oplus 0\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>⊕</mo> <mn>0</mn> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

A note on the problem of the parallelogram of forces (the axiomatization of vector addition)

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2025年9月3日

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Miklós Laczkovich

摘要

Suppose $n\ge 3$ $n \geq 3$ . We prove that if $({\mathbb {R}}^n,\oplus )$ $(R^{n}, \oplus)$ is a commutative semigroup such that $A(a\oplus b)=A(a)\oplus A(b)$ $A (a \oplus b) = A (a) \oplus A (b)$ for every $a,b \in {\mathbb {R}}^n$ $a, b \in R^{n}$ and $A\in SO_n$ $A \in S O_{n}$ , then S has a maximal subgroup G. The group G is invariant under $SO_n$ $S O_{n}$ , and either $G=\{ 0\}$ $G = {0}$ or G is isomorphic to the Abelian group $({\mathbb {R}}^n,+)$ $(R^{n}, +)$ . The latter case holds if and only if there is an $a\in {\mathbb {R}}^n$ $a \in R^{n}$ such that $a\oplus 0\ne 0$ $a \oplus 0 \neq 0$ .