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Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-}a))\},$$called the standard equiproportion relation on $L$. This relation seems to describe the affine geometry of $L$.

Observe that for any points $x,y,z\in L$ with $(x,y,z)\sim (x,y,z)$ the point $y$ lies between the points $x$ and $z$. Therefore the betweenness relation is encoded in the equiproportion relation.

Problem. Is there any characterization of the affine spaces $\mathbb{R}^n$ in terms of the equiproportion relation? More precisely, given a set $X$ endowed with a relation ${\sim}\subseteq X^3\times X^3$ I would like to have a reasonably short list of axioms guaranteeing that the relation algebra $(X,\sim)$ is isomorphic to the space $\mathbb R^n$ endowed with the standard equiproportion relation.

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  • $\begingroup$ "$y$ lies between $x$ and $y$" seems to include a typo. $\endgroup$
    – YCor
    Oct 30, 2022 at 16:51
  • $\begingroup$ If things are described by some (finite?) set of 1st order conditions, there will be models of all infinite cardinals. $\endgroup$
    – YCor
    Oct 30, 2022 at 16:53
  • $\begingroup$ @YCor Thank you for pointing out a typo. Concerning possible axioms, I suggest that one of them should be the standard continuity axiom of Dedekind (which is second-order): $\forall c\in X\;\forall A,B\subseteq X\;(\forall a\in A\;\forall b\in B\;(c,a,b)\sim(c,a,b))\to \exists z\in X\;(\forall a\in A\;\forall b\in B\;(c,a,z)\sim (c,a,z)\wedge (c,z,b)\sim (c,z,b))$. $\endgroup$ Oct 30, 2022 at 17:13
  • $\begingroup$ @YCor One can also try the first-order counterpart of the continuity axiom and try to prove that with some extra-axioms an equiproportion space $(X,\sim)$ will be isomorphic to $F^n$ for some ordered field $F$. $\endgroup$ Oct 30, 2022 at 17:18
  • $\begingroup$ Is there a reason why you write $y{-}x$ (coded as y{-}x instead of $y-x$ (coded as y-x)? That requires you to type two additional characters whose only effect is that you don't have standard spacing before an after the minus sign. $\endgroup$ Oct 30, 2022 at 18:19

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The midpoint of $a$ and $b$ can be defined from equiproportion as the unique $m$ for which $(a,m,b)$ and $(b,m,a)$ are equiproportional.

Similarly, the collinearity of $a,b,c$ can be defined as one of $(a,b,c)$, $(b,c,a)$ or $(c,a,b)$ being equiproportional to itself.

Using these, $\mathbb{R}^n$ should be characterized by axioms for midpoints and collinearity plus the second-order axiom of continuity suggested in the comments.

For axioms of affine geometry from midpoint and collinearity, see Patrick Suppes, “Quantifier-Free Axioms for Constructive Affine Plane Geometry”, and the book referred to there: Wanda Szmielew, From Affine to Euclidean Geometry: An Axiomatic Approach.

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