Geodesic
A dyadic view of rational convex sets
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 159-173.

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Let $F$ be a subfield of the field $\mathbb R$ of real numbers. Equipped with the binary arithmetic mean operation, each convex subset $C$ of $F^n$ becomes a commutative binary mode, also called idempotent commutative medial (or entropic) groupoid. Let $C$ and $C'$ be convex subsets of $F^n$. Assume that they are of the same dimension and at least one of them is bounded, or $F$ is the field of all rational numbers. We prove that the corresponding idempotent commutative medial groupoids are isomorphic iff the affine space $F^n$ over $F$ has an automorphism that maps $C$ onto $C'$. We also prove a more general statement for the case when $C,C'\subseteq F^n$ are barycentric algebras over a unital subring of $F$ that is distinct from the ring of integers. A related result, for a subring of $\mathbb R$ instead of a subfield $F$, is given in Czédli G., Romanowska A.B., Generalized convexity and closure conditions, Internat. J. Algebra Comput. 23 (2013), no. 8, 1805--1835.
Classification : 08A99, 52A01
Keywords: convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number 
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Czédli, Gábor; Maróti, Miklós; Romanowska, A. B. A dyadic view of rational convex sets. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 159-173. http://geodesic.mathdoc.fr/item/CMUC_2014__55_2_a2/