Gábor Czédli, Miklós Maróti, A. B. Romanowska
A dyadic view of rational convex sets

Comment.Math.Univ.Carolin. 55,2 (2014) 159-173.

Abstract: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 Cz\'edli G., Romanowska A.B., {\it Generalized convexity and closure conditions\/}, Internat. J. Algebra Comput. {\bf 23} (2013), no.~8, 1805--1835.

Keywords: convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number
AMS Subject Classification: 08A99 52A01