# Article

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Keywords:
convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number
Summary:
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.
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