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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2012
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1203.5054 |
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Table of Contents:
- Let F be a subfield of the field 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 considered barycentric algebras over a unital subring of F that is distinct from the ring of integers. A related result, for a subring of R instead of a subfield F, is given in \cite{rczgaroman2}.