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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2012
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| Accès en ligne: | https://arxiv.org/abs/1203.5054 |
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| _version_ | 1866913568566280192 |
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| author | Czédli, Gábor Maróti, Miklós Romanowska, A. B. |
| author_facet | Czédli, Gábor Maróti, Miklós Romanowska, A. B. |
| 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1203_5054 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | A dyadic view of rational convex sets Czédli, Gábor Maróti, Miklós Romanowska, A. B. Rings and Algebras 08A99 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}. |
| title | A dyadic view of rational convex sets |
| topic | Rings and Algebras 08A99 |
| url | https://arxiv.org/abs/1203.5054 |