Commentationes mathematicae Universitatis Carolinae | |
A dyadic view of rational convex sets | |
Gábor Czédli1  | |
关键词: convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number; | |
DOI : | |
学科分类:物理化学和理论化学 | |
来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics | |
【 摘 要 】
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.
【 授权许可】
CC BY
【 预 览 】
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RO201904032964566ZK.pdf | 56KB | download |