| Title:
|
A dyadic view of rational convex sets (English) |
| Author:
|
Czédli, Gábor |
| Author:
|
Maróti, Miklós |
| Author:
|
Romanowska, A. B. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
159-173 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
convex set |
| Keyword:
|
mode |
| Keyword:
|
barycentric algebra |
| Keyword:
|
commutative medial groupoid |
| Keyword:
|
entropic groupoid |
| Keyword:
|
entropic algebra |
| Keyword:
|
dyadic number |
| MSC:
|
08A99 |
| MSC:
|
52A01 |
| idZBL:
|
Zbl 06391534 |
| idMR:
|
MR3193922 |
| . |
| Date available:
|
2014-06-07T15:31:29Z |
| Last updated:
|
2016-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143798 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
