# A dyadic view of rational convex sets

• Volume: 55, Issue: 2, page 159-173
• ISSN: 0010-2628

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## Abstract

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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 ${C}^{\text{'}}$ 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 ${C}^{\text{'}}$. We also prove a more general statement for the case when $C,{C}^{\text{'}}\subseteq {F}^{n}$ are barycentric algebras over a unital subring of $F$ that is distinct from the ring of integers. A related result, for a subring of $ℝ$ 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.

## How to cite

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Czédli, Gábor, Maróti, Miklós, and Romanowska, Anna B.. "A dyadic view of rational convex sets." Commentationes Mathematicae Universitatis Carolinae 55.2 (2014): 159-173. <http://eudml.org/doc/261855>.

@article{Czédli2014,
abstract = {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 $C^\{\prime \}$ 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 $C^\{\prime \}$. We also prove a more general statement for the case when $C,C^\{\prime \}\subseteq F^n$ 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.},
author = {Czédli, Gábor, Maróti, Miklós, Romanowska, Anna B.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number; convex sets; modes; barycentric algebras; commutative medial groupoids; entropic groupoids; dyadic numbers},
language = {eng},
number = {2},
pages = {159-173},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A dyadic view of rational convex sets},
url = {http://eudml.org/doc/261855},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Czédli, Gábor
AU - Maróti, Miklós
AU - Romanowska, Anna B.
TI - A dyadic view of rational convex sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 2
SP - 159
EP - 173
AB - 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 $C^{\prime }$ 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 $C^{\prime }$. We also prove a more general statement for the case when $C,C^{\prime }\subseteq F^n$ 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.
LA - eng
KW - convex set; mode; barycentric algebra; commutative medial groupoid; entropic groupoid; entropic algebra; dyadic number; convex sets; modes; barycentric algebras; commutative medial groupoids; entropic groupoids; dyadic numbers
UR - http://eudml.org/doc/261855
ER -

## References

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8. Matczak K., Romanowska A.B., Smith J.D.H., 10.1142/S0218196711006248, Internat. J. Algebra Comput. 21 (2011), 387–408; DOI:10.1142/80218196711006248. Zbl1223.52008MR2804518DOI10.1142/S0218196711006248
9. Pszczoła K., Romanowska A., Smith J.D.H., 10.7151/dmgaa.1063, Discuss. Math. Gen. Algebra Appl. 23 (2003), 45–62. Zbl1060.08009MR2070045DOI10.7151/dmgaa.1063
10. Romanowska A.B., Smith J.D.H., Modal Theory, Heldermann, Berlin, 1985. Zbl0553.08001MR0788695
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12. Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002. Zbl1060.08009MR1932199

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