# A dyadic view of rational convex sets

Gábor Czédli; Miklós Maróti; Anna B. Romanowska

Commentationes Mathematicae Universitatis Carolinae (2014)

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

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topCzédli, 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

top- Burris S., Sankappanavar H.P., A Course in Universal Algebra, Graduate Texts in Mathematics, 78, Springer, New York–Berlin, 1981; The Millennium Edition, http://www.math.uwaterloo.ca/snburris/htdocs/ualg.html. Zbl0478.08001MR0648287
- Cox D.A., 10.4169/amer.math.monthly.118.01.003, Amer. Math. Monthly 118 (2011), no. 1, 3–21. Zbl1225.11002MR2795943DOI10.4169/amer.math.monthly.118.01.003
- Czédli G., Romanowska A.B., 10.1007/s00012-012-0195-y, Algebra Universalis 68 (2012), 111–143. Zbl1261.08002MR3008741DOI10.1007/s00012-012-0195-y
- Czédli G., Romanowska A.B., Generalized convexity and closure conditions, Internat. J. Algebra Comput. 23 (2013), no. 8, 1805–1835. MR3163609
- Fulton W., Algebraic Curves. An Introduction to Algebraic Geometry, 2008; . Zbl0681.14011MR1042981
- Ježek J., Kepka T., Medial Groupoids, Academia, Praha, 1983. MR0734873
- Matczak K., Romanowska A., 10.1007/s11225-005-1335-6, Studia Logica 78 (2004), 321–335. Zbl1092.08005MR2108032DOI10.1007/s11225-005-1335-6
- 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
- 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
- Romanowska A.B., Smith J.D.H., Modal Theory, Heldermann, Berlin, 1985. Zbl0553.08001MR0788695
- Romanowska A.B., Smith J.D.H., On the structure of semilattice sums, Czechoslovak Math. J. 41 (1991), 24–43. Zbl0793.08010MR1087619
- Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002. Zbl1060.08009MR1932199

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