Geodesic
Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 21-32 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Some algebras over the operad of hollow cubes have been studied in this paper. These algebras are defined in the non-negative quadrant of the Euclidean plane. The geometric description of subalgebras generated by two elements has been given. It has been proved that the subalgebras generated by two elements are the polylines. Using the structure of such subalgebras, a new metric function on the first quadrant has been constructed. The distance between two points in this metric is the length of the polyline, which is a subalgebra generated by two elements over the operad of hollow cubes. This work can be considered as a continuation of our previous paper: Gaynullina A. On one class of commutative operads. Asian-Eur. J. Math., 2017, vol. 10, no. 1, p. 1750007. doi: 10.1142/S1793557117500073.
Keywords: operad, algebra over operad, metric. 
@article{UZKU_2017_159_1_a2,
     author = {A. R. Gaynullina and S. N. Tronin},
     title = {Algebras over the operad of hollow cubes and a~new metric function on the non-negative quadrant of the {Euclidean} plane},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {21--32},
     year = {2017},
     volume = {159},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a2/}
}
TY  - JOUR
AU  - A. R. Gaynullina
AU  - S. N. Tronin
TI  - Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2017
SP  - 21
EP  - 32
VL  - 159
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a2/
LA  - ru
ID  - UZKU_2017_159_1_a2
ER  - 
%0 Journal Article
%A A. R. Gaynullina
%A S. N. Tronin
%T Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2017
%P 21-32
%V 159
%N 1
%U http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a2/
%G ru
%F UZKU_2017_159_1_a2
A. R. Gaynullina; S. N. Tronin. Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 21-32. http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a2/

[1] Gaynullina A., “On one class of commutative operads”, Asian-Eur. J. Math., 10:1 (2017), 1–34 | DOI | MR | Zbl

[2] Deza M., Deza E., Encyclopedia of Distances, Springer-Verlag, Berlin-Heidelberg, 2016, xxii, 756 pp. | MR | Zbl

[3] Deza M. M., Laurent M., Geometry of Cuts and Metrics, Springer, Berlin–Heidelberg, 1997, 588 pp. | MR | Zbl

[4] Loday J.-L., Vallette B., Algebraic Operads, Grundlehren der mathematischen Wissenschaften, 346, Springer-Verlag, Berlin–Heidelberg, 2012, xxiv, 636 pp. | DOI | MR | Zbl

[5] Markl M., Shnider S., Stasheff J., Operads in Algebra, Topology and Physics, Math. Surveys and Monographs, 96, Amer. Math. Soc., 2002, 349 pp. | MR | Zbl

[6] Bremner M. R., Dotsenko V., Algebraic Operads. An Algorithmic Companion, CRC Press, Boca Raton–London–N.Y., 2016, xvii, 361 pp. | MR | Zbl

[7] Tronin S. N., “Operads and varieties of algebras defined by polylinear identities”, Sib. Math. J., 47:3 (2006), 555–573 | DOI | MR | Zbl

[8] Tronin S. N., “Natural multitransformations of multifunctors”, Russ. Math., 55:11 (2011), 49–60 | DOI | MR | Zbl

[9] Tronin S. N., “Operads in the category of convexors. I”, Russ. Math., 46:3 (2002), 38–46 | MR | Zbl

[10] Tronin S. N., “Algebras over operad of spheres”, Russ. Math., 54:3 (2010), 63–71 | DOI | MR | Zbl