Algebras over the Operad of Hollow Cubes and a New Metric Function on the Non-Negative Quadrant of the Euclidean Plane

A.R. Gaynullina*, S.N. Tronin**

Kazan Federal University, Kazan, 420008 Russia

E-mail: *GaynullinaAlina@gmail.com, **Serge.Tronin@kpfu.ru

Received January 18, 2017

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Abstract

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

Acknowledgments. This study was funded by the subsidy allocated to Kazan Federal University as part of the state assignment in the sphere of scientific activity (project no. 1.1515.2017/PCh) and supported by the International Mathematical Center of Kazan Federal University.

Figure Captions

Fig. 1. Operads Cmax and Cmin on the plane R2.

Fig. 2. Subalgebra in the case a1,1 = a2,1, * a1,2 = a2,2.

Fig. 3. Subalgebra in the case a1,1 = a2,1, * a1,2 ≠ a2,2.

Fig. 4. Subalgebra in the case a1,2 = a2,2, * a1,1 ≠ a2,1.

Fig. 5. Subalgebra in the case a1,1 < a2,1, * a1,2 > a2,2.

Fig. 6. Subalgebra in the case ā2 = λā1,  * λ > 0.

Fig. 7. Subalgebra in the case a1,1 < a2,1, * a1,2 < a2,2, where a1,1/a2,1 < a1,2/a2,2.

Fig. 8. Subalgebra in the case a1,1 < a2,1, * a1,2 < a2,2, where a1,1/a2,1 > a1,2/a2,2.

Fig. 9. Subalgebra over the operad Cmin in the case a1,1 < a2,1, * a1,2 > a2,2.

Fig. 10. Subalgebra over the operad Cmin in the case a1,1 < a2,1, * a1,2 < a2,2, where a1,1/a2,1 < a1,2/a2,2.

Fig. 11. Subalgebra over the operad Cmin in the case a1,1 < a2,1, * a1,2 < a2,2, where a1,1/a2,1 > a1,2/a2,2.

Fig. 12. Case 1.

Fig. 13. Case 2.

Fig. 14. Case 3.

Fig. 15. Counterexample.

References

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For citation: Gaynullina A.R., Tronin S.N. Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 21–32. (In Russian)


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