Algebras over the operad of hollow cubes and a new metric function on the non-negative quadrant of the Euclidean plane

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.