Multiplication of quandle structures

TL;DR

This paper generalizes quandle constructions, introduces a composition operation, and proves that the resulting structure forms an abelian group, expanding the algebraic understanding of quandles.

Contribution

It extends the concept of quandle families, defines a new composition operation, and establishes the algebraic properties of the resulting structures.

Findings

Defined a composition of quandles on the same set

Proved the composition yields a quandle under certain conditions

Showed the resulting group from this operation is abelian

Abstract

We generalise the construction of $Q$-family of quandles and $G$-family of quandles which were introduced in the paper of A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro, and find connection with other constructions of quandles. We define a composition of quandl's structures, which are defined on the same set and find conditions under which this composition gives a quandle. Further we prove that under this multiplication we get a group and show that this group is abelian.