Verbal quandles with one parameter

TL;DR

This paper characterizes all words in three variables that define verbal quandles with one parameter, ensuring the algebraic system forms a quandle in any group and element.

Contribution

It provides a complete classification of words in the free group that generate verbal quandles with one parameter.

Findings

Identifies all words W(x,y,z) that produce quandles in any group G with element c.

Establishes conditions under which the algebraic system (G,*_{W,c}) is a quandle.

Contributes to the understanding of verbal constructions in quandle theory.

Abstract

We find all words $W(x,y,z)$ in the free group $F(x,y,z)$, such that for every group $G$ and an element $c\in G$ the algebraic system $(G,*_{W,c})$ with the binary operation $*_{W,c}$ given by $a*_{W,c}b=W(a,b,c)$ for $a,b\in G$ is a quandle. Such quandles are called verbal quandles with one parameter.