A construction of free digroup

TL;DR

This paper constructs a free digroup on a set, detailing its structure and proving isomorphism conditions based on set cardinality, thus advancing the algebraic understanding of digroups.

Contribution

It provides a concrete construction of free digroups and characterizes their isomorphism classes by set cardinality.

Findings

Construction of free digroup $F(X)$ on a set $X$

Identification of halo and group parts of $F(X)$

Isomorphism of $F(X)$ and $F(Y)$ iff $card(X)=card(Y)$

Abstract

We give a construction of a free digroup $F(X)$ on a set $X$ and formulate the halo and the group parts of $F(X)$. We prove that $F(X)$ is isomorphic to $F(Y)$ if and only if $card(X)=card(Y)$.