Half-isomorphisms whose inverses are also half-isomorphisms

TL;DR

This paper investigates special half-isomorphisms in groupoids and quasigroups, providing conditions for their existence and examples of non-special half-automorphisms in infinite loops.

Contribution

It establishes necessary and sufficient conditions for the existence of special half-isomorphisms and explores their properties in various algebraic structures.

Findings

Conditions for existence of special half-isomorphisms derived

Examples of non-special half-automorphisms in infinite loops provided

Characterization of half-isomorphisms in groupoids and quasigroups

Abstract

Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a half-isomorphism if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$, for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism $f$ is called special if $f^{-1}$ is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.