Half-isomorphisms whose inverses are also half-isomorphisms

Abstract

Let $(G,\ast )$ and $(G^{\prime },\cdot )$ be groupoids. A bijection $f:G\to G^{\prime }$ is called a half-isomorphism if $f(x\ast y)\in \{f(x)\cdot f(y),f(y)\cdot f(x)\}$, 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.

11 pages