On the existence of A-loops with some commutative inner mappings and others of order 2

TL;DR

This paper investigates specific types of A-loops with particular inner mapping commutativity and order properties, expanding understanding of their algebraic structure.

Contribution

It demonstrates the existence of A-loops with certain commutative inner mappings where some are of order 2, not fitting into known categories like extra or CC-loops.

Findings

Existence of A-loops with specified inner mapping properties

Inner mappings $R(x,y),L(x,y),T(x)$ exhibit particular commutativity patterns

Some inner mappings are of order 2, others are not

Abstract

The existence of A$_\rho$-loops, A$_\lambda$-loops and A$_\mu$-loops that are neither extra loops nor CC-loops such that any two of their inner mappings $R(x,y),L(x,y)$ and $T(x)$ commute while the other one is of order 2 is shown.