D-loops

TL;DR

This paper explores the properties and construction methods of D-loops, a class of loops with the antiautomorphic inverse property, highlighting differences from IP-loops and challenges in isotopy.

Contribution

It introduces new properties of D-loops, compares them with IP-loops, and presents methods for constructing D-loops from IP-loops, expanding understanding of their structure.

Findings

Smallest non-IP D-loop has six elements

D-loops are larger class than IP-loops

Isotopy does not preserve D-loop property

Abstract

$D$-loops are loops with the antiautomorphic inverse property. The class of such loops is larger than the class of IP-loops. The smallest $D$-loops which is not an IP-loop has six elements. We prove several basic properties of such loops and present methods of constructions of $D$-loops from IP-loops. Unfortunately, a loop isotopic to a $D$-loop may not be a $D$-loop.