Algebraic properties of some varieties of central loops

TL;DR

This paper explores the algebraic properties of certain central loops, demonstrating how isotopes of C-loops relate to other loop types and examining conditions under which C-loops exhibit specific inverse properties.

Contribution

It establishes new connections between C-loops, A-loops, and Steiner loops, and characterizes when C-loops become Osborn loops based on element squares.

Findings

Isotopes of C-loops with unique non-identity squares are C-loops and A-loops.

Relationship between C-loops and Steiner loops is clarified.

C-loops are Osborn loops if all elements are squares.

Abstract

Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also investigated. C-loops are found to be Osborn loops if every element in them are squares.