A variety of Steiner loops satisfying Moufang’s theorem: a solution to Rajah’s Problem

A loop X is said to satisfy Moufang’s theorem if for every x , y , z ∈ X such that x ( y z ) = ( x y ) z the subloop generated by x , y , z is a group. We prove that the variety V of Steiner loops satisfying the identity ( x z ) ( ( ( x y ) z ) ( y z ) ) = ( ( x z ) ( ( x y ) z ) ) ( y z ) is not co...

Full description

Saved in:
Bibliographic Details
Published in:Aequationes mathematicae 2020-02, Vol.94 (1), p.97-101
Main Authors: Drápal, Aleš, Vojtěchovský, Petr
Format: Article
Language:English
Subjects:
Analysis
Combinatorics
Mathematics
Mathematics and Statistics
Theorems
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A loop X is said to satisfy Moufang’s theorem if for every x , y , z ∈ X such that x ( y z ) = ( x y ) z the subloop generated by x , y , z is a group. We prove that the variety V of Steiner loops satisfying the identity ( x z ) ( ( ( x y ) z ) ( y z ) ) = ( ( x z ) ( ( x y ) z ) ) ( y z ) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang’s theorem. This solves a problem posed by Andrew Rajah.
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-019-00692-3