A note on loops

A. K. Austin

Vol. 18 (1966), No. 2, 209–212

DOI: 10.2140/pjm.1966.18.209

Abstract

An Associative Element of a quasigroup is defined to be an element a with the property that x(yz) = a implies (xy)z = a.

It is then shown that

(i) a quasigroup which contains an associative element is a loop,

(ii) if a loop contains an associative element then the nuclei coincide,

(iii) if a loop is weak inverse then the set of associative elements coincides with the nucleus,

(iv) if a loop is not weak inverse then no associative element is a member of the nucleus and the product of any two associative elements is not associative.

Mathematical Subject Classification

Primary: 20.95

Milestones

Received: 10 June 1965

Published: 1 August 1966