EVANS' NORMAL FORM THEOREM REVISITED

Abstract

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.