Finite embedding theorems for partial Latin squares, quasi-groups, and loops

Abstract

In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities { x( xy) = y, ( yx) x = y} and the quasi-group varieties defined by any non-empty subset of { x 2 = x, x( xy) = y, ( yx) x = y}, except possibly { x( xy) = y, ( yx) x = y}, have the strong finite embeddability property. It is then shown that the finitely presented algebras in these varities are residually finite, Hopfian, and have a solvable word problem.