The Embedding Theorems of Malcev and Lambek

Abstract

A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.