# Imbedding of semi-groups

into groups

A monomorphism of a semi-group into a group. A semi-group $S$ is imbeddable in a group $G$ if $S$ is isomorphic to a sub-semi-group of $G$. Necessary and sufficient conditions for imbeddability of a semi-group into a group were found by A.I. Mal'tsev  (see also ). These conditions form an infinite system of conditional identities (or quasi-identities, cf. Quasi-identity), in particular, the following ones:

$$ap=aq \Rightarrow p=q,\quad pa=qa \Rightarrow p=q$$

$$ap=bq,ar=bs,cp=dq \Rightarrow cr=ds,$$

where $a,b,c,d,p,q,r,s$ are elements of the semi-group. The class of semi-groups imbeddable in groups cannot be characterized by a finite system of conditional identities . A number of sufficient conditions for imbeddability of a semi-group into a group are known. The most important ones are the following. If $S$ is a semi-group with cancellation and if for any elements $a,b$ of $S$ there exist elements $x,y\in S$ such that $ax=by$ (Ore's condition), then $S$ is imbeddable in a group. If $S$ is a semi-group with cancellation in which it always follows from the equality $ab=cd$ that either $a=cx$ or $c=ax$ for some element $x\in S$, then $S$ is imbeddable in a group . Sufficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, ), are known.

How to Cite This Entry:
Imbedding of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_semi-groups&oldid=35848
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article