View source for Imbedding of semi-groups

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''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 [[#References|[1]]] (see also [[#References|[3]]]). These conditions form an infinite system of conditional identities (or quasi-identities, cf. [[Quasi-identity]]), in particular, the following ones:

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

([[cancellation law]]s);

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

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 [[#References|[2]]]. 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 [[#References|[4]]]. Sufficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, [[#References|[5]]]), are known.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "On inclusion of associative systems in a group"  ''Mat. Sb.'' , '''6 (48)''' :  2  (1939)  pp. 331–336  (In Russian)  (German abstract)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,   "On inclusion of associative systems in a group II"  ''Mat. Sb.'' , '''8 (50)''' :  2  (1940)  pp. 251–264  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  R. Doss,   "Sur l'immersion d'un semi-groupe dans un groupe"  ''Bull. Sci. Math. (2)'' , '''72'''  (1948)  pp. 139–150</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  S.I. [S.I. Adyan] Adjan,   "Defining relations and algorithmic problems for groups and semigroups"  ''Proc. Steklov Inst. Math.'' , '''85'''  (1967)  ''Trudy Mat. Inst. Steklov.'' , '''85'''  (1966)</TD></TR>
</table>

====Comments====
Instead of imbedding the word embedding is also used.

A semi-group $S$ that satisfies Ore's condition is also known as ''left [[Reversible semi-group|reversible]]'': that for all $a,b \in S$ the intersection of right [[ideal]]s $aS \cap bS$ is non-empty.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR>
</table>