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 [1] (see also [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 laws);

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

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

References

[1]	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)
[2]	A.I. Mal'tsev, "On inclusion of associative systems in a group II" Mat. Sb. , 8 (50) : 2 (1940) pp. 251–264 (In Russian)
[3]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[4]	R. Doss, "Sur l'immersion d'un semi-groupe dans un groupe" Bull. Sci. Math. (2) , 72 (1948) pp. 139–150 Zbl 0033.14703
[5]	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)

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: that for all $a,b \in S$ the intersection of right ideals $aS \cap bS$ is non-empty.

References

[a1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)