# Extension of a semi-group

A semi-group $S$ containing the given semi-group $A$ as a sub-semi-group. One is usually concerned with extensions that are in some way related to the given semi-group $A$. The most well-developed theory is that of ideal extensions (those semi-groups containing $A$ as an ideal). To each element $s$ of an ideal extension $S$ of a semi-group $A$ are assigned its left and right translations $\lambda _ {s}$, $\rho _ {s}$: $\lambda _ {s} x = sx$, $x \rho _ {s} = xs$( $x \in A$); let $\tau = \tau _ {s} = ( \lambda _ {s} , \rho _ {s} )$. The mapping $\tau$ is a homomorphism of $S$ into the translation hull $T ( A)$ of $A$, and is an isomorphism in the case when $A$ is weakly reductive (see Translations of semi-groups). The semi-group $\tau S$ is called the type of the ideal extension $S$. Among the ideal extensions $S$ of $A$, one can distinguish strong extensions, for which $\tau S = \tau A$, and pure extensions, for which $\tau ^ {-} 1 \tau A = A$. Every ideal extension of $A$ is a pure extension of one of its strong extensions.

An ideal extension $S$ of $A$ is called dense (or essential) if every homomorphism of $S$ that is injective on $A$ is an isomorphism. $A$ has a maximal dense ideal extension $D$ if and only if $A$ is weakly reductive. In this case, $D$ is unique up to an isomorphism and is isomorphic to $T ( A)$. Also, in this case, $A$ is called a densely-imbedded ideal in $D$. The sub-semi-groups of $T ( A)$ containing $\tau A$, and only these, are isomorphic to dense ideal extensions of a weakly reductive semi-group $A$.

If $S$ is an ideal extension of $A$ and if the quotient semi-group $S/A$ is isomorphic to $Q$, then $S$ is called an extension of $A$ by $Q$. The following cases have been studied extensively: ideal extensions of completely-simple semi-groups, of a group by a completely $O$- simple semi-group, of a commutative semi-group with cancellation by a group with added zero, etc. In general, the problem of describing all ideal extensions of a semi-group $A$ by $Q$ is far from being solved.

Among other types of extensions of $A$ one can mention semi-groups that have a congruence with $A$ as one of its classes, and in particular the so-called Schreier extensions of a semi-group with identity [1], which are analogues of Schreier extensions of groups. In studying the various forms of extensions of a semi-group (in particular, for inverse semi-groups), one uses cohomology of semi-groups.

Another broad area in the theory of extensions of semi-groups is concerned with various problems on the existence of extensions of a semi-group $A$ that belong to a given class. Thus, any semi-group $A$ can be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see Simple semi-group), and any finite or countable semi-group can be imbedded in a semi-group with two generators. Conditions are known under which a semi-group $A$ can be imbedded in a semi-group without proper left ideals, in an inverse semi-group (cf. Inversion semi-group), in a group (see Imbedding of semi-groups), etc.

#### References

 [1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961) [2] M. Petrich, "Introduction to semigroups" , C.E. Merrill (1973)
How to Cite This Entry:
Extension of a semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_semi-group&oldid=46881
This article was adapted from an original article by L.M. Gluskin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article