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)