Ideal extensions of semigroups
Pierre Antoine Grillet, Mario Petrich · Pacific Journal of Mathematics · 1968
An ideal extension (here called an extension) of a semigroup S by a semigroup with zero Q is a semigroup V such that S is an ideal of V and the Rees quotient semigroup V/S is isomorphic to Q.To study the structure of these extensions, special kinds of extensions are introduced, called strict and pure extensions.It is proved that any extension of S is a pure extension of a strict extension of S; also, if Q has no proper nonzero ideals, any extension of S by Q is either strict or pure.Dense extensions, closely related to Ljapin's *'densely embedded ideals", are special cases of pure extensions.When S is weakly reductive, constructions of strict, pure, and arbitrary extensions of S are given, including descriptions of the ramification function.