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.
