Ideal extensions of semigroups

Abstract

sions, 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 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. Extensions were first systematically studied by Clifford [1] who gave the first general structure theorem in the case when S is weakly reductive (Theorem 4.21 of [2]) (later extended to arbitrary S by Yoshida [7]). In this theorem the multiplication in the extension V of S by Q is described in terms of the action of V on S and a ramification function. Our structure theorems are a refinement of this in that the ramification function is not used explicitly, or, equivalently, is described in terms of other functions. Our methods are not essentially new; except in § 3, we use exclusively the action of the extension V on S, this gives rise also to the notions of strict and pure extensions: the extension V of S is strict if every element of V—S has same action on S as some element of S, pure if no element of V—S has this property. In the introductory §1 we establish some preliminary results