Vol. 26, No. 3, 1968

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Ideal extensions of semigroups

Pierre A. Grillet and Mario Petrich

Vol. 26 (1968), No. 3, 493–508

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.

Mathematical Subject Classification
Primary: 20.93
Received: 1 April 1966
Published: 1 September 1968
Pierre A. Grillet
Mario Petrich