On some semigroup compactifications

TLDR: In this paper, it was shown that these properties are also true in UG for a large class of locally compact groups, and the method used is to transfer the information from {3N to (3G where G is an infinite discrete group (or a cancellative commutative semigroup), and then to UG where g is not necessarily discrete.

Abstract: The LUG-compactification UG of a locally com­ pact group is a semigroup with an operation which extends that of G and which is continuous (only) in one variable. When G is discrete, UG and the Stone-Cech compactification {3G are identi­ cal. Some algebraic properties, such as the num­ ber of left ideals and cancellation, are known to hold in the semigroup {3N where N is the additive semigroup of the integers. We show that these properties are also true in UG for a large class of locally compact groups. The method used is to transfer the information from {3N to (3G where G is an infinite discrete group (or a cancellative commutative semigroup), and then to UG where G is not necessarily discrete.