Let
A be a commutative semigroup which has either a
greatest
regular image or a
greatest
group image. Then for any commutative semigroup
B,
A⊗
B has a
greatest
image of the same type and it is describable by standard constructions based on
A and
B. If a commutative semigroup
A has a
greatest
group-with-zero image then
A⊗
B has such an image if and only if
B is archi-medean, in which case this image is again describable by standard constructions based on
A and
B. A handy elementary tool is the fact that the Grothendieck group of a commutative semigroup
A may be regarded as the direct limit of the directed system of groups provided by
Z⊗
A where
Z is the additive group of integers.
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