Abstract
This paper characterizes commutative semigroups which admit a greatest group-homomorphism in various ways. One of the important theorems is that a commutative semigroup S S has a greatest group-homomorphic image if and only if for every a ∈ S a \in S there are b , c ∈ S b,c \in S such that a b c = c abc = c . Further the authors study a relationship between S S and a certain cofinal subsemigroup and discuss the structure of commutative separative semigroups which have a greatest group-homomorphic image.
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