Let (X, +) be a group, and (S, +) a subsemigroup (i.e., a semigroup such that SG and the operation + is the same as in X; cf. 4.5). Let (Y, +) ba another group, and let g : SY be a homomorphism1. The problem with which we shall deal in 18.1–18.4 is the following. Does there exist a homomorphism f : XY such that f Ç S = g? The main result in this section (cf. Dhombres-Ger [70], Balcerzyk [18]) reads as follows.


Commutative Group Convex Function Normal Subgroup Arbitrary Function Additive Function 
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