Abstract

The construction of the integers introduced by Dedekind is an algebraic one. Subtraction can not be done without restriction
in natural numbers N. If we consider the definition of multiplication of integral domain Z, N with respect to
subtraction is needed. It is necessary to give the definition of subtraction in N. Instead of starting from natural numbers,
one could begin with any commutative semi-group and construct from it as the construction of the integers to obtain a
commutative group. If the cancellation law does not hold in the commutative semi-group, some modifications are required.
The mapping from the commutative semi-group to the commutative group is not injective and compatible with
addition. In the relation between real numbers and decimals, N also plays an important role.

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