# Article

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Keywords:
alternative set theory; commutative $\pi$-group; free group; inverse system of Sd-classes and Sd-maps; prolongation; set-definable; tensor product; total homomorphism
Summary:
The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called {\bf commutative $\pi$-group}), is introduced. Commutative $\pi$-groups are studied from the set-theoretical point of view and from the point of view of free groups. Commutativity of the operator which is a special kind of inverse limit and factorization, is proved. Tensor product is defined, commutativity of direct product (also a free group construction and tensor product) with the special kind of inverse limit is proved. Some important examples of tensor product are computed.
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