## Jaroslav Guri\v can

*Homology theory in the alternative set theory I. Algebraic preliminaries *

Comment.Math.Univ.Carolinae 32,1 (1991) 75-93. **Abstract:**The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called {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.

**Keywords:** alternative set theory, commutative $\pi $-group, free group, inverse system of Sd-classes and Sd-maps, prolongation, set-definable, tensor product, total homomorphism

**AMS Subject Classification:** 55N99, 20F99, 18G99

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