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Homology theory in the alternative set theory I. Algebraic preliminaries
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 75-93
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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.
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.
Classification : 03E70, 03H05, 18G99, 20F99, 55N99
Keywords: alternative set theory; commutative $\pi $-group; free group; inverse system of Sd-classes and Sd-maps; prolongation; set-definable; tensor product; total homomorphism 
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Guričan, Jaroslav. Homology theory in the alternative set theory I. Algebraic preliminaries. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 75-93. http://geodesic.mathdoc.fr/item/CMUC_1991_32_1_a8/

[C] McCord M.C.: Non-standard analysis and homology. Fund. Math. 74 (1972), 21-28. | MR | Zbl

[G] Garavaglia S.: Homology with equationally compact coefficients. Fund. Math. 100 (1978), 89-95. | MR | Zbl

[E-S] Eilenberg S., Steenrod N.: Foundations of algebraic topology. Princeton Press, 1952. | MR | Zbl

[H-W] Hilton P.J., Wylie S.: Homology Theory. Cambridge University Press, Cambridge, 1960. | MR | Zbl

[S-V] Sochor A., Vopěnka P.: Endomorphic universes and their standard extensions. Comment. Math. Univ. Carolinae 20 (1979), 605-629. | MR

[V1] Vopěnka P.: Mathematics in the alternative set theory. Teubner-Texte, Leipzig, 1979. | MR

[V2] Vopěnka P.: Mathematics in the alternative set theory (in Slovak). Alfa, Bratislava, 1989.

[W] Wattenberg F.: Non-standard analysis and the theory of shape. Fund. Math. 98 (1978), 41-60. | MR

[Ž1] Živaljevič R.T.: Infinitesimals, microsimplexes and elementary homology theory. AMM 93 (1986), 540-544. | MR

[Ž2] Živaljevič R.T.: On a cohomology theory based on hyperfinite sums of microsimplexes. Pacific J. Math. 128 (1987), 201-208. | MR