# Article

 Title: Homology theory in the alternative set theory I. Algebraic preliminaries (English) Author: Guričan, Jaroslav Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 32 Issue: 1 Year: 1991 Pages: 75-93 . Category: math . 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. (English) Keyword: alternative set theory Keyword: commutative $\pi$-group Keyword: free group Keyword: inverse system of Sd-classes and Sd-maps Keyword: prolongation Keyword: set-definable Keyword: tensor product Keyword: total homomorphism MSC: 03E70 MSC: 03H05 MSC: 18G99 MSC: 20F99 MSC: 55N99 idZBL: Zbl 0735.03032 idMR: MR1118291 . Date available: 2008-10-09T13:11:07Z Last updated: 2012-08-08 Stable URL: http://hdl.handle.net/10338.dmlcz/116944 . Related article: http://dml.cz/handle/10338.dmlcz/118504 Related article: http://dml.cz/handle/10338.dmlcz/118551 . Reference: [C] McCord M.C.: Non-standard analysis and homology.Fund. Math. 74 (1972), 21-28. Zbl 0233.55005, MR 0300270 Reference: [G] Garavaglia S.: Homology with equationally compact coefficients.Fund. Math. 100 (1978), 89-95. Zbl 0377.55006, MR 0494066 Reference: [E-S] Eilenberg S., Steenrod N.: Foundations of algebraic topology.Princeton Press, 1952. Zbl 0047.41402, MR 0050886 Reference: [H-W] Hilton P.J., Wylie S.: Homology Theory.Cambridge University Press, Cambridge, 1960. Zbl 0163.17803, MR 0115161 Reference: [S-V] Sochor A., Vopěnka P.: Endomorphic universes and their standard extensions.Comment. Math. Univ. Carolinae 20 (1979), 605-629. MR 0555178 Reference: [V1] Vopěnka P.: Mathematics in the alternative set theory.Teubner-Texte, Leipzig, 1979. MR 0581368 Reference: [V2] Vopěnka P.: Mathematics in the alternative set theory (in Slovak).Alfa, Bratislava, 1989. Reference: [W] Wattenberg F.: Non-standard analysis and the theory of shape.Fund. Math. 98 (1978), 41-60. MR 0528354 Reference: [Ž1] Živaljevič R.T.: Infinitesimals, microsimplexes and elementary homology theory.AMM 93 (1986), 540-544. MR 0856293 Reference: [Ž2] Živaljevič R.T.: On a cohomology theory based on hyperfinite sums of microsimplexes.Pacific J. Math. 128 (1987), 201-208. MR 0883385 .

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