Groups – Additive Notation

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Abstract

We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25].

In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented.

Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

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  • [1] Jonathan Backer Piotr Rudnicki and Christoph Schwarzweller. Ring ideals. Formalized Mathematics 9(3):565-582 2001.

  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics 1(2):377-382 1990.

  • [3] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics 1(3):543-547 1990.

  • [4] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics 1(3):563-567 1990.

  • [5] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics 1(1):41-46 1990.

  • [6] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics 1(1):91-96 1990.

  • [7] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics 1 (2):265-267 1990.

  • [8] Józef Białas. Group and field definitions. Formalized Mathematics 1(3):433-439 1990.

  • [9] Richard E. Blahut. Cryptography and Secure Communication. Cambridge University Press 2014.

  • [10] Sylvie Boldo Catherine Lelay and Guillaume Melquiond. Formalization of real analysis: A survey of proof assistants and libraries. Mathematical Structures in Computer Science pages 1-38 2014.

  • [11] Nicolas Bourbaki. General Topology: Chapters 1-4. Springer Science and Business Media 2013.

  • [12] Czesław Bylinski. Binary operations. Formalized Mathematics 1(1):175-180 1990.

  • [13] Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics 1(4):643-649 1990.

  • [14] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics 1(1): 55-65 1990.

  • [15] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics 1(1):153-164 1990.

  • [16] Czesław Bylinski. Partial functions. Formalized Mathematics 1(2):357-367 1990.

  • [17] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics 1(1):47-53 1990.

  • [18] Agata Darmochwał. Compact spaces. Formalized Mathematics 1(2):383-386 1990.

  • [19] Agata Darmochwał. Finite sets. Formalized Mathematics 1(1):165-167 1990.

  • [20] Agata Darmochwał. Families of subsets subspaces and mappings in topological spaces. Formalized Mathematics 1(2):257-261 1990.

  • [21] Yuichi Futa Hiroyuki Okazaki and Yasunari Shidama. Z-modules. Formalized Mathematics 20(1):47-59 2012. doi:10.2478/v10037-012-0007-z.

  • [22] Edwin Hewitt and Kenneth A. Ross. Abstract Harmonic Analysis: Volume I. Structure of Topological Groups. Integration. Theory Group Representations volume 115. Springer Science and Business Media 2012.

  • [23] Johannes Hölzl Fabian Immler and Brian Huffman. Type classes and filters for mathematical analysis in Isabelle/HOL. In Interactive Theorem Proving pages 279-294. Springer 2013.

  • [24] Teturo Inui Yukito Tanabe and Yositaka Onodera. Group theory and its applications in physics volume 78. Springer Science and Business Media 2012.

  • [25] Artur Korniłowicz. The definition and basic properties of topological groups. Formalized Mathematics 7(2):217-225 1998.

  • [26] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics 1(5):829-832 1990.

  • [27] Christopher Norman. Basic theory of additive Abelian groups. In Finitely Generated Abelian Groups and Similarity of Matrices over a Field Springer Undergraduate Mathematics Series pages 47-96. Springer 2012. ISBN 978-1-4471-2729-1. doi:10.1007/978-1-4471-2730-7 2.

  • [28] Beata Padlewska. Locally connected spaces. Formalized Mathematics 2(1):93-96 1991.

  • [29] Beata Padlewska. Families of sets. Formalized Mathematics 1(1):147-152 1990.

  • [30] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics 1(1):223-230 1990.

  • [31] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics 5(2):233-236 1996.

  • [32] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics 2(4): 535-545 1991.

  • [33] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics 1(1): 115-122 1990.

  • [34] Andrzej Trybulec. Enumerated sets. Formalized Mathematics 1(1):25-34 1990.

  • [35] Andrzej Trybulec. Tuples projections and Cartesian products. Formalized Mathematics 1(1):97-105 1990.

  • [36] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics 11(4): 341-347 2003.

  • [37] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics 1 (2):369-376 1990.

  • [38] Andrzej Trybulec. Baire spaces Sober spaces. Formalized Mathematics 6(2):289-294 1997.

  • [39] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics 1 (1):187-190 1990.

  • [40] Michał J. Trybulec. Integers. Formalized Mathematics 1(3):501-505 1990.

  • [41] Wojciech A. Trybulec. Groups. Formalized Mathematics 1(5):821-827 1990.

  • [42] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics 1(5): 855-864 1990.

  • [43] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics 1(5):955-962 1990.

  • [44] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics 1(2):291-296 1990.

  • [45] Zinaida Trybulec. Properties of subsets. Formalized Mathematics 1(1):67-71 1990.

  • [46] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics 1 (1):73-83 1990.

  • [47] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics 1(1):181-186 1990.

  • [48] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics 1(1):231-237 1990.

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