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Formalized Mathematics

2016 | 24 | 1 | 81-94

Conservation Rules of Direct Sum Decomposition of Groups

EN

Abstrakty

EN
In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization.

EN

81-94

wydano
2016-03-01
otrzymano
2015-12-31
online
2016-08-19

Twórcy

autor
• Shinshu University Nagano, Japan
autor
• Shinshu University Nagano, Japan
autor
• Shinshu University Nagano, Japan
autor
• Shinshu University Nagano, Japan

Bibliografia

• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• [2] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.
• [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
• [4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
• [5] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.
• [6] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989.
• [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
• [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [10] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.
• [11] Serge Lang. Algebra. Springer, 3rd edition, 2005.
• [12] Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, and Yasunari Shidama. Definition and properties of direct sum decomposition of groups. Formalized Mathematics, 23 (1):15-27, 2015. doi:10.2478/forma-2015-0002.
• [13] D. Robinson. A Course in the Theory of Groups. Springer New York, 2012.
• [14] J.J. Rotman. An Introduction to the Theory of Groups. Springer, 1995.
• [15] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
• [16] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.
• [17] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
• [18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.