References
- [1] Jesse Alama. The rank+nullity theorem., 15():137–142, 2007. doi:10.2478/v10037-007-0015-6.
- [2] Grzegorz Bancerek. Cardinal numbers., 1():377–382, 1990.
- [3] Grzegorz Bancerek. Cardinal arithmetics., 1():543–547, 1990.
- [4] Grzegorz Bancerek. The fundamental properties of natural numbers., 1():41–46, 1990.
- [5] Grzegorz Bancerek. The ordinal numbers., 1():91–96, 1990.
- [6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences., 1():107–114, 1990.
- [7] 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,, volume 9150 of, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.
- [8] Czesław Byliński. Binary operations., 1():175–180, 1990.
- [9] Czesław Byliński. Functions and their basic properties., 1(): 55–65, 1990.
- [10] Czesław Byliński. Functions from a set to a set., 1():153–164, 1990.
- [11] Czesław Byliński. Partial functions., 1():357–367, 1990.
- [12] Czesław Byliński. Some basic properties of sets., 1():47–53, 1990.
- [13] Agata Darmochwał. Finite sets., 1():165–167, 1990.
- [14] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules., 20():47–59, 2012. doi:10.2478/v10037-012-0007-z.
- [15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module., 20():205–214, 2012. doi:10.2478/v10037-012-0024-y.
- [16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module., 20():275–280, 2012. doi:10.2478/v10037-012-0033-x.
- [17] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers., 21():115–125, 2013. doi:10.2478/forma-2013-0013.
- [18] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Submodule of free ℤ-module., 21():273–282, 2013. doi:10.2478/forma-2013-0029.
- [19] Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free ℤ-module., 22():277–289, 2014. doi:10.2478/forma-2014-0028.
- [20] Andrzej Kondracki. Basic properties of rational numbers., 1(): 841–845, 1990.
- [21] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces., 1():335–342, 1990.
- [22] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes., 1():829–832, 1990.
- [23] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients., 261(4), 1982.
- [24] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective., 2002.
- [25] Michał Muzalewski. Rings and modules – part II., 2():579–585, 1991.
- [26] Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Rank of submodule, linear transformations and linearly independent subsets of ℤ-module., 22():189–198, 2014. doi:10.2478/forma-2014-0021.
- [27] Christoph Schwarzweller. The binomial theorem for algebraic structures., 9():559–564, 2001.
- [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers., 8():29–34, 1999.
- [29] Andrzej Trybulec. Domains and their Cartesian products., 1(): 115–122, 1990.
- [30] Michał J. Trybulec. Integers., 1():501–505, 1990.
- [31] Wojciech A. Trybulec. Operations on subspaces in real linear space., 1():395–399, 1990.
- [32] Wojciech A. Trybulec. Vectors in real linear space., 1():291–296, 1990.
- [33] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space., 1():865–870, 1990.
- [34] Wojciech A. Trybulec. Operations on subspaces in vector space., 1():871–876, 1990.
- [35] Wojciech A. Trybulec. Linear combinations in vector space., 1 ():877–882, 1990.
- [36] Wojciech A. Trybulec. Basis of vector space., 1():883–885, 1990.
- [37] Zinaida Trybulec. Properties of subsets., 1():67–71, 1990.
- [38] Edmund Woronowicz. Relations and their basic properties., 1 ():73–83, 1990.
- [39] Edmund Woronowicz. Relations defined on sets., 1():181–186, 1990.
Language: English
Page range: 297 - 307
Submitted on: Aug 14, 2015
Published on: Mar 25, 2016
Published by: University of Białystok
In partnership with: Paradigm Publishing Services
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© 2016 Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama, published by University of Białystok
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