Dual Lattice of ℤ-module Lattice

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Search in Google Scholar

[2] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.Search in Google Scholar

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Search in Google Scholar

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Search in Google Scholar

[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: 10.1007/978-3-319-20615-817.10.1007/978-3-319-20615-817Apri DOISearch in Google Scholar

[6] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Search in Google Scholar

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Search in Google Scholar

[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Search in Google Scholar

[9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Search in Google Scholar

[10] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.10.1007/978-3-658-00360-9Search in Google Scholar

[11] Yuichi Futa and Yasunari Shidama. Lattice of ℤ-module. Formalized Mathematics, 24 (1):49–68, 2016. doi: 10.1515/forma-2016-0005.10.1515/forma-2016-0005Apri DOISearch in Google Scholar

[12] Yuichi Futa and Yasunari Shidama. Embedded lattice and properties of Gram matrix. Formalized Mathematics, 25(1):73–86, 2017. doi: 10.1515/forma-2017-0007.10.1515/forma-2017-0007Apri DOISearch in Google Scholar

[13] Yuichi Futa and Yasunari Shidama. Divisible ℤ-modules. Formalized Mathematics, 24 (1):37–47, 2016. doi: 10.1515/forma-2016-0004.10.1515/forma-2016-0004Apri DOISearch in Google Scholar

[14] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi: 10.2478/v10037-012-0007-z.10.2478/v10037-012-0007-zApri DOISearch in Google Scholar

[15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205–214, 2012. doi: 10.2478/v10037-012-0024-y.10.2478/v10037-012-0024-yApri DOISearch in Google Scholar

[16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29–49, 2015. doi: 10.2478/forma-2015-0003.10.2478/forma-2015-0003Apri DOISearch in Google Scholar

[17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Search in Google Scholar

[18] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Search in Google Scholar

[19] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982. doi: 10.1007/BF01457454.10.1007/BF01457454Apri DOISearch in Google Scholar

[20] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.10.1007/978-1-4615-0897-7_8Search in Google Scholar

[21] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495–500, 1990.Search in Google Scholar

[22] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Search in Google Scholar

[23] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.Search in Google Scholar

[24] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877–882, 1990.Search in Google Scholar

[25] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.Search in Google Scholar

[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Search in Google Scholar

[27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Search in Google Scholar

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