Ordered Rings and Fields

Open access

Summary

We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].

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

[2] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.

[3] Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363-371, 2016. doi: 10.15439/2016F520.

[4] Nathan Jacobson. Lecture Notes in Abstract Algebra, III. Theory of Fields and Galois Theory. Springer-Verlag, 1964.

[5] Manfred Knebusch and Claus Scheiderer. Einf¨uhrung in die reelle Algebra. Vieweg-Verlag, 1989.

[6] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.

[7] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.

[8] Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984.

[9] Knut Radbruch. Geordnete K¨orper. Lecture Notes, University of Kaiserslautern, Germany, 1991.

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

Formalized Mathematics

(a computer assisted approach)

Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237



Target Group

researchers in the fields of formal methods and computer-checked mathematics

Cited By

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 97 97 7
PDF Downloads 30 30 2