Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Ordered Rings - Part II

Michal Muzalewski
Warsaw University, Bialystok
Leslaw W. Szczerba
Siedlce University

### Summary.

This series of papers is devoted to the notion of the ordered ring, and one of its most important cases: the notion of ordered field. It follows the results of [5]. The idea of the notion of order in the ring is based on that of positive cone i.e. the set of positive elements. Positive cone has to contain at least squares of all elements, and has to be closed under sum and product. Therefore the key notions of this theory are that of square, sum of squares, product of squares, etc. and finally elements generated from squares by means of sums and products. Part II contains classification of sums of such elements.

#### MML Identifier: O_RING_2

The terminology and notation used in this paper have been introduced in the following articles [2] [1] [6] [3] [4]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[4] Michal Muzalewski and Leslaw W. Szczerba. Ordered rings - part I. Journal of Formalized Mathematics, 2, 1990.
[5] Wanda Szmielew. \em From Affine to Euclidean Geometry, volume 27. PWN -- D.Reidel Publ. Co., Warszawa -- Dordrecht, 1983.
[6] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.