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Paweł Gładki

, Orderings of higher level in multifields and multirings , Ann. Math. Sil. 24 (2010) 15–25. [6] P. Gładki and M. Marshall, Orderings and signatures of higher level on multirings and hyperfields , J. K-Theory 10 (2012) 489–518. doi:10.1017/is012004021jkt189 [7] P. Gładki and M. Marshall, Witt equivalence of function fields over global fields , Trans. Amer. Math. Soc. 369 (2017) 7861–7881. doi:10.1090/tran/6898 [8] J. Jun, Algebraic geometry over hyperfields , Adv. Math. 323 (2018) 142–192. doi:10.1016/j.aim.2017.10.043 [9] M. Krasner

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Ole Jørgen Benedictow

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Biological Invasions in Changing Ecosystems

Vectors, Ecological Impacts, Management and Predictions

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Paweł Gładki

References [1] E. Becker, Hereditarily Pythagorean Fields and Orderings of Higher Level , Monografías de Matemática, 29, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978. [2] E. Becker, Summen n-ter Potenzen in Körpern , J. Reine Angew. Math. 307/308 (1979), 8–30. [3] P. Berrizbeitia, Additive properties of multiplicative subgroups of finite index in fields , Proc. Amer. Math. Soc. 112 (1991), 365–369. [4] J. Königsmann, Half-ordered fields , PhD thesis, Universität Konstanz, Konstanz, 1993. [5] T.Y. Lam, The

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Christoph Schwarzweller

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

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Christoph Schwarzweller

Summary

We extend the algebraic theory of ordered fields [7, 6] in Mizar [1, 2, 3]: we show that every preordering can be extended into an ordering, i.e. that formally real and ordered fields coincide.We further prove some characterizations of formally real fields, in particular the one by Artin and Schreier using sums of squares [4]. In the second part of the article we define absolute values and the square root function [5].

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International Journal of Area Studies

A Journal of Vytautas Magnus University