Directed partial orders over non-archimedean o -fields
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Positivity
Directed partial orders over non-archimedean o-fields Zhipeng Xu2 · Yuehui Zhang1,2 Received: 18 November 2019 / Accepted: 12 December 2019 © Springer Nature Switzerland AG 2019
Abstract Let F be a non-archimedean o-field, C = F(i) the imaginary quadratic extension field of F with i 2 = −1. In this paper, all directed partial orders on C are classified via the new concept of doubly convex set consisting of some infinitesimals. This unifies the previous work Ma et al. (Order 34(1):37–44, 2017; Order 35(3):461–466, 2018; Positivity 24(3):1001–1007, 2019). It is surprising that this new theory applies well to the quaternions H = F + Fi + F j + Fk over F and all directed partial orders on H are classified. As an application, the Fuchs’ problem is answered negatively for H . Keywords Non-archimedean o-field · Directed partially ordered algebra · Directed partial order · Doubly convex set of infinitesimals Mathematics Subject Classification Primary 06F25; Secondary 20M25
1 Introduction In 1956, Birkhoff and Pierce raised in [2] the following problem (so-called Birkhoff– Pierce problem) Does the complex number field C admit any lattice orderings? Birkhoff and Pierce prove that C does not admit any such orderings as an algebra over the real number field R. But no one knows any such orderings on C as a ring.
Supported by NSFC under Grant Nos. 11771280 and 11671258, by NSF of Shanghai Municipal under Grant No. 17ZR1415400.
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Yuehui Zhang [email protected] Zhipeng Xu [email protected]
1
School of Mathematics and Statistics, Ningxia University, 489 Helanshan West Road, Yinchuan 750021, China
2
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China
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Z. Xu, Y. Zhang
There seems less hope to find any lattice orderings on C or R, so in 1963, Fuchs [3, problem 31, p. 212] generalized Birkhoff–Pierce problem as to describe the directed partial orders of C, R and other rings. In 1976, R. Wilson surprisingly proved that there are infinitely many lattice orderings on R, which inspires the hope of a positive answer to Birkhoff–Pierce problem. In 1986, Schwartz [8] proved that algebraic number fields admitting no total order do not allow a lattice order. But it is still not known whether there are fields admitting no total order allow a lattice order. In 2011, Schwartz and Yang [9] proved that C admits directed partial orders and they further proved that none of those orders constructed from their method are lattice orders, ad hoc Birkhoff–Pierce problem is still open. Recently, J. Ma, L. Wu and the second author of the present paper classified all directed partial orders over an imaginary quadratic extension of a non-archimedean o-field. In [4], they use the notion admissible semigroup to classify directed partial orders with 1 > 0, while in [5,6] they use the notion special convex set to classify directed partial orders with 1 > 0 . In the present paper, we use a new concept doubly convex set consisting of some infinitesimals to uniformly classify all
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