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Some examples of denumerable pseudo-ordered fields and their applications to geometry

Jakóbowski J.

Demonstratio Mathematica

2013

Vol. 46, nr 2

247--256

The paper deals with the following problems concerning pseudo-ordered denumerable fields: (i) a connection between Fermat’s two squares theorem and the unique pseudo-order in a finite field; (ii) properties of a proper pseudo-order determined by any prime number in the field of rational numbers; (iii) existence of a proper pseudo-order in every subfield of the sequence used to obtain the field of constructible numbers; (iv) some brief of applications of the latter pseudo-orders to construct new algebraic and geometric structures. In particular, we extend the known construction of finite nearfields or quasifields given by e.g. W. A. Pierce or P. Dembowski – to infinite cases.

Nearaffine planes related to pseudo-ordered fields

Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 3

345-360

The constructions of affine planes and Minkowski planes related to pseudo-ordered fields are given in [3] and [2], respectively. We here give some analogous construction for nearaffine planes. Like before, we shall use some functions f, g and determine some conditions on f, g, necessary and suffcient to get the required plane. The Veblen postulate has a particular meaning in nearaffine planes, so it is also considered in the work. Some special cases like the field of the reals and finite fields of odd order are investigated, too. We give some examples of such nearaffine planes and consider their particular automorphisms. Every Minkowski plane related to pseudo-ordered field F determines a nearaffine plane connected with F [2, Proposition 1, p. 187]. But only weaker version of the reciprocal statement is true, i.e. a nearaffine plane related to a pseudo-ordered field determines a hyperbola structure (i.e. Minkowski plane without touching axiom).