On the existence of directed rings and algebras with negative squares

Abstract

We show that there exist many directed rings and algebras with negative squares.

Directed Partial Orders on Complex Numbers and Quaternions over Non-Archimedean Linearly Ordered Fields

“…We show now that directed partial orders constructed in [2,3] are special cases of P x,y (S) constructed above.…”

“…In Section 2, we construct a class of directed partial orders on the complex numbers C via positive additive semigroups of F . This class of directed partial orders includes all directed partial orders on C constructed by Yang using negative valuations [3] and by Rump and Yang using multiplicative segment [2], and it contains the largest one among all directed partial orders on C. We conjecture that the directed partial orders that we have constructed give all possible directed partial orders on C such that 1 > 0. In Section 3, we use the similar construction to get directed partial orders on the quaternions over F .…”

“…In [3], Yang has constructed directed partial orders on C = F (i) by using negative valuation. The reader is referred to [3] for more information on negative valuations.…”

“…Yang has produced some directed partial orders on C by using negative valuations [3], and recently Rump and Yang have constructed a class of directed partial orders on C using multiplicative segment, which is a convex additive subgroup of F containing multiplicative identity element 1 [2].…”

A Lattice-Ordered Skew Field Is Totally Ordered If Squares Are Positive

“…In [12], Schwartz showed that an Archimedean lattice-ordered (commutative) field that has 1 > 0 and that is algebraic over its maximal totally ordered subfield cannot have negative squares, and in [13], DeMarr and Steger showed that in a partially ordered finite dimensional real linear algebra no square can be the negative of a strong unit. Furthermore, in [15], we guarantee the existence of directed commutative fields with negative squares. In 1969, Conrad and Dauns [6] raised the following problem (this is Question (b) of their list in [6] [11], p. 124).…”