On elements with negative squares

Abstract

We prove that in a partially ordered linear algebra no element can have a square which is the negative of an order unit. In particular, the square of a real matrix cannot consist entirely of negative entries. We generalize the well-known theorem that the complex numbers admit no lattice order.

On the existence of directed rings and algebras with negative squares

“…And in [6], DeMarr and Steger proved that if A is a finite-dimensional nontrivial algebra over R whose center contains a square root of −1, then A admits no partial order with respect to which it is a directed algebra over R. We first note that the proof of DeMarr and Steger may be easily generalized to prove the following result.…”

“…Let T be a totally ordered ring and recall (see [6]) that a directed T -algebra is an algebra D over T with a partial order that makes it a directed ring with the following compatibility property: if 0 τ ∈ T and 0 d ∈ D, then 0 τ d.…”

“…In [12], Schwartz showed that an archimedean lattice-ordered field that has 1 > 0 and that is algebraic over its maximal totally ordered subfield cannot have negative squares, and in [6], DeMarr and Steger showed that in a partially ordered finite-dimensional real linear algebra no square can be the negative of a strong unit. In [4], Birkhoff and Pierce asked whether the complex numbers have a compatible lattice-order.…”

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

“…In [1], Artin and Schreier observed that a totally ordered commutative field cannot have negative squares, and Johnson [7] and Fuchs [9] extended this result to totally ordered domains with unit element. 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.…”

Fields with directed partial orders

“…(Note that DeMarr and Steger [3], have shown that C cannot be made into a directed algebra over the reals. )…”