On Elements with Negative Squares

Abstract

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.It is a simple matter, using the Perron-Frobenius theorem [3], to show that the square of a real (finite) matrix cannot consist entirely of negative entries. In this paper we give an alternate proof of this result which makes use only of some elementary order properties. It is appropriate, therefore, to construct a proof within the more general framework of partially ordered linear algebras. The main theorem will then be valid not only for finite matrices but also for (row-finite) infinite matrices as well as operators on a Banach space [1]. We also generalize a result due to Birkhoff and Pierce [2] which states that the complex number field regarded as a linear algebra over the reals admits no lattice order.Definition. A partially ordered linear algebra (p.o.l.a.) P is a real associative linear algebra on which there is defined a partial ordering which satisfies the following conditions (x, y, z denote elements of P and a. denotes a real number):(a) if x ^ y then x + z ^ y + z,

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.…”