1972
DOI: 10.2307/2038512
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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 resu… Show more

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“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%