On \(f\)-rings that are not formally real (Q1959929)

The paper consists of seven sections. In the first one, the author summarizes basic information about ordered rings. In the second one, he recalls Hiron's work from 1957 about generalized valuation defined on totally ordered rings and taking values in totally ordered monoids. In the third one, he reviews the work of Birkhoff and Isbell from1952 in which they undertook a deep study of the equational theory of \(f\)-rings. They essentially showed that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings). The author ends the section with his own example, which seems to be simpler. In Sections 5 and 6, the author gives some generalizations of his example. In the last section, he proves a new theorem that exhibits a base for the equational theory of totally ordered fields in the language of lattice-ordered rings consisting of equations of a particularly simple form.