Abstract
Henriksen and Isbell showed in 1962 that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings) that are satisfied by all totally-ordered fields. In this paper, we review the work of Henriksen and Isbell on this topic, construct and classify some examples that illustrate this phenomenon using the valuation theory of Hion (in the process, answering a question posed in [E]) and, finally, prove that a base for the equational theory of totally-ordered fields consists of the f-ring identities of the form 0=0∨(f 1 ∧⋯∧f n ), n=1,2,..., where {f 1 ,...,f n }⊆ℤ[X 1 ,X 2 ,...] is not a subset of any positive cone.
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