Old and New Unsolved Problems in Lattice-Ordered Rings that need not be f-Rings

Abstract

Recall that a lattice-ordered ring or l-ring A(+, •, ∨, ∧) is a set together with four binary operations such that A(+, •) is a ring, A(∨, ∧) is a lattice, and letting P = {a ∈ A : a ∨ 0 = a{, we have both P + P and P • P contained in P. For a ∈ A, we let a + = a ∨ 0, a - = (-a) and |a| = a ∨ (-a). It follows that a = a + - a -, |a| = a + + a -, and for any a, b ∈ A, |aa+b| < |a|+ |b| and |ab| < |a| |b|. As usual a < b means (b–a) ∈ P. We leave it to the reader to fill in what is meant by a lattice-ordered algebra over a totally ordered field.