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

• Melvin Henriksen
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 7)

## 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 = {aA : a ∨ 0 = a{, we have both P + P and PP contained in P. For aA, we let a + = a ∨ 0, a - = (-a) and |a| = a ∨ (-a). It follows that a = a + - a -, |a| = a + + a -, and for any a, bA, |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.

## Keywords

Structure Space Division Algebra Division Ring Laurent Series Subdirect Product
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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