Ordered Algebraic Structures pp 11-17 | Cite as

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

Chapter

## 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*, |*a*a+*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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