#### Title

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

#### Document Type

Book Chapter

#### Department

Mathematics (HMC)

#### Publication Date

2002

#### 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.

#### Rights Information

© 2002 Springer

#### DOI

10.1007/978-1-4757-3627-4_2

#### Recommended Citation

Henriksen, Melvin Old and new unsolved problems in lattice-ordered rings that need not be f-rings. Ordered algebraic structures, 11–17, Dev. Math., 7, Kluwer Acad. Publ., Dordrecht, 2002. DOI: 10.1007/978-1-4757-3627-4_2