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, |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.
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
