Torsion Classes of Vector Lattices

P. F. Conrad, S. M. Lin and D. G. Nelson
Additional contact information
P. F. Conrad: University of Kansas
S. M. Lin: Bethel College
D. G. Nelson: Mercer University

A chapter in Ordered Algebraic Structures, 1993, pp 11-30 from Springer

Abstract: Abstract Let Vl be the class of all vector lattices, and let S and T be torsion classes of ℓ-groups. T ∩ Vl is a torsion class if and only if each divisible abelian ℓ -group in T contains a largest ℓ -ideal that is a vector lattice. Moreover, if T ∩ Vl is a torsion class, so is S ∩ T ∩ Vl. The following classes of vector lattices form torsion classes: the hyperarchimedean vector lattices; the finite-valued vector lattices; the class of all vector lattices of the form Σ(Δ,R). In particular, the principal torsion class $$ \tilde \sum (\Delta, R) $$ determined by Σ(Δ,R) consists of vector lattices; it consists of all cardinal sums of ℓ-groups Σ(Λ, R) where Λ is a direct limit of connected, convex subsets of Λ. The following classes of vector lattices form pseudo torsion classes: the archimedean ℓ -groups; the special-valued and conditionally laterally complete ℓ -groups. Underlying this theory is the fact that if K is a finite-valued ℓ -group or a conditionally laterally complete ℓ -group, then K is a vector lattice if and only if each K(k) is a vector lattice, which is true if and only if each K(k), with k a special element, is a vector lattice.

Keywords: Vector Lattice; Scalar Multiplication; Special Element; Direct Limit; Special Basis (search for similar items in EconPapers)
Date: 1993
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DOI: 10.1007/978-94-011-1723-4_2

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