Ideals and bands in pre-Riesz spaces

Abstract

In a vector lattice, ideals and bands are well-investigated subjects. We study similar notions in a pre-Riesz space. The pre-Riesz spaces are exactly the order dense linear subspaces of vector lattices. Restriction and extension properties of ideals, solvex ideals and bands are investigated. Since every Archimedean directed partially ordered vector space is pre-Riesz, we establish properties of ideals and bands in such spaces. Mathematics Subject Classification (2000). Primary 46A40; Secondary 06F20.

Bands in pervasive pre-Riesz spaces

“…The restriction property (R) for bands is not true in general [6,Example 4.16]. In the present paper, we provide sufficient conditions on X such that (R) is satisfied.…”

“…We refer to [6,Example 3.4], where subspaces of the space C(R) of continuous functions on R are considered, ordered by the natural cone {x ∈ C(R) : x(t) ≥ 0 for all t ∈ R} .…”

“…Further, β j := lim i→∞ b ij = a 0j and ∞ j=1 |β j | = ∞ j=1 |a 0j | < ∞, so we have (4). To infer (5) and (6), use that ∞ j=1 |a ij | i is bounded and (a i0 ) i ∈ c 00 . It is clear that the map F is linear and that F (A) is positive if and only if A is a positive element of R.…”

“…The notions of ideal and band in a partially ordered vector space have been investigated in [6]. The method used there is embedding of the space as an order dense subspace of a vector lattice.…”

“…The extension property (E) for bands is shown in [6,Proposition 3.15]. The restriction property (R) for bands is not true in general [6,Example 4.16].…”