Bands in partially ordered vector spaces with order unit

Abstract

In an Archimedean directed partially ordered vector space X , one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover Y of X . If X has an order unit, Y can be represented as a subspace of C( ), where is a compact Hausdorff space. We characterize bands in X , and their disjoint complements, in terms of subsets of . We also analyze two methods to extend bands in X to C( ) and show how the carriers of a band and its extensions are related. We use the results to show that in each n-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by 1 4 2 2 n for n ≥ 2. We also construct examples of (n + 1)-dimensional partially ordered vector spaces with 2n n + 2 bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when n ≥ 4.

Inverses of disjointness preserving operators in finite dimensional pre-Riesz spaces

“…As a consequence of the previous lemma, the main result on the inverses of disjointness preserving bijections in finite dimensions is obtained next. The number b of bands in (R n , K) is less or equal 1 4 2 2 n , see [6]. In the subsequent theorem, P(b) denotes the set of orders of permutations on b symbols.…”

“…Kalauch, B. Lemmens and O. van Gaans and collect properties of p in the subsequent lemmas. The set B of bands in (R n , K) is finite, see [6]. Let T : R n → R n be a disjointness preserving linear bijection, and let T be defined as in (2).…”

“…In partially ordered vector spaces, bands are defined in [4] as sets that equal their double-disjoint complement. In [6] it is shown that the number b of bands in A. Kalauch, B. Lemmens and O. van Gaans (R n , K) does not exceed 1 4 2 2 n . Using this fact, for a disjointness preserving bijection T : R n → R n we will show that there is k ∈ N such that T k is band preserving.…”

Order closed ideals in pre-Riesz spaces and their relationship to bands

“…By [8, Propositions 5.12, 5.3 and 5.1(iii)] every band has (E) and every ideal and o-closed ideal has (R). By [7,Proposition 17 (a)] for a band B an extension band is given by B i(B) . By [8, Propositions 5.5(i) and 5.6] every solvex ideal has both (E) and (R).…”

“…They were introduced 1993 in [16] by van Haandel. Later pre-Riesz spaces and structures therein were thoroughly investigated by Kalauch, Lemmens and van Gaans in [5][6][7][8][9][10][11][12] and [15]. The definition of disjointness in pre-Riesz spaces was first given in [8].…”