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

Abstract

In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal I in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of I is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal I equals its double disjoint complement, provided the double disjoint complement of I is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples.

Pervasive and weakly pervasive pre-Riesz spaces

“…Note that a pervasive pre-Riesz space which does not have the RDP is given in [11,Example 23]. The next example shows that RDP does not imply pervasiveness, in general.…”

“…We start with a recollection of two (non-intrinsic) characterizations of pervasiveness, which we include for the sake of completeness. For the following result from [15, Theorem 4.15, Corollary 4.16] a short proof can also be found in [11,Lemma 1].…”

“…In [4,Example 22] the authors prove that L oc ( ∞ 0 ) is pervasive. In [11,Theorem 14] it is shown that in a pervasive pre-Riesz space with the Riesz decomposition property every directed order closed ideal with a directed double disjoint complement is a band. In [5,Theorem 27] it is established that for a directed ideal I in X a vector lattice cover of I is given by the smallest extension ideal of I in Y , provided X is pervasive.…”

Vector lattice covers of ideals and bands in Pre-Riesz spaces

“…The result in Theorem 22 below is a generalization of a well-known statement for Archimedean vector lattices. First we recall a technical result for Archimedean pervasive pre-Riesz spaces, which is given in [16,Theorem 10]. Clearly, in the setting of Theorem 22 we obtain for two sets S 1 , S 2 ⊆ X + , for which the suprema sup S 1 and sup S 2 exist in X, that the relation S 1 ⊥ S 2 implies sup S 1 ⊥ sup S 2 .…”

Order Continuous Operators on Pre-Riesz Spaces and Embeddings

“…If X is an Archimedean directed ordered vector space, then every vector lattice cover of X is Archimedean. For the following result see [11,Corollary 5].…”