Positivity 12 (2008), 591–611
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040591-21, published online May 27, 2008
Ideals and bands in pre-Riesz spaces
Onno van Gaans and Anke Kalauch
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.
Keywords. Band, disjointness, ideal, order dense subspace, partially ordered
vector space, pre-Riesz space, solvex ideal.
In the study of vector lattices one naturally encounters partially ordered vector
spaces that are not lattices. For instance, spaces of operators between vector lat-
tices often lack lattice structure. This problem is usually avoided by assuming
Dedekind completeness of the codomain. Alternatively, one could try to extend the
required notions from lattice theory to a more general class of partially ordered
vector spaces. One approach is to reformulate relations involving absolute values
or positive or negative parts as relations of suitable sets of upper bounds. Intrinsic
deﬁnitions of several notions can thus be obtained, but it turns out to be diﬃcult
to set up their usual properties known from the vector lattice setting. Another ap-
proach is to embed the partially ordered vector space in a vector lattice and use the
lattice structure of the ambient space. The latter approach is taken in tostudy
disjointness in partially ordered vector spaces. As a rule, the embedding method
succeeds if the embedding is order dense. The partially ordered vector spaces that
can be embedded order densely in a vector lattice have been characterized in 
as the pre-Riesz spaces. Every directed Archimedean space is pre-Riesz, but there
exist also non-Archimedean pre-Riesz spaces. Often a lattice notion in a vector
lattice can be stated in several ways and their reformulation in partially ordered
vector spaces may lead to diﬀerent notions. It may be expected that the most useful