d-independence and d-bases in vector lattices

This article contains the results of two types. In Section 3 we give a complete characterization of band preserving projection operators on Dedekind complete vector lattices. These operators were instrumental in our work [AK2], and now we have obtained their description. This is done in Theorem 3.4. Let us mention also Theorem 3.2 that contains a description of such operators on arbitrary laterally complete vector lattices. The central role in these descriptions is played by d-bases, one of two principal tools utilized in [AK2]. The concept of a d-basis, originally considered in this context in [AVK], has been applied so far only to vector lattices with a large amount of projection bands. The absence of the projection bands has been the major obstacle for extending, otherwise very useful concept of d-bases, to arbitrary vector lattices. In Section 4 we will be able to overcome this obstacle by finding a new way to introduce d-independence in an arbitrary vector lattice. This allows us to produce a new definition of a d-basis which is free of the existence of projection bands. We illustrate this by proving several results devoted to cardinality of d-bases. Theorems 4.13 and 4.15 are the main of them and they assert that, under very general conditions, a vector lattice either has a singleton d-basis of else this d-basis must be infinite. This extends some of our work in [AK2, Section 6]. To make the reading of the article as much independent of [AK2] as possible we collect in the next section some necessary definitions and facts about d-bases. Most of this preliminary material, as well as some appropriate history regarding the subject, can be found in [AK2].