## Abstract

In vector lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space *X*. We relate them to projection bands in a vector lattice cover *Y* of *X*. If *X* is pervasive, then a projection band in *X* extends to a projection band in *Y*, whereas the restriction of a projection band *B* in *Y* is not a projection band in *X*, in general. We give conditions under which the restriction of *B* is a projection band in *X*. We introduce atoms and discrete elements in *X* and show that every atom is discrete. The converse implication is true, provided *X* is pervasive. In this setting, we link atoms in *X* to atoms in *Y*. If *X* contains an atom \(a>0\), we show that the principal band generated by *a* is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space *X*, we establish that *X* is pervasive if and only if it is a vector lattice.

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## Notes

- 1.
There are several alternative definitions of order convergence of nets in partially ordered vector spaces. However, by [6, Theorem 4.4] an operator

*T*is order continuous if and only if*T*is continuous with respect to any of these alternative notions. - 2.
We stress that in [2, Definition 1.42] the authors introduce this concept using a different term. They call an element \(a\in X_+\) for which \(0\leqslant x\leqslant a\) implies that \(x=\lambda a\) for some real \(\lambda \geqslant 0\) an

*extremal vector*or a*discrete vector*of \(X_+\). However, a similar concept is well-known in the less general setting of vector lattices. Indeed, [20, § 3] gives a definition of an atom in a vector lattice which differs from our notion by the fact that the considered element need not be positive. However, by [24, III.13.1 a)] for every atom \(a\in Y\) it follows \(a>0\) or \(a<0\). Moreover, if \(a<0\) is an atom, so is \(-a\). These circumstances maybe clarify that, based on [2, Definition 1.42], we define atoms as positive elements. Proposition 26 justifies our choice in Archimedean pervasive pre-Riesz spaces. - 3.
For instance, in [16, Definition 26.1] the terms

*atom*and*discrete element*are used in a reversed way with respect to our terminology. - 4.
Minkowski’s theorem states that in a finite-dimensional normed vector space every element of a compact convex set \(S\subseteq X\) is a convex combination of extreme points of

*S*, see e.g. [19, p. 1].

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Kalauch, A., Malinowski, H. Projection bands and atoms in pervasive pre-Riesz spaces.
*Positivity* **25, **177–203 (2021). https://doi.org/10.1007/s11117-020-00757-7

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### Keywords

- Order projection
- Projection band
- Band projection
- Principal band
- Atom
- Discrete element
- Extremal vector
- Pervasive
- Weakly pervasive
- Archimedean directed ordered vector space
- Pre-Riesz space
- Vector lattice cover

### Mathematics Subject Classification

- 46A40
- 06F20
- 47B65