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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.11212 |
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| _version_ | 1866915935594479616 |
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| author | Jokela, Jani |
| author_facet | Jokela, Jani |
| contents | The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces. We then apply our results to establish the equivalence of pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on disjointness and discrete elements in partially ordered vector spaces Jokela, Jani Functional Analysis 06F20 The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces. We then apply our results to establish the equivalence of pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces. |
| title | A note on disjointness and discrete elements in partially ordered vector spaces |
| topic | Functional Analysis 06F20 |
| url | https://arxiv.org/abs/2509.11212 |