## On finitely generated vector sublattices

### Authors:

- Lech Drewnowski,
- Witold Wnuk

### Abstract

We investigate various questions concerning vector sublattices, vlt S, generated by subsets S of an Archimedean vector lattice E. We first prove a distributivity law: vlt(X, Y,Z) = vlt(X, Y )+ vlt(X,Z) if X, Y,Z ⊂ E and Y ⊥ Z, and derive a number of its consequences. We next show that in a topological vector lattice the dimension of the sublattice generated by an analytic set is either ≤ N_{0} or 2^{N0}, and that the same is true for sublattices generated by at most countable sets in arbitrary vector lattices. In a vector lattice, we characterize those sets that generate n-dimensional sublattices and prove that a finite set generates a finite-dimensional sublattice if so does each pair of its elements. We also show that in a uniformly complete vector lattice every principal ideal of infinite dimension contains pairs of positive elements generating N_{0}- as well as 2^{N0}-dimensional sublattices. The special case of lattices C(K) is also treated in this respect. Moreover, for a compact set K ⊂ ^{n} with a nonempty interior, it is shown that the minimal number of functions in C(K) or C(K)+ generating a dense sublattice is n + 1. We also prove that every (separable) Banach lattice C(K) can be embedded in a discrete (separable) Banach lattice of the same type. Finally, we prove that in a discrete and σ-Dedekind complete separable F-lattice one can always find a pair of positive elements generating a dense sublattice, and we use that result to show that, in general, this is far from being true even in the case of discrete separable C(K) lattices.

- Record ID
- UAM1a3cc3658f5c4032b598c72b6eb11ef3
- Author
- Journal series
- Studia Mathematica, ISSN 0039-3223
- Issue year
- 2019
- Vol
- 245
- Pages
- 129-167
- Publication size in sheets
- 1.90
- ASJC Classification
- DOI
- DOI:10.4064/sm170524-23-12 opening in a new tab
- Language
- en English
- Score (nominal)
- 100
- Score source
- journalList
- Score
- = 100.0, 26-04-2020, ArticleFromJournal
- Publication indicators
- = 0; = 0; : 2017 = 1.076; : 2018 = 0.750 (2) - 2018=0.663 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAM1a3cc3658f5c4032b598c72b6eb11ef3/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.