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On sets of vectors of a finite vector space in which every subset of basis size is a basis

Simeon Ball — 2012

Journal of the European Mathematical Society

It is shown that the maximum size of a set S of vectors of a k -dimensional vector space over 𝔽 q , with the property that every subset of size k is a basis, is at most q + 1 , if k p , and at most q + k p , if q k p + 1 4 , where q = p k and p is prime. Moreover, for k p , the sets S of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a k × ( p + 2 ) matrix, with k p and entries from 𝔽 p , has k columns which are linearly dependent. Another is...

Structural aspects of truncated archimedean vector lattices: good sequences, simple elements

Richard N. Ball — 2021

Commentationes Mathematicae Universitatis Carolinae

The truncation operation facilitates the articulation and analysis of several aspects of the structure of archimedean vector lattices; we investigate two such aspects in this article. We refer to archimedean vector lattices equipped with a truncation as truncs. In the first part of the article we review the basic definitions, state the (pointed) Yosida representation theorem for truncs, and then prove a representation theorem which subsumes and extends the (pointfree) Madden representation theorem....

C- and C*-quotients in pointfree topology

We generalize a major portion of the classical theory of C- and C*-embedded subspaces to pointfree topology, where the corresponding notions are frame C- and C*-quotients. The central results characterize these quotients and generalize Urysohn's Extension Theorem, among others. The proofs require calculations in CL, the archimedean f-ring of frame maps from the topology of the reals into the frame L. We give a number of applications of the central results.

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