ON THE MAXIMAL CARDINALITY OF AN INFINITE CHAIN OF VECTOR SUBSPACES

Year 2013, Volume: 13 Issue: 13, 63 - 68, 01.06.2013

David E. Dobbs

Abstract

For each infinite cardinal number κ, let Ω(κ) be the supremum of
the cardinalities of chains of subsets of a set of cardinality κ. (Ω(κ) is equal
to what has been called ded(κ) in the literature.) Let K be a field and V a
vector space over K. Let Λ(V ) be the supremum of the cardinalities of chains
of vector subspaces of V . Let the dimension of V as a vector space over K be
the infinite cardinal number κ. Then Ω(κ) ≤ Λ(V ) ≤ Ω(|V |), and so Λ(V ) > κ,
contrary to a result of Menth. If, in addition, K is either finite or infinite with
|K| ≤ κ, then Ω(κ) = Ω(|V |) (= Λ(V )).

Keywords

vector space, vector subspace, chain, infinite cardinal number, dimension, field extension