ON THE MAXIMAL CARDINALITY OF AN INFINITE CHAIN OF VECTOR SUBSPACES

Authors : David E DOBBS

Pages : 63-68

View : 18 | Download : 13

Publication Date : 2013-06-01

Article Type : Research Paper

Abstract :For each infinite cardinal number κ, let Ωinsert ignore into journalissuearticles values(κ); be the supremum of the cardinalities of chains of subsets of a set of cardinality κ. insert ignore into journalissuearticles values(Ωinsert ignore into journalissuearticles values(κ); is equal to what has been called dedinsert ignore into journalissuearticles values(κ); in the literature.); Let K be a field and V a vector space over K. Let Λinsert ignore into journalissuearticles values(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 Ωinsert ignore into journalissuearticles values(κ); ≤ Λinsert ignore into journalissuearticles values(V ); ≤ Ωinsert ignore into journalissuearticles values(|V |);, and so Λinsert ignore into journalissuearticles values(V ); > κ, contrary to a result of Menth. If, in addition, K is either finite or infinite with |K| ≤ κ, then Ωinsert ignore into journalissuearticles values(κ); = Ωinsert ignore into journalissuearticles values(|V |); insert ignore into journalissuearticles values(= Λinsert ignore into journalissuearticles values(V ););.
Keywords : vector space, vector subspace, chain, infinite cardinal number, dimension, field extension