d 3.pdf

The cardinal n um b er of algebras
b y
An ton C e d i l n i k
Ljubljana
Abstract
If the cardinal n um b er of a
eld I F is c
the cardinal n um b er of the set of
3
d
nonisomorphic t yp es of d
dimensional algebras o v er I F is less or equal to c
the equalit y
holds if at least one of c and d is in
nite
In this note w e shall assume the v alidit y of the Axiom of c hoice
W e
shall understand the designation x
as x is an in
nite cardinal n um b er
I F will alw a ys mean m utativ e
eld with c haracteristic c hr I F
p and
cardinalit y crd I F
c
Let V b e a v ector space o v er I F with dimension dim V
d and A a family
of all
nonasso ciativ e
algebras o v er V
A can b e in terpreted also as the set
2
of all bilinear maps from V in to V
T o b e isomorphic is an equiv alence

relation in A
for whic h w e shall use the sym b ol

The elemen ts of the


will b e called algebr aic typ es and
quotien t set A



nat
c
d
crd
A

will b e the cardinalit y of this set
Our in ten tion is to determine nat
c
d
at
least in the in
nite case
The de
nition of the sym b ol nat
c
d
at the
rst sigh t is not correct
since
it is not ob vious that to t w o di
eren t
elds I F and I F with the same cardinal
1 2
n um b er c there b elongs the same nat
c
d
F or the
nite
elds it is trivia