In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let
be a set. A (binary) relation
between an element
of
and a subset
of
is called a dependence relation, written
, if it satisfies the following properties:
- if
, then
;
- if
, then there is a finite subset
of
, such that
;
- if
is a subset of
such that
implies
, then
implies
;
- if
but
for some
, then
.
Given a dependence relation
on
, a subset
of
is said to be independent if
for all
If
, then
is said to span
if
for every
is said to be a basis of
if
is independent and
spans
Remark. If
is a non-empty set with a dependence relation
, then
always has a basis with respect to
Furthermore, any two bases of
have the same cardinality.