In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let
X
be a set. A (binary) relation
◃
between an element
a
of
X
and a subset
S
of
X
is called a dependence relation, written
a
◃
S
, if it satisfies the following properties:
if
a
∈
S
, then
a
◃
S
;
if
a
◃
S
, then there is a finite subset
S
0
of
S
, such that
a
◃
S
0
;
if
T
is a subset of
X
such that
b
∈
S
implies
b
◃
T
, then
a
◃
S
implies
a
◃
T
;
if
a
◃
S
but
a
⧸
◃
S
−
{
b
}
for some
b
∈
S
, then
b
◃
(
S
−
{
b
}
)
∪
{
a
}
.
Given a dependence relation
◃
on
X
, a subset
S
of
X
is said to be independent if
a
⧸
◃
S
−
{
a
}
for all
a
∈
S
.
If
S
⊆
T
, then
S
is said to span
T
if
t
◃
S
for every
t
∈
T
.
S
is said to be a basis of
X
if
S
is independent and
S
spans
X
.
Remark. If
X
is a non-empty set with a dependence relation
◃
, then
X
always has a basis with respect to
◃
.
Furthermore, any two bases of
X
have the same cardinality.
