A partial order on a semigroup
(
S
,
⋅
)
(S, \cdot )
is called natural if it is defined by means of the multiplication of
S
S
. It is shown that for any semigroup
(
S
,
⋅
)
(S, \cdot )
the relation
a
≤
b
a \leq b
iff
a
=
x
b
=
b
y
a = xb = by
,
x
a
=
a
xa = a
for some
x
x
,
y
∈
S
1
y \in {S^1}
, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup
(
T
X
,
∘
)
({T_X}, \circ )
of all maps on the set
X
X
are investigated.