Let
X
X
and
Y
Y
be compact Hausdorff spaces, and
E
E
,
F
F
be Banach lattices. Let
C
(
X
,
E
)
C(X,E)
denote the Banach lattice of all continuous
E
E
-valued functions on
X
X
equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism
Φ
:
C
(
X
,
E
)
→
C
(
Y
,
F
)
\Phi : C(X,E)\rightarrow C(Y,F)
such that
Φ
f
\Phi f
is non-vanishing on
Y
Y
if and only if
f
f
is non-vanishing on
X
X
, then
X
X
is homeomorphic to
Y
Y
, and
E
E
is Riesz isomorphic to
F
F
. In this case,
Φ
\Phi
can be written as a weighted composition operator:
Φ
f
(
y
)
=
Π
(
y
)
(
f
(
φ
(
y
)
)
)
\Phi f(y)=\Pi (y)(f(\varphi (y)))
, where
φ
\varphi
is a homeomorphism from
Y
Y
onto
X
X
, and
Π
(
y
)
\Pi (y)
is a Riesz isomorphism from
E
E
onto
F
F
for every
y
y
in
Y
Y
. This generalizes some known results obtained recently.