Are all adjoints lattice homomorphisms?

Question

Obviously something must be wrong in the following reasoning proving that any linear operator $T:X\to Y$ between Banach lattices has a lattice homomorphic adjoint: $\forall a,b\in E':$ $$T'(a\wedge b)=(T'a)\wedge(T'b)$$ if and only if $\forall x\in E:$ $$(T'(a\wedge b))x=((T'a)\wedge(T'b))x$$ if and only if (by definition adjoint) $$(a\wedge b)T x=((T'a)\wedge(T'b))x$$ if and only if (by definition $\wedge$ on dual $E'$) $$(a T x)\wedge(b T x)=((T'a)x)\wedge((T'b)x)$$ if and only if (by definition adjoint) $$(a T x)\wedge(b T x)=(a T x)\wedge(b T x)$$ which is always true (and the same logic holds for $\vee$). Is this correct?

Answer 1

The definition of the lattice structure on the dual is (see [1], Proposition II.4.2):

$$(a\wedge b)(x)=\sup\left\{f(y)+g(z):y+z=x,y\geq 0,z\geq 0\right\}$$

which is not the 'pointwise meet' as I assumed in the question.

[1] Banach Lattices and Positive Operators, Schaefer