Is the adjoint of a linear operator $T:X\to Y$ between Banach lattices, always lattice homomorphic if $T$ is surjective?
This is my proof but I really doubt this is true: $\forall a,b\in Y'$ and $\forall x\in X$:
$$T'(a\wedge b)x=((T'a)\wedge(T'b))x$$ iff (definition adjoint) $$(a\wedge b)T x=((T'a)\wedge(T'b))x$$ iff (definition $\wedge$ on $Y'$ and definition $\wedge$ on $X'$) $$\sup\{a y+b z:T x=y+z\}=\sup\{(T'a)y+(T'b)z:x=y+z\}$$ iff (definition adjoint) $$\sup\{a y+b z:T x=y+z\}=\sup\{a T y+b T z:x=y+z\}$$ iff (change of variables) $$\sup\{a y +b(T x - y):y \in Y\}=\sup\{a T y+b T (x-y):y \in X\}$$ iff (remove constant term) $$\sup\{a y+b y:y \in Y\}=\sup\{a T y+b T y:y \in X\}$$ which holds if $T$ is surjective. Or even when $T$ only has dense range?
(I made a small mistake in this question, but this result still seems way too strong)