If $(L, \land, \lor, 0, 1)$ is a lattice, and there exists a unary operation $'$ on $L$ such that
$(x \lor x')=1$, and
$(x \land x')=0$
both hold, is the unary operation $'$ an isomorphism between the semilattices $(L, \land)$ and $(L, \lor)$? If we add the condition that $'$ is an involution (i. e. $x''=x$), is $'$ an isomorphism?