 We could add the requirement of empty intersection between different domains. Structures obeying this requirement are called “strict”, otherwise “lax” (or “liberal”). For strict structures we may consider the possibility of having an equality symbol for each sort, \(\approx _{i}\), each of them with \(\Rank(\approx _{i})=\langle i,i,0\rangle\). We require equality symbols (either strict or liberal) to be interpreted as genuine identity, the prototypical relation that holds between an object and itself and fails to hold between any two other objects.