thermodynamics - Entropy function additivity

I am reading Herbert Callen's book Thermodynamics, which proposes a postulatory treatment to the subject.

The postulate number 3 states the properties of the "entropy" function, one of which is the additive property: the entropy of a system composed of multiple subsystems separated via internal constraints is sum of the entropy of the individual subsystems.

Consider two systems with internal energies $U_1, U_2$, volumes $V_1$, $V_2$ and number of molecules $N_1$, $N_2$. Doesn't the above statement mean

$S(U_1+U_2,V_1+V_2,N_1+N_2) = S(U_1,V_1,N_1)+S(U_2,V_2,N_2)$

? In which case the entropy function is linear. But the entropy function for an ideal gas is nonlinear in the preceding sense.

For two identical systems, it would mean the entropy function must be a homogenous function of order one, which makes sense. But this doesn't imply linearity.

So, I would like to know the lapse in my understanding. Thanks!

Edit: The above equation doesn't qualify for the entropy function to be linear. I think the additivity of entropy can mathematically be represented by this equation.

Answer

I believe it rather means: $$ S_{tot}(U_1,U_2,V_1,V_2,N_1,N_2) = S_1(U_1,V_1,N_1) + S_2(U_2,V_2,N_2) $$ Think, for instance, of two isolated containers. If the statement, as you wrote it down, would be true, the mixing of two ideal gases would be reversible.