The number of antichains in multilayered ranked sets

We find the asymptotic behavior of the number of antichains in some multilayered partially ordered sets with a rank function given on them. As a consequence we obtain the asymptotic behavior of the number of monotone Boolean functions, the number of self-dual monotone Boolean functions, the number of monotone fuzzy $(0,1)$-functions, and the number of monotone Boolean functions that possess the property $\langle A^{(2)}\rangle$.