Irreducible generating sets of complete semigroups of unions $B_X(S)$ defined by semilattices of the class $\Sigma_1(X,4)$

In complete semigroups of unions $B_X(D)$ defined by semilattices of the class $\Sigma_1(X,4)$, we describe the set of all external elements and show that it is a generating (and, therefore, irreducible) set of the semigroup $B_X(D)$. For a finite semigroup $B_X(D)$, we give a formula for calculating the number of elements of the generating set.