Idempotent elements of the semigroup $B_{X}(D)$ defined by semilattices of the class $\Sigma_{2}(X,8)$

A complete semigroup of binary relations is defined by semilattices of the class $\Sigma _{2}(X,8)$. A description of idempotent elements of this semigroup is given. For the case where $X$ is a finite set and $Z_7\cap Z_6\neq \varnothing$, formulas are derived by calculating the number of idempotent elements of the semigroup.