Idempotent elements of the semigroup $B_{X}(D)$ defined by semilattices of the class $\Sigma_{3}(X,8)$ when $Z_{7}=\varnothing$

The paper presents a full description of idempotent elements of the semigroup of binary relations $B_{X}(D)$, which are defined by semilattices of the class $\Sigma_{3}(X,8)$. For the case where $X$ is a finite set and $Z_{7}=\varnothing$, we derive formulas for calculating the number of idempotent elements of the respective semigroup.