As we know if D is a complete X-semilattice of unions then semigroup Bx(D)possesses a right unit iff D is an XI-semilattice of unions. The investigation of those a-idempotent and regular elements of semigroups B_{x}(D) requires an investigation of XI-subsemilattices of semilattice D for which V(D,a)=Q∈∑_{2}(X,8) . Because the semilattice Q of the class ∑_{2}(X,8) are not always XI -semilattices, there is a need of full description for those idempotent and regular elements when V(D,a)=Q . For the case where X is a finite set we derive formulas by calculating the numbers of such regular elements and right units for which V(D,a)=Q .
Diasamidze, Ya., Makharadze, Sh. and Diasamidze, Il. (2008) Idempotents and Regular Elements of Complete Semigroups of Binary Relations. Journal of Mathematical Sciences, 153, 481-499.
Diasamidze, Ya. (2009) The Properties of Right Units of Semigroups Belonging to Some Classes of Complete Semigroups of Binary Relations. Proceedings of A. Razmadze Mathematical Institute, 150, 51-70.