Regular elements of the complete semigroups of binary relations of the class Σ6(X,8)

In this paper, let X be a finite set, D be a complete X-semilattice of unions and Q={T1,T2,T3,T4,T5,T6,T7,T8} be an X-subsemilattice of D where T1⊂T3⊂T5⊂T6⊂T8, T1⊂T3⊂T5⊂T7⊂T8, T2⊂T3⊂T5⊂T6⊂T8, T2⊂T3⊂T5⊂T7⊂T8, T2⊂T4⊂T5⊂T6⊂T8, T2⊂T4⊂T5⊂T7⊂T8, T2∖T1≠∅, T1∖T2≠∅, T4∖T3≠∅, T3∖T4≠∅, T6∖T7≠∅, T7∖T6≠∅, T2∪T1=T3, T4∪T3=T5, T6∪T7=T8. Using the characteristic family of sets, the characteristic mapping and base sources of Q, we characterize the class whose elements are each isomorphic to Q. We generate some advanced formulas in order to calculate the number of regular elements α of BX(D) satisfying V(D,α)=Q, in an efficient way.