Regular Elements of Semigroups BX(D) Defined by the Generalized X-Semilattice

In this paper, we take Q = {T sub 1 /sub , T sub 2 /sub , ..., T sub m-3 /sub , T sub m-2 /sub , T sub m-1 /sub , T sub m /sub subsemilattice of X -- semilattice of unions D where the elements T sub i /sub s are satisfying the following properties, T sub 1 /sub ⊂ T sub 3 /sub ⊂ ... ⊂ T sub m-3 /sub ⊂ T sub m-2 /sub ⊂ T sub m /sub , T sub 1 /sub ⊂ T sub 3 /sub ⊂ ... ⊂ T sub m-3 /sub ⊂ T sub m-1 /sub ∅, T sub 2 /sub \T sub 1 /sub ≠∅, T sub m-2 /sub \T sub m-1 /sub ≠∅, T sub m-1 /sub \T sub m-2 /sub ≠∅mm T sub 1 /sub ∪T sub 2 /sub = T sub 3 /sub , T sub m-2 /sub ∪T sub m-1 /sub = T sub m /sub . We will unvestigate the properties of regular element of α ∈ B sub X /sub (D) satisfying V(D, α) = Q. Moreover, we will calculate the number of regular elements of B sub X /sub (D) for a finite set X.