Some Regular Elements, Idempotents and Right Units of Semigroup BX (D) Defined by X- Semilattices which is Union of a Chain and Two Rhombus

In this paper we take Q={T sub 0 /sub ,T sub 1,... /sub ,T sub m-6 /sub ,T sub m-5 /sub ,T sub m-4 /sub ,T sub m-3 /sub ,T sub m-2 /sub ,T sub m-1 /sub ,T sub m /sub } (m≥6) subsemilattice of X- semilattice of unions D where the elements ... s are satisfying the following properties, T sub 0 /sub ⊂ T sub 1 /sub ⊂ ... ⊂ T sub m-5 /sub ⊂ T sub m-4 /sub ⊂ T sub m-2 /sub ⊂ T sub m /sub , T sub 0 /sub ⊂ T sub 1 /sub ⊂ ... ⊂ T sub m-5 /sub ⊂ T sub m-3 /sub ⊂ T sub m-2 /sub ⊂ T sub m /sub , T sub 0 /sub ⊂ T sub 1 /sub ⊂ ... ⊂ T sub m-5 /sub ⊂ ⊂ T sub m-3 /sub ⊂ T sub m-1 /sub ⊂ T sub m /sub , T sub m-4 /sub \ T sub m-3 /sub ≠ ∅, T sub m-3 /sub \ T sub m-4 /sub ≠ ∅, T sub m-4 /sub \ T sub m-1 /sub ≠ ∅, T sub m-1 /sub \ T sub m-4 /sub ≠ ∅, T sub m-2 /sub \ T sub m-1 /sub ≠ ∅, T sub m-1 /sub \ T sub m-2 /sub ≠ ∅ T sub m-4 /sub ∪ T sub m-3 /sub = T sub m-2 /sub , T sub m-4 /sub ∪ T sub m-1 /sub = T sub m-2 /sub ∪ T sub m-1 /sub = T sub m /sub . We will investigate the properties of regular elements and idempotents of the complete semigroup of binary relations B sub x /sub (D) satisfying V (D, α)=Q. Also we investigate right units of the semmigroup B sub x /sub (Q). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup.