Regular Elements of the Complete Semigroups BX(D) of Binary Relations of the Class ∑2(X,8)

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 Bx(D) requires an investigation of XI-subsemilattices of semilattice D for which V(D,a)=Q2(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 .

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Tsinaridze, N. and Makharadze, S. (2015) Regular Elements of the Complete Semigroups BX(D) of Binary Relations of the Class ∑2(X,8). Applied Mathematics, 6, 447-455. doi: 10.4236/am.2015.63042.

 [1] Diasamidze, Ya. and Makharadze, Sh. (2013) Complete Semigroups of Binary Relations. Monograph. Kriter, Turkey, 1-520. [2] Diasamidze, Ya. and Makharadze, Sh. (2010) Complete Semigroups of Binary Relations. Monograph. M., Sputnik+, 657 p. (In Russian) [3] Lyapin, E.S. (1960) Semigroups. Fizmatgiz, Moscow. (In Russian) [4] Diasamidze, Ya., Makharadze, Sh. and Rokva, N. (2008) On XI-Semilattices of Union. Bull. Georg. Nation. Acad. Sci., 2, 16-24. [5] Diasamidze, Ya.I. (2003) Complete Semigroups of Binary Relations. Journal of Mathematical Sciences, 117, 4271-4319. [6] 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. [7] 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.