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

Nino Tsinaridze; Shota Makharadze

doi:10.4236/am.2015.63042

Home > Journals > Physics & Mathematics > AM

Applied Mathematics > Vol.6 No.3, March 2015

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

Department of Mathematics, Faculty of Mathematics, Physics and Computer Sciences, Shota Rustaveli Batumi State University, Batumi, Georgia.

DOI: 10.4236/am.2015.63042 PDF HTML XML 2,954 Downloads 3,334 Views Citations

Abstract

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 .

Keywords

Semilattice, Semigroup, Binary Relation

Share and Cite:

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.