Regular Elements of the Complete Semigroups *B*_{X}(D) of Binary Relations of the Class ∑_{2}(*X*,8)

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* .

As we know if

Cite this paper

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.

Tsinaridze, N. and Makharadze, S. (2015) Regular Elements of the Complete Semigroups

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.

[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.