Regular Elements of the Complete Semigroups BX(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 Bx(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 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)=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 .
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.
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.
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.