Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class (original) (raw)

https://doi.org/10.4236/AM.2015.62029

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Abstract

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idem-potent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

REGULAR ELEMENTS OF THE COMPLETE SEMIGROUPS OF BINARY RELATIONS OF THE CLASS 7 (X, 8)

In this paper let Q = {T1, T2, T3, T4, T5, T6, T7, T8} be a subsemilattice of X−semilattice of unions D where T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T3 ⊂ T5 ⊂ T7 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T6 ⊂ T8, T1 ⊂ T2 ⊂ T4 ⊂ T5 ⊂ T7 ⊂ T8, T1 6= ∅, T4\T3 6= ∅, T3\T4 6= ∅, T6\T7 6= ∅, T7\T6 6= ∅, T3 ∪T4 = T5, T6 ∪T7 = T8, then we characterize the class each element of which is isomorphic to Q by means of the characteristic family of sets, the characteristic mapping and the generate set of Q. Moreover, we calculate the number of regular elements of BX(D) for a finite set X.

Generating Set of the Complete Semigroups of Binary Relations

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set () () { } X B B D V X D α α * = ∈ = , .

On the Semigroup of Difunctional Binary Relations

FUDMA JOURNAL OF SCIENCES

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

Regular elements of the complete semigroups of binary relations of the class Σ 6 ((, 8

In this paper, let be a nite set, be a complete-semilattice of unions and = {{ 1 \ 1 = , 1 \ 2 = , 4 \ 3 = , 3 \ 4 = , 6 \ 7 = , 7 \ 6 = , 2 ∪ 1 = 3 , 4 ∪ 3 = 5 , 6 ∪ 7 = 8. Using the characteristic family of sets, the characteristic mapping and base sources of , we characterize the class whose elements are each isomorphic to. We generate some advanced formulas in order to calculate the number of regular elements of (() satisfying ((,) = , in an eecient way.

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References (3)

  1. Diasamidze, Ya. and Makharadze, Sh. (2013) Complete Semigroups of Binary Relations. Kriter Yayınevi, İstanbul, 524 p.
  2. Albayrak, B., Aydin, N. and Diasamidze, Ya. (2013) Reguler Elements of the Complete Semigroups of Binary Rela- tions of the Class ( ) 7 ,8
  3. X ∑ . International Journal of Pure and Applied Mathematics, 86, 199-216. http://dx.doi.org/10.12732/ijpam.v86i1.13

Regular Elements of Semigroups BX(D) Defined bythe Generalized X-Semilattice

2014

In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilattice of unions D where the elements Ti’ s are satisfying the following properties, T1⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T1⊂ T 3⊂ · · · ⊂ Tm−3 ⊂ Tm−1⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−1 ⊂ Tm, T1\T 2 6= ∅, T2\T 1 6= ∅, Tm−2\Tm−1 6= ∅, Tm−1\Tm−2 6= ∅, T1∪T 2= T 3, Tm−2∪Tm−1= Tm. We will investigate the properties of regular element α ∈ BX(D) satisfying V (D,α) = Q. Moreover, we will calculate the number of regular elements of BX(D) for a finite set X.

Idempotents, band and Green's relations on ternary semigroups

2016

This paper is for one part a generalization of some results obtained by Miyuki Yamada [20] in the case of binary semigroups to ternary semigroups. We prove analogous of almost all the results previously cited. We prove in particular that the set of the idempotents in regular ternary semigroup is a band (that is, a semigroup). In a second part we continue our investigations started in [13; 14] on these semigroups, as on the structure of the set E(S) of idempotents of the ternary semigroup S. The particular case of ternary inverse semigroup has been studied and a relationship between the existence of idempotents and the inverse elements has been caracterized. The documents [5]; [9] and [10] have been intensively used. We asked two questions and the answer for the second one will be the subject of a forcoming paper. We use many references in our work the most important are those used as bibliography.