Idempotent Elements of the Semigroups B_X(D) Defined by Semilattices of the Class ∑₃ (X,8) When Z₇=Ø

G. Tavdgiridze, Ya. Diasamidze, O. Givradze

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

DOI: 10.4236/am.2016.79085                

Abstract

In this paper, complete semigroup binary relation is defined by semilattices of the class . We give a full description of idempotent elements of given semigroup. For the case where X is a finite set and , we derive formulas by calculating the numbers of idempotent elements of the respective semigroup.

Keywords

Semilattice, Semigroup, Binary Relation, Idempotent Element

Received 18 March 2016; accepted 24 May 2016; published 27 May 2016

1. Introduction

Definition 1.1. Let. If or for any, then is called an idem­potent element or called right unit of the semigroup respectively.

Definition 1.2. We say that a complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:

a) for any;

b) for any nonempty element Z of D (see [1] , Definition 1.14.2 or see [2] , Definition 1.14.2).

Definition 1.3. Let D be an arbitrary complete X-semilattice of unions,. If

then it is obvious that any binary relation of a semigroup can always be written in the form

the sequel, such a representation of a binary relation will be called quasinormal.

Note that for a quasinormal representation of a binary relation, not all sets can be different from an empty set. But for this representation the following conditions are always fulfilled:

a), for any and;

b) (see [1] , Definition 1.11 or see [2] , Definition 1.11).

Theorem 1.1. Let D, , and I denote respectively the complete X-semilattice of unions D, the set of all XI-subsemilattices of the semilattice D, the set of all right units of the semigroup and the set of all idempotents of the semigroup. Then for the sets and I the following statements are true:

a) if and, then

1) for any elements and of the set that satisfy the condition;

2);

3) the equality is fulfilled for the finite set X.

b) if, then

1) for any elements and of the set that satisfy the condition;

2);

3) the equality is fulfilled for the finite set X (see [1] [2] Theorem 6.2.3).

2. Results

Lemma 2.1. Let and. Then the following sets are all XI-subsemilattices of the given semilattice D:

1) (see diagram 1 of the Figure 1);

2) (see diagram 2 of the Figure 1);

3) (see di-

agram 3 of the Figure 1);

4) (see diagram

4 of the Figure 1);

5) (see diagram

5 of the Figure 1);

6) (see diagram 6 of the Figure 1);

7) (see diagram 7 of the Figure 1);

8) (see diagram 8 of the Figure 1);

9) (see diagram 9 of the Figure 1);

10) (see diagram 10 of the Figure 1);

11) (see diagram 11 of the Figure 1);

12) (see diagram 12 of the Figure 1);

13) (see diagram 13 of the Figure 1);

14) (see diagram 14 of the Figure 1);

15) (see diagram 15 of the Figure 1);

16) (see diagram 16 of the Figure 1);

Proof: This lemma immediately follows from the ( [3] , lemma 2.4).

Lemma is proved.

We denote the following semitattices as follows:

1), where;

2) where;

3) where;

4) where;

5) where;

6) where, , , ,;

7) where, , , ,;

8) where;

9)

10) where, , , ,;

11) where;

12) where, , , , ,

, ,;

13)

14)

15)

16)

Theorem 2.1. Let, and. Binary relation is an idempotent relation of the semigroup iff binary relation satisfies only one conditions of the following conditions:

1);

2), where, , and satisfies the conditions:,;

3), where, , and satisfies the conditions:, , ,;

4), where, , and satisfies the conditions:, , , , ,;

Figure 1. All Diagrams XI-subsemilattices of the semilattice D.

5), where,

, and satisfies the conditions:, , ,

, , , ,;

6), where, ,

and satisfies the conditions:, , ,;

7), where, ,

, , , and satisfies the conditions:

, , , , ,;

8), where,

and satisfies the conditions:, , , , , , , , ;

9), where, ,

, , and satisfies the conditions:, ,

, , , , ,;

10), where, ,

, , and satisfies the conditions:, ,

, ,;

11), where,

and satisfies the conditions:, ,

, , , ,;

12),

where, , , , ,

and satisfies the conditions:, , ,

, ,;

13), where,

, , , , , , and satisfies the conditions:, , , , , , , ,;

14), where, ,

and satisfies the conditions:, ,

, , , ,;

15), where

and satisfies the conditions:, ,

, , , ,

,;

16),

where, and satisfies the conditions:, ,

, , , , ,

,.

Proof. This Theorem immediately follows from the ( [3] , Theorem 2.1]).

Theorem is proved.

Lemma 2.2. If X be a finite set, then the following equalities are true:

a);

b);

c);

d);

e);

f);

g);

h)

i)

j);

k);

l);

m)

n)

o);

p).

Proof. This lemma immediately follows from the ( [3] , lemma 2.6).

Lemma is proved.

Lemma 2.3. Let and. If X is a finite set, then the number may be calculated by the formula.

Proof. By definition of the given semilattice D we have

.

If the following equalities are hold

,

then

.

[See Theorem 1.1] Of this equality we have:.

[See statement a) of the Lemma 2.2.]

Lemma 2.4. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

if

.

Then

.

[See Theorem 1.1] Of this equality we have:

[See statement b) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.5. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

[See Theorem 1.1]. Of this equality we have:

[See statement c) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.6. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

[See Theorem 1.1] Of this equality we have:

[See statement d) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.7. Let and.If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

[See Theorem 1.1] Of this equality we have:

[See statement e) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.8. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

[See Theorem 1.1] Of this equality we have:

[See statement f) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.9. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

[See Theorem 1.1] Of this equality we have:

[See statement g) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.10. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

[See Theorem 1.1] Of this equality we have:

[See statement h) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.11. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have.

If the following equality is hold then.

[See Theorem 1.1] Of this equality we have:

[See statement i) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.12. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice we have

If

[See Theorem 1.1] Of this equality we have:

[See statement j) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.13. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

[See Theorem 1.1] Of this equality we have:

[See statement k) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.14. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

[See Theorem 1.1] Of this equality we have:

[See statement l) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.15. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have. If the following

equality is hold then.

[See Theorem 1.1] Of this equality we have:

[See statement m) of the Lemma 2.2.]

Lemma is proved.

Lemma 2.16. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have. If the following

equality is hold then.

[See Theorem 1.1] Of this equality we have:

[See statement n) of the Lemma 2.2).]

Lemma is proved.

Lemma 2.17. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have. If the following

equality is hold then.

[See Theorem 1.1] Of this equality we have:

[See statement o) of the Lemma 2.2).]

Lemma is proved.

Lemma 2.18. Let and. If X is a finite set, then the number may be calculated by the formula

.

Proof. By definition of the given semilattice D we have. If the fol-

lowing equality is hold then.

[See Theorem 1.1] Of this equality we have:

.

[See statement p) of the Lemma 2.2).]

Lemma is proved.

Theorem 2.2. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. This Theorem immediately follows from the Theorem 2.1.

Theorem is proved.

Example 2.1. Let, , , , , , , , ,

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Diasamidze, Ya. and Makharadze, Sh. (2013) Complete Semigroups of Binary Relations. Kriter, Turkey.
[2]	Diasamidze, Ya. and Makharadze, Sh. (2010) Complete Semigroups of Binary Relations. Sputnik+, Moscow. (In Russian)
[3]	Diasamidze, Ya., Givradze, O. and Tavdgiridze, G. (2016) Idempotent Elements of the Semigroups Bx(D) Defined by Semilattices of the Class ∑3(X,8) When Z7≠Ø. Applied Mathematics, 7, 193-218. http://dx.doi.org/10.4236/am.2016.73019