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

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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, , , , , , , , ,

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 B_{x}(D) Defined by Semilattices of the Class ∑_{3}(X,8) When Z_{7}≠Ø. Applied Mathematics, 7, 193-218.

http://dx.doi.org/10.4236/am.2016.73019