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In the 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.

Let X be an arbitrary nonempty set, D be a X-semilattice of unions, i.e. a nonempty set of subsets of the set X that is closed with respect to the set-theoretic operations of unification of elements from D, f be an arbitrary mapping from X into D. To each such a mapping f there corresponds a binary relation

easy to prove that

which is called a complete semigroup of binary relations defined by a X-semilattice of unions D (see 2.1 p. 34 of [

By

By symbol

Definition 1.1. Let

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)

b)

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

then it is obvious that any binary relation

Note that for a quasinormal representation of a binary relation

a)

b)

Definition 1.4. We say that a nonempty element T is a nonlimiting element of the set

Definition 1.5. Let us assume that by the symbol

Further, let

Further, if Q ia a XI-subsemilattice of unions, then by the symbol

Theorem 1.1. A binary relation

Theorem 1.2. Let D be a complete X-semilattice of unions. The semigroup

Theorem 1.3. Let X be a finite set and

representation

a)

b)

c)

Theorem 1.4. Let D,

a) if

1)

2)

3) The equality

b) if

1)

2)

3) The equality

Lemma 1.1. Let

Lemma 1.2. Let

X in the set

Lemma 1.3. Let D by a complete X-semilattice of unions. If a binary relation

unit of that semigroup (see [

Theorem 1.5. Let

In the sequel these equalities will be called formal.

It is proved that if the elements of the semilattice D are represented in the form (1.1), then among the parameters

The number the basis sources we denote by symbol

It is proved that under the mapping

Theorem 1.6. Let X be a finite set;

where

we give complete classification all XI-subsemilattices of the semilatticeopf the class

we derive formulas by calculation the numbers of the semilattices of the given class.

In this subsection it is assumed that

By the symbol

The semilattice satisfying the conditions (2.1) is shown in

It is further assumed that

where

Lemma 2.1. Let

Proof. In this case we have:

where

The Lemma is proved.

Example 2.1. Let

The number obtained show that if, for instance

Let us define all subsemilattice of the semilattice D.

Lemma 2.2. Let

1)

(see diagram 1 of the

2)

(see diagram 2 of the

3)

(see diagram 3 of the

4)

(see diagram 4 of the

5)

(see diagram 5 of the

6)

(see diagram 6 of the

7)

(see diagram 7 of the

8)

(see diagram 8 of the

9)

(see diagram 9 of the

10)

(see diagram 10 of the

11)

(see diagram 11 of the

12)

(see diagram 12 of the

13)

(see diagram 13 of the

14)

(see diagram 14 of the

15)

(see diagram 15 of the

16)

(see diagram 16 of the

17)

(see diagram 17 of the

18)

(see diagram 18 of the

19)

(see diagram 19 of the

20)

(see diagram 20 of the

21)

(see diagram 21 of the

22)

(see diagram 22 of the

23)

(see diagram 23 of the

24)

(see diagram 24 of the

25)

(see diagram 25 of the

26)

(see diagram 26 of the

27)

(see diagram 27 of the

28)

(see diagram 28 of the

29)

(see diagram 29 of the

30)

Proof. It is easy to see that, the sets

The number subsets of the semilattise D, which contain two element is equal to

It is easy to see that, last five sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain tree element is equal to

It is easy to see that, last twenty sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain four element is equal to

is easy to see that, last 33 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain five element is equal to

is easy to see that, last 29 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain six element is equal to

is easy to see that, last 13 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain seven element is equal to

is easy to see that, last 3 sats are not subsemilattices of the semilattice D.

From the proven lemma it follows that diagrams shown in

Lemma 2.3. Let

Proof: Remark, that the all subsemilattices of semilattice D which has diagrams of form 17 - 30 are never XI-semilattices. For example we consider the semilatticesuchis defined by the diagram of the form 30 of the

Let

ping of the semilattice

Here, the elements

We have

Lemma is proved.

Lemma 2.4. Let

1)

(see diagram 1 of the

2)

(see diagram 2 of the

3)

(see diagram 3 of the

4)

(see diagram 4 of the

5)

(see diagram 5 of the

6)

(see diagram 6 of the

7)

(see diagram 7 of the

8)

(see diagram 8 of the

9)

(see diagram 9 of the

10)

(see diagram 10 of the

11)

(see diagram 11 of the

12)

(see diagram 12 of the

13)

(see diagram 13 of the

14)

(see diagram 14 of the

15)

(see diagram 15 of the

16)

Proof: The statements 1), 2), 3), 4), 5) immediately follows from the Theorems 11.6.1 in [

We denote the following semitattices

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

Theorem 2.1. Let

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

Proof. The statements 1), 2), 3), 4) and 5) immediately follows from the Corollary 13.1.1 in [

Lemma 2.6. 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. The statements 1), 2), 3), 4), 5) immediately follows from the Corollary 13.1.5 in [

13.1.5 in [

Theorem is proved.

Lemma 2.7. Let

Proof. By definition of the given semilattice D we have

If the following equalities are hold

then

(see Theorem 1.4). Of this equality we have:

(see statement a) of the Lemma 2.6).

Lemma 2.8. Let

Proof. By definitionof the given semilattice D we have

if

Then

(see Theorem 1.4). Of this equality we have:

(see statement b) of the Lemma 2.6).

Lemma is proved.

Lemma 2.9. Let

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement c) of the Lemma 2.6).

Lemma is proved.

Lemma 2.10. Let

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement d) of the Lemma 2.6).

Lemma is proved.

Lemma 2.11. Let

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement e) of the Lemma 2.6).

Lemma is proved.

Lemma 2.12. Let

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement f) of the Lemma 2.6).

Lemma is proved.

Lemma 2.13. Let

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement g) of the Lemma 2.6).

Lemma is proved.

Lemma 2.14. Let

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement h) of the Lemma 2.6).

Lemma is proved.

Lemma 2.15. Let

Proof. By definition of the given semilattice D we have

If the following equality is hold

(see Theorem 1.4). Of this equality we have:

(see statement i) of the Lemma 2.6).

Lemma is proved.

Lemma 2.16. Let

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement j) of the Lemma 2.6).

Lemma is proved.

Lemma 2.17. Let

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement k) of the Lemma 2.6).

Lemma is proved.

Lemma 2.18. Let

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement l) of the Lemma 2.6).

Lemma is proved.

Lemma 2.19. Let

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement m) of the Lemma 2.6).

Lemma is proved.

Lemma 2.20. Let

Proof. Bydefinitionof the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement n) of the Lemma 2.6).

Lemma is proved.

Lemma 2.21. Let

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement o) of the Lemma 2.6).

Lemma is proved.

Lemma 2.22. Let

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement p) of the Lemma 2.6).

Lemma is proved

Theorem 2.2. Let

Proof. This Theorem immediately follows from the Theorem 2.1.

Theorem is proved.

Example 2.1. Let

Then

Then we have that following equality are hold:

Giuli Tavdgiridze,Yasha Diasamidze,Omari Givradze, (2016) Idempotent Elements of the Semigroups B_{x}(D) Defined by Semilattices of the Class ∑_{3}(x,8) When Z_{7}‡ Ø. Applied Mathematics,07,193-218. doi: 10.4236/am.2016.73019