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The paper gives description of regular elements of the semigroup
B
_{ X }
* (*
*D*
*) which are defined by semilattices of the class Σ*
_{2}
* (*
*X*
*, 8), for which intersection the minimal elements is not empty. When*
* X*
* is a finite set, the formulas are derived, by means of which the number of regular elements of the semigroup is calculated. In this case the set of all regular elements is a subsemigroup of the semigroup *
*B*
_{ X }
* (*
*D*
*) which is defined by semilattices of the class Σ*
_{2}
* (*
*X*
*, 8).*

An element

Definition 1.1. We say that a complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:

1)

2)

Definition 1.2. The one-to-one mapping

set D_{1} of the semilattice D' (see ( [

Definition 1.3. Let

1)

2)

Theorem 1.1. Let

1)

2)

3) If X is a finite set, then

By the symbol

(see [

Now assume that

1)

2)

3)

4)

5)

(see diagram 5 in

6)

(see diagram 6 in

7)

(see diagram 7 in

8)

(see diagram 8 in

9)

(see diagram 9 in

10)

11)

12)

13)

14)

15)

16)

Denote by the symbol

Definition 1.4. Let the symbol

Let, further,

Let the symbol

4)

5)

6)

7)

8)

9)

10)

11)

12)

13)

14)

15)

16)

Proof. The statements 1)-4) immediately follows from the Theorem 13.1.2 in [

The lemma is proved.

Lemma 1.2. Let

1)

2)

(see diagram 2 of the

3)

(see diagram 3 of the

4)

(see diagram 4 of the

5)

6)

7)

8)

Proof. The statements 1)-4) immediately follows from the Theorems 11.6.1 in [

The lemma is proved.

Theorem 2.1. Let

1)

2)

3)

4)

5)

6)

7)

8)

Proof. In this case, when

The theorem is proved.

1) Lemm 2.1. Let

Proof. According to the definition of the semilattice D we have

Assume that

Then from Theorem 1.1 we obtain

From this and by the statement 1) of Lemma 1.1 we obtain

The lemma is proved.

2) Now let binary relation

If the equalities

Then from Theorem 1.1 we obtain:

Lemma 2.2. Let

Proof. Let

On the other hand,

i.e.,

Now, let

Of this we have that

Of the other hand if

that

is fulfilled. Now of the equalities (2.2) and (2.4) follows the following equality

The lemma is proved.

Lemma 2.3. Let

Proof: It is easy to see

The lemma is proved.

3) Let binary relation

Now if

Then from Theorem 1.1 we obtain:

Lemma 3.1. Let

Proof. Let

i.e.

Now we show that the following equalities are true:

For this we consider the following case.

a) If

It follows that

The similar way we can show that the following equalities are hold:

b) If

It follows that

The similar way we can show that the following equalities are hold:

c) If

It follows that

i.e.,

Of the other hand, if

and

The similar way we can show that the following equality is hold:

d) If

It follows that

i.e.,

Of the other hand, if

The similar way we can show that the following equalities are hold:

We have that all equalities of (3.3) are true. Now, by the equalities of (3.2) and (3.3) we obtain the validity of Lemma 3.1.

The lemma is proved.

Lemma 3.2. Let

Proof. If

Of the last condition we have

since

Of the other hand, if the conditions of (3.9) are hold, then also hold the conditions of (3.8), i.e.

The lemma is proved.

Lemma 3.3. Let

Proof. Let

Let

We are going to find properties of the maps

1)

2)

Preposition we have that

3)

Preposition we have that

4)

Therefore for every binary relation

Now, let

5)

6)

7)

8)

Now we define a map f of a set X in the semilattice D, which satisfies the condition:

Let

(By suppose

Therefore for every binary relation

By ( [

1,

Note that the number

Note that the number

The lemma is proved.

Lemma 3.4. Let X be a finite set,

Proof: It is easy to see

The lemma is proved.

4) Now let binary relation

Now if

Then from Theorem 1.1 we obtain

Lemma 4.1. Let

Proof. First we show that the following equalities are hold:

For this we consider the following case.

a) Let

Then by statement 4) of the Theorem 2.1, we have

It follows that

The similar way we can show that the following equalities are hold:

b) Let

Then by statement 4) of the Theorem 2.1, we have

It follows that

The similar way we can show that the following equalities are hold:

By equalities (4.1) and (4.2) follows, that

It is easy to see

The lemma is proved.

5) Now let binary relation

Now if

Then from Theorem 1.1 we obtain

Lemma 5.1. Let X be a finite set,

Proof. Let

where

Of this we have that the inclusions

Now we show that the following equalities are hold:

a) Let

has a form

Of this conditions follows that

But the inequality

The similar way we can show that the following equalities are hold:

b) Let

has a form

Of this conditions follows that

The similar way we can show that the following equalities are hold:

c) If

It follows that

i.e.,

Of the other hand, if

The similar way we can show that the following equalities are hold:

Now by equalities (5.2) and (5.3) we obtain the validity of Lemma 5.1.

The lemma is proved.

Lemma 5.2. Let

for some

Proof. If

Of the last condition we have

since

Of the other hand, if the conditions of (5.7) are hold, then, also hold the conditions of (5.6) i.e.

The lemma is proved.

Lemma 5.3. Let X be a finite set,

Proof. Let

for some

Let f_{α} is a mapping of the set X in the semilattice D satisfying the conditions _{0α}, f_{1α}, f_{2α} and f_{3α} are the restrictions of the mapping f_{α} on the sets

We are going to find properties of the maps f_{0α}, f_{1α}, f_{2α} and f_{3α}.

1)

2)

Preposition we have that

Therefore

3)

Preposition we have that

4)

Therefore for every binary relation

Now let

5)

6)

7)

8)

Now we define a map f of a set X in the semilattice D, which satisfies the condition:

Now let

and satisfying the conditions:

(By suppose

Therefore for every binary relation

By ( [

1,

Note that the number

elements

set

The lemma is proved.

Lemma 5.4. Let X be a finite set,

Proof. It is easy to see

The lemma is proved.

6) Let binary relation

If

Lemma 6.1. Let X be a finite set,

Proof. First we show that the following equalities are hold:

For this we consider the following case.

a) Let

The similar way we can show that the following equality is hold:

b) Let

The similar way we can show that the following equalities are hold:

By equalities (6.1) and (6.2) follows that

It is easy to see

The lemma is proved.

7) Let binary relation

If

Lemma 7.1. Let X be a finite set,

Proof. First we show that the following equalities are hold:

For this we consider the following case.

a) Let

The similar way we can show that the following equality is hold:

b) Let

It follows that

The similar way we can show that the following equalities are hold:

By equalities (7.1) and (7.2) follows that

It is easy to see

The lemma is proved.

8) Let binary relation

If

Lemma 8.1. Let X be a finite set,

Proof. First we show that the following equalities are hold:

Let

where

By equalities (8.1) and (8.2) follows that

It is easy to see

The lemma is proved.

Let X be a finite set and

Theorem 2.2. Let X is a finite set,

Proof. This Theorem immediately follows from the Theorem 2.1.

The theorem is proved.

I was seen in ( [

Theorem 2.3. Let

Proof. This Theorem immediately follows from the Theorem 2 in [

The theorem is proved.

NinoTsinaridze,ShotaMakharadze,GuladiFartenadze, (2015) Regular Elements of the Semigroup B_{ X } (D) Defined by Semilattices of the Class Σ_{2} (X, 8) and Their Calculation Formulas. Applied Mathematics,06,2257-2278. doi: 10.4236/am.2015.614199