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In order to obtain some results in the theory of semigroups, the concept of regularity, introduced by J. V. Neumann for elements of rings, is useful. In this work, all regular elements of semigroup defined by semilattices of the class ∑<sub>1</sub>(X,10)-<i>I</i> are studied. When X has finitely many elements, we have given the number of regular elements.

Let D be a nonempty set of subsets of a given set X, closed under union. Such a set D is called a complete X-semilattice of unions. For any map f from X to D, we define a binary relation.

The set of all

All notations, symbol and required definitions used in this work can be found in [

Lemma 1. [

Theorem 1. [

of mappings from X to

Let X be a nonempty set, D a X-semilattice of union with the conditions (see

The class of X-semilattices where each element is isomorphic to D is denoted by

An element

Definition 1. The complete X-semilattice of unions is called an XI-semilattice of unions if

and

The following Lemma is well known (see [

Lemma 2. All semilattices in the form of the diagrams in

Definition 2. Let

for

Definition 3. [

a)

b)

The following subsemilattices are all XI-semilattices of the X-semilattices of unions D.

a)

b)

c)

d)

e)

f)

(see diagram 6 of the

g)

h)

For each

One can see that

Assume that

(see [

The following results have the key role in this study.

Theorem 2. Let

a)

b)

c) if X is a finite set, then

Lemma 3. Let

a)

b)

c)

d)

e)

f)

g)

h)

Proof. The propositions a), b), c) and d) immediately follow from ( [

Theorem 3. Let

of the semigroup

regular element of this semigroup iff there exist a complete a-isomorphism

a)

b)

c)

d)

e)

f)

g)

h)

Proof. In this case from Lemma 2 it follows that diagrams 1-8 given in

Lemma 4. Let

Proof. Let binary relation a of the semigroup

Now let binary relation a of the semigroup

It is easy to see that there is only one isomorphism from

then

Lemma 5. Let X be a finite set,

and

Proof. Let

From this equality and by statement b) of Lemma 3 it immediately follows that

Let binary relation a of the semigroup

It is easy to see

then

Lemma 6. Let X be a finite set,

and

where

Proof. Let

i.e.

Therefore the following equality

holds. Now, let

for the binary relation a we have

From the last condition it follows that

1)

2)

and

respectively, i.e.,

Therefore the equality

3)

and

respectively, i.e.,

Therefore the equality

for all

Now, by equality (4) and conditions 1), 2) and 3) it follows that the following equality is true

where

Lemma 7. Let

sentation of binary relation a of the semigroup

some

Proof. If

From the last condition we have

since

On the other hand, if the conditions of (7) holds, then (6) immediately follows, i.e.

Lemma 8. Let

Proof. Let

Further, let

1)

2)

By suppose we have that

Therefore

3)

By suppose we have that

Therefore

4)

We have seen that for every binary relation

Further, let

be such mappings that satisfy the conditions:

Now we define a map f from X to the semilattice D, which satisfies the condition:

Further, let

binary relation

and satisfies the conditions

(By suppose

Therefore for every binary relation

By Lemma 1 and by Theorem 1 the number of the mappings

Note that the number

Therefore we obtain

Lemma 9. Let X be a finite set,

Proof. The given Lemma immediately follows from Lemma 6 and from the Equalities (5).

Now let a binary relation a of the semigroup

of the

finition of the semilattice D it follows that

It is easy to see

then

Lemma 10. Let X be a finite set,

and

Proof. Let

From this it follows that

We consider the following cases.

1)

the equality

2)

contradicts the condition that representation of binary relation a is quasinormal. So, the equality

By conditions (10a) and (10b) it follows that

From the last equality we have that the given Lemma is true. □

Now let a binary relation a of the semigroup

It is easy to see

then

Lemma 11. Let X be a finite set,

and

where

Proof. Let

where

From this we have that the inclusions

are fulfilled. Therefore from the Equality (1) it follows that

Let

where

Then by statement e) of Theorem 3 we have

From this conditions it follows that

For

1)

respectively. But the inequalities

2)

follows that the inequalities

and

the equality

that

3) If

Then by definition of the semilattice D it follows that the inequalities

are true. But the inequalities

By similar way one can prove that

4)

and

respectively, i.e.,

Therefore, the equality

From the equalities

where

□

Lemma 12. Let

of the semigroup

Proof. If

From the last condition we have

since

On the other hand, if the conditions of (14) hold, then the conditions of (13) follow, i.e.

□

Lemma 13. Let

Proof. Let

for some

Now, let

We are going to find properties of the maps

(1)

since

(2)

By suppose we have that

Therefore

(3)

By suppose we have that

Therefore

(4)

Therefore for every binary relation

Further, let

be such mappings, which satisfy the conditions

Now we define a map f from X to the semilattice D, which satisfies the condition

Further, let

and satisfies the conditions

(By suppose

The number of the mappings

Note that the number

elements

Lemma 14. Let X be a finite set,

□

and

and

Proof. The given Lemma immediately follows from Lemma 11 and Lemma 13.

□Let binary relation a of the semigroup

It is easy to see

(see

Then

Lemma 15. Let X be a finite set,

Proof. Let

where

From this conditions it follows that

For

or

But the inequality

Now by Equalities (16) and by condition (17) it follows that

By statement f) of Lemma 3 the given Lemma is true.

□Now let binary relation a of the semigroup

It is easy to see

(see

Then

Lemma 16. Let X be a finite set,

Proof. Let

where

From this conditions it follows that

For

1)

or

But the inequalities

2)

or

But the inequalities

3)

or

But the inequalities

Now by conditions 1), 2) and 3) it follows that

By statement (g) of Lemma 3 the given Lemma is true.

□Let binary relation a of the semigroup

It is easy to see

(see

Then

Lemma 17. Let X be a finite set,

Proof. Let

binary relation a of the semigroup

where

From this conditions it follows that

For

the equality

Therefore we have

By statement h) of Lemma 3 the given Lemma is true. □

Let us assume that

Theorem 4. Let

Proof. This Theorem immediately follows from Theorem 2 and Theorem 3. □

Example 1. Let

Then

We have

Yasha Diasamidze,Nino Tsinaridze,Neşet Aydn,Neşet Aydn,Ali Erdoğan, (2016) Regular Elements of B_{x}(D) Defined by the Class ∑_{1}(X,10)-I. Applied Mathematics,07,867-893. doi: 10.4236/am.2016.79078