^{1}

^{*}

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As we know if
D is a complete
*X*-semilattice of unions then semigroup
*Bx*(
*D)*
* *possesses a right unit iff
*D* is an
*XI*-semilattice of unions. The investigation of those
*a*-idempotent and regular elements of semigroups
*B _{x}*(

*D*) requires an investigation of

*XI*-subsemilattices of semilattice

*D*for which

*V*(

*D*,

*a*)=

*Q*∈∑

_{2}(

*X*,8) . Because the semilattice

*Q*of the class ∑

_{2}(

*X*,8) are not always

*XI*-semilattices, there is a need of full description for those idempotent and regular elements when

*V*(

*D*,

*a*)=

*Q*. For the case where

*X*is a finite set we derive formulas by calculating the numbers of such regular elements and right units for which

*V*(

*D*,

*a*)=

*Q*.

In this paper we characterize the elements of the class

Let

Let

Let

Let

In general, a representation of a binary relation

Note that for a quasinormal representation of a binary relation

a)

b)

Let

In [

A complete

(a)

(b)

Let

Let

In the sequel these equalities will be called formal.

It is proved that if the elements of the semilattice

meters

It is proved that under the mapping

The one-to-one mapping

Is fulfilled for each nonempty subset

(a)

(b)

Lemma 1.1. Let

unit of that semigroup (see ( [

Theorem 1.1. Let

mapping of the set

mappings

Theorem 1.2. Let

(a)

(b)

(c)

Theorem 1.3. Let

Let

The semilattice

Let

is a mapping of the semilattice

here the elements

Theorem 2.1. Let

Proof. Let

We have

If

Of the other hand, if

The Theorem is proved.

Lemma 2.1. Let

Proof. The given Lemma immediately follows from the formal equalities (2) of the semilattice

The lemma is proved.

Lemma 2.2. Let

is the largest right unit of the semigroup

Proof. By preposition and from Theorem 2.1 follows that

is the largest right unit of the semigroup

The lemma is proved.

Lemma 2.3. Let

where

Proof. It is easy to see, that the set

By Statement b) of the Theorem 1.2 follows that the following conditions are true:

i.e., the inclusions

i.e.,

Therefore the following conditions are hold:

The lemma is proved.

Definition 2.1. Assume that

It is easy to see the number

Theorem 2.2. Let

Proof. Assume that

where

Let

We are going to find properties of the maps

1)

2)

3)

Preposition we have that

4)

Preposition we have that

5)

Therefore for every binary relation

Let

are such mappings, which satisfying the conditions:

Now we define a map

Now let

and satisfying the conditions:

From this and by Lemma 2.3 we have that

Therefore for every binary relation

By Theorem 1.1 the number of the mappings

(see ( [

The number of ordered system

(see ( [

The theorem is proved.

Corollary 2.1. Let

Proof: This corollary immediately follows from Theorem 2.2 and from the ( [

The corollary is proved.

NinoTsinaridze,ShotaMakharadze, (2015) Regular Elements of the Complete Semigroups B_{X}(D) of Binary Relations of the Class∑_{2}(X,8). Applied Mathematics,06,447-455. doi: 10.4236/am.2015.63042