^{1}

^{*}

^{1}

^{*}

In this paper we take
subsemilattice of
*X*-semilattice of unions
*D* which satisfies the following conditions:<br/>
We will investigate the properties of regular elements of the complete semigroup of binary relations
*B*
* _{x}*(

*D*) satisfying

*V*(

*D*, а)=

*Q*. For the case where

*X*is a finite set we derive formulas by means of which we can calculate the numbers of regular elements and right units of the respective semigroup.

Let X be an arbitrary nonempty set and D be an X-semilattice of unions, which means 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. Let’s denote an arbitrary mapping from X into D by f. For each f there exists a binary relation

satisfies the condition

is not hard to prove that

An empty binary relation or an empty subset of the set X is denoted by

And

Definition 1.1. Let

Definition 1.2. An element

Definition 1.3. 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 1.4. 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

1)

2)

Definition 1.5. We say that a nonempty element T is a nonlimiting element of the set D' if

Definition 1.6. The one-to-one mapping

is fulfilled for each nonempty subset D_{1} of the semilattice D' (see [

Definition 1.7. Let

1)

2)

Lemma 1.1. Let _{j} such that

Lemma 1.2. Let D by a complete X-semilattice of unions. If a binary relation

unit of that semigroup (see [

Theorem 1.1. Let_{j}, for which

set X in the set D_{j} is equal to

Theorem 1.2. 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 (*), then among the parameters P_{i} _{i} _{i}

It is proved that under the mapping

Theorem 1.3. Let D be a complete X-semilattice of unions. The semigroup

Theorem 1.4. Let

1)

2)

Theorem 1.5. Let D be a finite X-semilattice of unions and

unions and for

1)

2)

3)

Let D be arbitrary X-semilattice of unions and

P_{7}, P_{6}, P_{5}, P_{4}, P_{3}, P_{2}, P_{1}, P_{0} are pairwise disjoint subsets of the set X and let

is a mapping from the semilattice Q into the family sets

Note that the elements P_{1}, P_{2}, P_{3}, P_{6} are basis sources, the element P_{0}, P_{4}, P_{5}, P_{7} is sources of completenes of the semilattice Q. Therefore

Theorem 2.1. Let

Proof. Let_{t} in Q. Then from the formal equalities (2) we get that

We have

Theorem is proved.

Lemma 2.1. Let

Proof. This Lemma follows directly from the formal equalities (2) of the semilattice Q.

Lemma is proved.

Lemma 2.2. Let

is the largest right unit of the semigroup

Proof. From preposition and from Theorem 2.1 we get that Q is XI-semilattice. To prove this Lemma we will use Lemma 1.2, lemma 2.1, and Theorem 1.3, from where we have that the following binary relation

is the largest right unit of the semigroup

Lemma is proved.

Lemma 2.3. Let

where

on some X-subsemilattice

Proof. It is easy to see, that the set

If we follow statement b) of the Theorem 1.5 we get that followings are true:

From the last conditions we have that following is true:

Moreover, the following conditions are true:

The elements

respectively. The proof of condition

Therefore the following conditions are hold:

Lemma is proved.

Definition 2.1. Assume that

Note that,

Theorem 2.2. Let

XI-semilattice Q and

Proof. Assume that

where

Father, let

the set

We are going to find properties of the maps

1)

and

2)

definition of the set

By suppose we have that

Therefore

3)

i.e.,

By suppose we have, that

Therefore

4)

By suppose we have, that

Therefore

5)

By suppose we have, that

Therefore

6)

the sets

Therefore for every binary relation

Father, let

are such mappings, which satisfying the conditions:

7)

8)

9)

10)

11)

12)

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

Father, let

and satisfying the conditions:

(By suppose

Therefore for every binary relation

By Theorem 1.1 the number of the mappings

(see Lemma 1.1). The number of ordered system

Theorem is proved.

Corollary 2.1. Let

Proof: This Corollary directly follows from the Theorem 2.2 and from the [2, 3 Theorem 6.3.7].

Corollary is proved.