^{1}

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In this paper we give a full description of idempotent elements of the semigroup
*B*
* _{X}* (

*D*), which are defined by semilattices of the class ∑

_{1}(

*X*, 10). For the case where

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

Let X be an arbitrary nonempty set, D be an X-semilattice of unions, i.e. such 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 of the set X in the set D. To each such a mapping f we put into correspondence a binary relation

The set of all such

Recall that we denote by

By symbol

Definition 1. We say that the complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:

a)

b)

Definition 2. We say that a nonempty element T is a nonlimiting element of the set D' if

Definition 3. Let

Note that, if

1)

2)

Let

The following Theorems are well know (see [

Theorem 4. Let X be a finite set; δ and q be respectively the number of basic sources and the number of all automorphisms of the semilattice D. If

where

Theorem 5. Let D be a complete X-semilattice of unions. The semigroup

Theorem 6. Let X be a finite set and

representation

a)

b)

c)

Theorem 7. Let D,

1) if

a)

b)

c) the equality

2) if

a)

b)

c) the equality

Corollary 1. Let

Let X and

An X-semilattice that satisfies conditions (1) is shown in

Let _{0}, P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}, P_{8}, P_{9}

are pairwise disjoint subsets of the set X and

ping of the semilattice D onto the family sets

Here the elements P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}, P_{8} are basis sources, the elements P_{0}, P_{6}, P_{9} are sources of completeness of the semilattice D. Therefore

Lemma 1. Let

Proof. In this case we have: m = 10, δ = 7. Notice that an X-semilattice given in

where

Example 8. Let

Lemma 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)

(see diagram 30 of the

31)

(see diagram 31 of the

32)

(see diagram 32 of the

33)

(see diagram 33 of the

34)

(see diagram 34 of the

35)

(see diagram 35 of the

36)

(see diagram 36 of the

37)

(see diagram 37 of the

38)

(see diagram 38 of the

39)

(see diagram 39 of the

40)

(see diagram 40 of the

41)

(see diagram 41 of the

42)

(see diagram 42 of the

43)

(see diagram 43 of the

44)

(see diagram 44 of the

45)

(see diagram 45 of the

46)

(see diagram 46 of the

47)

(see diagram 47 of the

48)

(see diagram 48 of the

49)

(see diagram 49 of the

50)

(see diagram 50 of the

51)

(see diagram 51 of the

52)

(see diagram 52 of the

Diagrams of subsemilattices of the semilattice D.

Lemma 3. 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

Proof. It is well know (see [

(see diagram 9 of the

(see diagram 10 of the

are XI-semilattices iff the intersection of minimal elements of the given semilattices is empty set. From the formal equalities (1) of the given semilattice D we have

From the equalities given above it follows that the semilattices 9 and 10 are not XI-semilattices.

The semilattices 11

(see diagram 1-8 of the

are not XI-semilattice since we have the following inequalities

The semilattices 12 to 52 are never XI-semilattices. We prove that the semilattice, diagram 52 of the

be a family of sets, where

be a mapping of the semilattice Q onto the family of sets

Here the elements

We have, that _{7}, T_{6}, T_{5}, T_{4}, T_{3}, T_{2}, T_{1}, T_{0} are not union of some elements of the set

We denoted the following semitattices by symbols:

a)

b)

c)

d)

e)

f)

g)

h)

Note that the semilattices in

Definition 9. Let us assume that by the symbol

Further, let

By the symbol

Let D' be an XI-subsemilattice of the semilattice D. By

where

Lemma 4. If X is a finite set, then the following equalities hold

a)

b)

c)

d)

e)

f)

g)

h)

Proof. This lemma immediately follows from Theorem 13.1.2, 13.3.2, and 13.7.2 of the [

Theorem 10. Let

a)

b)

c)

tisfies the conditions:

d)

e)

f)

g)

h)

Proof. By Lemma 3 we know that 1 to 8 are an XI-semilattices. We prove only statement g. Indeed, if

where

By statement a of the Theorem 6.2.1 (see [

Further, one can see, that the equalities are true:

We have the elements Z_{6}, T, T' are nonlimiting elements of the sets

By statement b of the Theorem 6.2.1 [

Lemma 5. Let

Lemma 6. Let

Lemma 7. Let

Lemma 8. Let

Lemma 9. Let

Lemma 10. Let

Lemma 11. Let

Lemma 12. Let

Theorem 11. Let

Example 12. Let

Then

We have

Lemma 13. 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

Theorem 13. Let

a)

b)

c)

conditions:

d)

e)

f)

g)

h)

Lemma 14. Let

Lemma 15. Let

Lemma 16. Let

Lemma 17. Let

Lemma 18. Let

Lemma 19. Let

Lemma 20. Let

Lemma 21. Let

Theorem 14. Let

the semigroup

Example 15. Let

Then

We have

It was seen in ([