Received 10 January 2015; accepted 28 January 2015; published 4 February 2015

ABSTRACT

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 S₁ (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.

Keywords:

Semilattice, Semigroup, Binary Relation

1. Introduction

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 on the set X that satisfies the condition

The set of all such is denoted by. It is easy to prove that is a semigroup with respect to the operation of multiplication of binary relations, which is called a complete semigroup of binary relations defined by an X-semilattice of unions D.

Recall that we denote by an empty binary relation or empty subset of the set X. The condition will be written in the form xαy. Further let, , , , , and. Then by symbols we denoted the following sets:

By symbol is denoted an exact lower bound of the set D' in the semilattice D.

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) for any;

b) for any nonempty element Z of the semilattice D.

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

Definition 3. Let, ,. A representation of a binary relation

of the form is called quasinormal.

Note that, if is a quasinormal representation of the binary relation, then the following conditions are true:

1);

2) for and.

Let denote the class of all complete X-semilattices of unions where every element is isomorphic to a fixed semilattice D.

The following Theorems are well know (see [1] and [3] ).

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 and, then

where (see Theorem 11.5.1 [1] ).

Theorem 5. Let D be a complete X-semilattice of unions. The semigroup possesses right unit iff D is an XI-semilattice of unions (see Theorem 6.1.3 [1] ).

Theorem 6. Let X be a finite set and be the set of all those elements T of the semilattice

which are nonlimiting elements of the set. A binary relation having a quasinormal

representation is an idempotent element of this semigroup iff

a) is complete XI-semilattice of unions;

b) for any;

c) for any nonlimiting element of the set (see Theorem 6.3.9 [1] ).

Theorem 7. Let D, , and I denote respectively the complete X-semilattice of unions, the set of all XI-subsemilatices of the semilattice D, the set of all right units of the semigroup and the set of all idempotents of the semigroup. Then for the sets and I the following statements are true:

1) if and then

a) for any elements and of the set that satisfy the condition;

b)

c) the equality is fulfilled for the finite set X.

2) if, then

a) for any elements and of the set that satisfy the condition;

b)

c) the equality is fulfilled for the finite set X (see Theorem 6.2.3 [1] ).

Corollary 1. Let and be some sets, where and. Then the number of all possible mappings of the set Y into any such subset of the set that can be calculated by the formula (see Corollary 1.18.1 [1] ).

2. Idempotent Elements of the Semigroups Defined by Semilattices of the Class

Let X and be respectively an arbitrary nonempty set and a class X-semilattices of unions, where each element is isomorphic to some X-semilattice of unions that satisfies the conditions:

(1)

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

Let be a family of sets, where P₀, P₁, P₂, P₃, P₄, P₅, P₆, P₇, P₈, P₉

are pairwise disjoint subsets of the set X and be a map-

ping of the semilattice D onto the family sets. Then for the formal equalities of the semilattice D we have a form:

(2)

Here the elements P₁, P₂, P₃, P₄, P₅, P₆, P₇, P₈ are basis sources, the elements P₀, P₆, P₉ are sources of completeness of the semilattice D. Therefore and (see [2] ).

Lemma 1. Let, and. If X is a finite set, then

.

Proof. In this case we have: m = 10, δ = 7. Notice that an X-semilattice given in Figure 1 has eight automorphims. By Theorem 1.1 it follows that

,

where and that

.

Example 8. Let Then:

.

Lemma 2. Let. Then the following sets are all proper subsemilattices of the semilattice:

1)

(see diagram 1 of the Figure 2);

2)

(see diagram 2 of the Figure 2);

3)

(see diagram 3 of the Figure 2);

4)

(see diagram 4 of the Figure 2);

5)

(see diagram 5 of the Figure 2);

6)

(see diagram 6 of the Figure 2);

7)

(see diagram 7 of the Figure 2);

8)

(see diagram 8 of the Figure 2);

9)

(see diagram 9 of the Figure 2);

10)

(see diagram 10 of the Figure 3);

11)

(see diagram 11 of the Figure 2);

12)

(see diagram 12 of the Figure 2);

13)

(see diagram 13 of the Figure 2);

14)

(see diagram 14 of the Figure 2);

15)

(see diagram 15 of the Figure 2);

16)

(see diagram 16 of the Figure 2);

17)

(see diagram 17 of the Figure 2);

18)

(see diagram 18 of the Figure 2);

19)

(see diagram 19 of the Figure 2);

20)

(see diagram 20 of the Figure 2);

21)

(see diagram 21 of the Figure 2);

22)

(see diagram 22 of the Figure 2);

23)

(see diagram 23 of the Figure 2);

24)

(see diagram 24 of the Figure 2);

25)

(see diagram 25 of the Figure 2);

26)

(see diagram 26 of the Figure 2);

27)

(see diagram 27 of the Figure 2);

28)

(see diagram 28 of the Figure 2);

29)

(see diagram 29 of the Figure 2);

30)

(see diagram 30 of the Figure 2);

31)

(see diagram 31 of the Figure 2);

32)

(see diagram 32 of the Figure 2);

33)

(see diagram 33 of the Figure 2);

34)

(see diagram 34 of the Figure 2);

35)

(see diagram 35 of the Figure 2);

36)

(see diagram 36 of the Figure 2);

37)

(see diagram 37 of the Figure 2);

38)

(see diagram 38 of the Figure 2);

39)

(see diagram 39 of the Figure 2);

40)

(see diagram 40 of the Figure 2);

41)

(see diagram 41 of the Figure 2);

42)

(see diagram 42 of the Figure 2);

43)

(see diagram 43 of the Figure 2);

44)

(see diagram 44 of the Figure 2);

45)

(see diagram 45 of the Figure 2);

46)

(see diagram 46 of the Figure 2);

47)

(see diagram 47 of the Figure 2);

48)

(see diagram 48 of the Figure 2);

49)

(see diagram 49 of the Figure 2);

50)

(see diagram 50 of the Figure 2);

51)

(see diagram 51 of the Figure 2);

52)

(see diagram 52 of the Figure 2);

Diagrams of subsemilattices of the semilattice D.

Lemma 3. Let. Then the following sets are all XI-subsemi-lattices of the given semilattice D:

1)

(see diagram 1 of the Figure 2);

2)

(see diagram 2 of the Figure 2);

3)

(see diagram 3 of the Figure 2);

4)

(see diagram 4 of the Figure 2);

5)

(see diagram 5 of the Figure 2);

6)

(see diagram 6 of the Figure 2);

7)

(see diagram 7 of the Figure 2);

8)

(see diagram 8 of the Figure 2);

Proof. It is well know (see [1] ), that the semilattices 1 to 8, which are given by lemma 2 are always XI-semi- lattices. The semilattices 9 and 10 which are given by Lemma 2

(see diagram 9 of the Figure 2);

(see diagram 10 of the Figure 2);

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 Figure 3);

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 Figure 2, is not an XI-semilattice (see Figure 4). Indeed, let and

be a family of sets, where are pairwise disjoint subsets of the set X. Let

be a mapping of the semilattice Q onto the family of sets. Then for the formal equalities of the semilattice Q we have a form:

(3)

Here the elements are basis sources, the elements, , are sources of completeness of the semilattice D. Therefore and (see [2] ). Then of the formal equalities we have:

We have, that and for any. But elements T₇, T₆, T₅, T₄, T₃, T₂, T₁, T₀ are not union of some elements of the set. Therefore from the Definition 1 it follows that Q is not an XI-semilattice of unions. Statements 12 to 51 can be proved analogously.

We denoted the following semitattices by symbols:

a), where (see diagram 1 of the Figure 5);

b), where and (see diagram 2 of the Figure 5);

c), where and (see diagram 3 of the Figure 5);

d), where and (see diagram 4 of the Figure 5);

e) where, , , , , (see dia- gram 5 of the Figure 5);

f), where, , , , (see diagram 6 of the Figure 5);

g), where, , , , , (see diagram 7 of the Figure 5);

h), where, , , , (see diagram 8 of the Figure 5);

Note that the semilattices in Figure 5 are all XI-semilattices (see [1] and Lemma 1.2.3).

Definition 9. Let us assume that by the symbol denote a set of all XI-subsemilatices of X-semila- tices of unions D that every element of this set contains an empty set if or denotes a set of all XI-sub- semilatices of D.

Further, let and. It is assumed that iff there exists some complete isomorphism between the semilatices and. One can easily verify that the binary relation is an equivalence relation on the set.

By the symbol denote the -equivalence class of the set, where every element is iso- morphic to the X-semilattice.

Let D' be an XI-subsemilattice of the semilattice D. By we denoted the set of all right units of the semigroup, and

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 [1] .

Theorem 10. Let and. Binary relation is an idempotent relation of the semmigroup iff binary relation satisfies only one conditions of the following conditions:

a), where;

b), where, , , and satisfies the conditions: T,;

c), where, , , and sa-

tisfies the conditions:, , ,;

d), where, , , , ,

, and satisfies the conditions:, , , , ,;

e), where, , , ,

, , and satisfies the conditions:, , ,;

f), where, ,

, , and satisfies the conditions:, , , ,;

g), where, , ,

, , , and satisfies the conditions:, , , , , ,;

h), where,

, , , , and satisfies the condi- tions:, , , ,.

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

,

where, then it is easy to see, that the set is a generating set of the semilattice. Then the following equalities hold

By statement a of the Theorem 6.2.1 (see [1] ) we have:

.

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

We have the elements Z₆, T, T' are nonlimiting elements of the sets, , respectively.

By statement b of the Theorem 6.2.1 [1] it follows, that the conditions, , hold. Therefore, the statement g is proved. Rest of statements can be proved analogously.

Lemma 5. Let and. If X is a finite set, then the number may be calculated by the formula.

Lemma 6. Let and. If X is a finite set, then the number may be calculated by formula

Lemma 7. Let and. If X is a finite set, then the number may be calculated by formula

Lemma 8. Let and. If X is a finite set, then the number may be calculated by formula

Lemma 9. Let and. If X is a finite set, then the number may be calculated by formula

Lemma 10. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 11. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 12. Let and. If X is a finite set, then the number may be calcu- lated by formula

Figure 6 shows all XI-subsemilattices with six elements.

Theorem 11. Let,. If X is a finite set and is a set of all idempotent elements of the semigroup. Then.

Example 12. Let,

Then, , , , , , , , and.

We have. Where, , , , , , , ,.

3. Results

Lemma 13. Let and. Then the following sets exhaust all subsemilattices of the semilattice which contains the empty set:

1)

(see diagram 1 of the Figure 2);

2)

(see diagram 2 of the Figure 2);

3)

(see diagram 3 of the Figure 2);

4)

(see diagram 4 of the Figure 2);

5)

(see diagram 5 of the Figure 2);

6)

(see diagram 6 of the Figure 2);

7)

(see diagram 7 of the Figure 2);

8)

(see diagram 8 of the Figure 2);

Theorem 13. Let, and. Binary relation is an idempotent relation of the semmigroup iff binary relation satisfies only one conditions of the following conditions:

a);

b), where, , , and satisfies the conditions:;

c), where, , , and satisfies the

conditions:, ,;

d), where, , , ,

, and satisfies the conditions:, , , , ;

e), where, , , ,

and satisfies the conditions:, , ,;

f), where, , ,

, and satisfies the conditions:, , , ,.

g), where, , ,

, , , and satisfies the conditions:, , , , ,;

h), where,

, , , , and satisfies the conditions:, , , , ,;

Lemma 14. Let and. If X is a finite set, then.

Lemma 15. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 16. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 17. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 18. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 19. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 20. Let and. If X is a finite set, then the number may be calcu- lated by formula

Lemma 21. Let and. If X is a finite set, then the number may be calcu- lated by formula

Theorem 14. Let,. If X is a finite set and is a set of all idempotent elements of

the semigroup, then.

Example 15. Let,

.

Then, , , , , , , , and.

We have. Where, , , , , , , ,.

It was seen in ([4] , Theorem 2) that if and are regular elements of then is an XI-subsemilattice of D. Therefore is regular elements of. That is the set of all regular elements of is a subsemigroup of.

References