Received 21 June 2015; accepted 22 February 2016; published 25 February 2016

ABSTRACT

In the paper, complete semigroup binary relation is defined by semilattices of the class. We give a full description of idempotent elements of given semigroup. For the case where X is a finite set and, we derive formulas by calculating the numbers of idempotent elements of the respective semigroup.

Keywords:

Semilattice, Semigroup, Binary Relation, Idempotent Element

1. Introduction

Let X be an arbitrary nonempty set, D be a X-semilattice of unions, i.e. 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 from X into D. To each such a mapping f there corresponds 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 a X-semilattice of unions D (see 2.1 p. 34 of [1] ).

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

By symbol we mean an exact lower bound of the set in the semilattice D.

Definition 1.1. Let. If or for any, then is called an idem­potent element or called right unit of the semigroup respectively.

Definition 1.2. We say that a 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 D. (see [2] , definition 1.14.2) or see ( [3] , definition 1.14.2)

Definition 1.3. Let D be an arbitrary complete X-semilattice of unions,. If

then it is obvious that any binary relation of a semigroup can always be written in the form

the sequel, such a representation of a binary relation will be called quasinormal.

Note that for a quasinormal representation of a binary relation, not all sets can be different from an empty set. But for this representtation the following conditions are always fulfilled:

a), for any and;

b). (see [2] , definition 1.11 or see [3] , definition 1.11)

Definition 1.4. We say that a nonempty element T is a nonlimiting element of the set if and a nonempty element T is a limiting element of the set if

Definition 1.5. Let us assume that by the symbol denote a set of all XI-subsemilatices of X-semilatices of unions D every element of this set contain an empty set if or denotes a set of all XI- subsemilatices of D.

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

Further, if Q ia a XI-subsemilattice of unions, then by the symbol we denote that -equivalence classes of the set for each element of which there exists a complete isomorphism on the semilattice Q. (see [2] , definition 6.3.5 or see [3] , definition 6.3.5)

Theorem 1.1. A binary relation is a right units of this semigroupiff is idempotent and (see [2] Theorem 4.1.3 or [3] Theorem 4.1.3 or [4] Theorem 2.1).

Theorem 1.2. Let D be a complete X-semilattice of unions. The semigroup possesses right unit iff D is an XI-semilattice of unions. (see [2] Theorem 6.1.3 or [3] Theorem 6.1.3 or [4] Theorem 2.6).

Theorem 1.3. 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 semigroupiff

a) is complete XI-semilattise of unions;

b) for any;

c) for any nonlimiting element of the set (see [2] Theorem 6.3.9 or [3] Theorem 6.3.9).

Theorem 1.4. 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 allidempotents of the semigroup. Then for the sets and I the following statements are true:

a) if and then

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

2);

3) The equality is fulfilled for the finite set X.

b) if, then

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

2);

3) The equality is fulfilled for the finite set X. (see [2] Theorem 6.2.3 or [3] Theorem 6.2.3 or [4] Theorem 6).

Lemma 1.1. Let and be some sets, where and. Then the number of all possible mappings of the set Y on any such subset of the set that can be calculated by the formula (see [2] Corollary 1.18.1 or [3] , Corollary 1.18.1 or [4] equality 6.9).

Lemma 1.2. Let, X, Y are tree nonempty set and. f be a mapping of the set X in the set which satisfies the conditions for some, Then number such mappings of the set

X in the set is equal to (see [2] Theorem 1.18.2 or [3] Theorem 1.18.2).

Lemma 1.3. Let D by a complete X-semilattice of unions. If a binary relation of the form is right unit of the semigroup, then is the greatest right

unit of that semigroup (see [2] , Lemma 12.1.2 or [3] , lemma 1.1.2).

Theorem 1.5. Let be some finite X-semilattice of unions and be the family of sets of pairwise nonintersecting subsets of the set X. If is a mapping of the semilattice D on the family of sets which satisfies the condition and for any and, then the following equali­ties are valid:

(1.1)

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 (1.1), then among the parameters there exist such parameters that cannot be empty sets. Such sets are called basis sources, whereas sets which can be empty sets too are called completeness sources.

The number the basis sources we denote by symbol.

It is proved that under the mapping the number of covering elements of the pre-image of a basis source is always equal to one, while under the mapping the number of covering elements of the pre-image of a completeness source either does not exist or is always greater than one (see [2] , 11.4 or [3] , 11.4 or [5] ).

Theorem 1.6. Let X be a finite set; and q are respectively the number of basic sources and the number of all automorphisms of the semilattice D. If and, then

,

where (see [2] Theorem 11.5.1 or [3] Theorem 11.5.4).

we give complete classification all XI-subsemilattices of the semilatticeopf the class

we derive formulas by calculation the numbers of the semilattices of the given class.

2. Results

In this subsection it is assumed that and we characterize the idemtpotent elements of the complete semigroup of binary relations which are defined by semilattices of the class.

By the symbol we denote the class of all X-semilattices of unions whose every element is isomor­phic to X-semilattice of the form, where

(2.1)

The semilattice satisfying the conditions (2.1) is shown in Figure 1.

It is further assumed that is some set of pairwise nonitersecting subsets of the set X, then formal equalities for the element of the considered semilattice have the form

(2.2)

where thus the elements are the sources of completeness, while the elements are the basis sources of the X-semilattice of unions D

Lemma 2.1. Let, and. If X be a finite set, then

.

Proof. In this case we have:, and, while the given semilattice D has only one identity automorphism. Of this and by Theorem 1.6 follows

,

where. Therefore the equality is true.

The Lemma is proved.

Example 2.1. Let then: and

The number obtained show that if, for instance, than the number of elements in the class of semigroups, where each element is defined by some semilattice of the class is equal to 327284760, while the number of elements in each semigroup belonging to this class is equal to 1072741824.

Let us define all subsemilattice of the semilattice D.

Lemma 2.2. Let. Then the following sets exhaust all 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 2);

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);

Proof. It is easy to see that, the sets are subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain two element is equal to. They are:

It is easy to see that, last five sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain tree element is equal to. They are:

It is easy to see that, last twenty sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain four element is equal to. They are:

is easy to see that, last 33 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain five element is equal to. They are:

is easy to see that, last 29 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain six element is equal to. They are:

is easy to see that, last 13 sats are not subsemilattices of the semilattice D.

The number subsets of the semilattise D, which contain seven element is equal to. They are:

is easy to see that, last 3 sats are not subsemilattices of the semilattice D.

From the proven lemma it follows that diagrams shown in Figure 2, exhaust all diagrams of subsemilattices of the semilattice D.

Lemma 2.3. Let and. Then any subsemilattices of the semilattice D having diagram 17 - 30 are never XI-semilattices.

Proof: Remark, that the all subsemilattices of semilattice D which has diagrams of form 17 - 30 are never XI-semilattices. For example we consider the semilatticesuchis defined by the diagram of the form 30 of the Figure 2.

Let and is a family sets, where

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

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

Here, the elements are basis sources, the element is sources of completenes of the semilattice. Therefore and Then of the formal equalities follows, that

We have and for all. But element is not union of some elements of the set. So, from the Definition 1.2 follows that semilattice which has diagram 41 of the Figure 3 never is XI-semilattice.

Lemma is proved.

Lemma 2.4. Let and. Then the following sets are all XI-subsemilattices of the given semilattice D:

1)

(see diagram 1 of the Figure 4);

2)

(see diagram 2 of the Figure 4);

3)

(see diagram 3 of the Figure 4);

4)

(see diagram 4 of the Figure 4);

5)

(see diagram 5 of the Figure 4);

6)

(see diagram 6 of the Figure 4);

7)

(see diagram 7 of the Figure 4);

8)

(see diagram 8 of the Figure 4);

9)

(see diagram 9 of the Figure 4);

10)

(see diagram 10 of the Figure 4);

11)

(see diagram 11 of the Figure 4);

12)

(see diagram 12 of the Figure 4);

13)

(see diagram 13 of the Figure 4);

14)

(see diagram 14 of the Figure 4);

15)

(see diagram 15 of the Figure 4);

16) (see diagram 16 of the Figure 4);

Proof: The statements 1), 2), 3), 4), 5) immediately follows from the Theorems 11.6.1 in [2] , 11.6.1 in [3] , the statements 6), 7), 8), 9), 10), 11) immediately follows from the Theorems 11.6.3in [2] , 11.6.3 in [3] ; the statement 12) immediately follows from the Theorems 11.7.2 in [2] ; the statement 13) immediately follows from the Theorema 2.1 in [4] , the statement 14) immediately follows from the lemma 2.1. in [5] , the statements 15) immediately follows from the Theorems 13.11.1 in [2] and the statement 16) immediately follows from the theorem 2.1. in [6] .

We denote the following semitattices as follows:

1), where

2) where,;

3) where,;

4) where,;

5) where;

6) where, , , ,;

7) where, , , ,;

8) where;

9) where, , ,;

10) where, , , ,;

11) where;

12) where, , , ,; , ,;

13) where, , , ,; ,;

14) where;

15)

16)

Theorem 2.1. Let, and. Binary relation is an idempotent relation of the semigroup iff binary relation satisfies only one conditions of the following conditions:

1), where;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16), where, and satisfies the conditions:, , , , , , , ,.

Proof. The statements 1), 2), 3), 4) and 5) immediately follows from the Corollary 13.1.1 in [2] , 13.1.1 in [3] , the statements 6) - 11) immediately follows from the Corollary 13.3.1 in [2] , 13.3.1 in [3] ; the statement 12) immediately follows from the Theorems 13.7.2 in [2] ; the statement 13) immediately follows from the corollary 2.1 in [4] , the statement 14) immediately follows from the lemma 2.1. in [5] , the statements 15) immediately follows from the Theorems 13.11.1 in [2] and the statement 16) immediately follows from the theorem 2.1. in [6] .

Lemma 2.6. If X be a finite set, then the following equalities are true:

a);

b);

c);

d);

e);

f);

g);

h)

i)

j);

k);

l);

m)

n)

o)

p)

Proof. The statements 1), 2), 3), 4), 5) immediately follows from the Corollary 13.1.5 in [2] ,

13.1.5 in [3] , the statements 6)-12) immediately follows from the Corollary 13.3.3 in [2] , 13.3.3 in [3] , the statement 13 immediately follows corollary 1.5 in [4] and corollary 6.3.6 in [3] , the statement 14 immediately follows from corollary 2.1 in [5] and corollary 6.3.6 in [3] , the statement 15) immediately follows from the Corollary 13.11.1 in [2] and the statement 16 immediately follows from the Corollary 2.1 in [6] .

Theorem is proved.

Lemma 2.7. Let and. If X is a finite set, then the number may be calculated by the formula.

Proof. By definition of the given semilattice D we have

.

If the following equalities are hold

,

then

(see Theorem 1.4). Of this equality we have:

(see statement a) of the Lemma 2.6).

Lemma 2.8. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definitionof the given semilattice D we have

if

Then

(see Theorem 1.4). Of this equality we have:

(see statement b) of the Lemma 2.6).

Lemma is proved.

Lemma 2.9. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement c) of the Lemma 2.6).

Lemma is proved.

Lemma 2.10. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement d) of the Lemma 2.6).

Lemma is proved.

Lemma 2.11. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

Then

(see Theorem 1.4). Of this equality we have:

(see statement e) of the Lemma 2.6).

Lemma is proved.

Lemma 2.12. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement f) of the Lemma 2.6).

Lemma is proved.

Lemma 2.13. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement g) of the Lemma 2.6).

Lemma is proved.

Lemma 2.14. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement h) of the Lemma 2.6).

Lemma is proved.

Lemma 2.15. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have.

If the following equality is hold then

(see Theorem 1.4). Of this equality we have:

(see statement i) of the Lemma 2.6).

Lemma is proved.

Lemma 2.16. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement j) of the Lemma 2.6).

Lemma is proved.

Lemma 2.17. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

If

(see Theorem 1.4). Of this equality we have:

(see statement k) of the Lemma 2.6).

Lemma is proved.

Lemma 2.18. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have

(see Theorem 1.4). Of this equality we have:

(see statement l) of the Lemma 2.6).

Lemma is proved.

Lemma 2.19. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have If the following equality is hold, then

(see Theorem 1.4). Of this equality we have:

(see statement m) of the Lemma 2.6).

Lemma is proved.

Lemma 2.20. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. Bydefinitionof the given semilattice D we have If the following equality is hold, then

(see Theorem 1.4). Of this equality we have:

(see statement n) of the Lemma 2.6).

Lemma is proved.

Lemma 2.21. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have If the following equality is hold, then

(see Theorem 1.4). Of this equality we have:

(see statement o) of the Lemma 2.6).

Lemma is proved.

Lemma 2.22. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. By definition of the given semilattice D we have. If the following equality is hold, then

(see Theorem 1.4). Of this equality we have:

(see statement p) of the Lemma 2.6).

Lemma is proved

Theorem 2.2. Let and. If X is a finite set, then the number may be calculated by the formula

Proof. This Theorem immediately follows from the Theorem 2.1.

Theorem is proved.

Example 2.1. Let,

.

Then, , , , , , , , and

Then we have that following equality are hold:

, , , , , , , , , , , , , ,

, ,.

Cite this paper

Giuli Tavdgiridze,Yasha Diasamidze,Omari Givradze, (2016) Idempotent Elements of the Semigroups B_x(D) Defined by Semilattices of the Class ∑₃(x,8) When Z₇‡ Ø. Applied Mathematics,07,193-218. doi: 10.4236/am.2016.73019

References