Regular Elements of the Semigroup BX ( D ) Defined by Semilattices of the Class Σ 2 ( X , 8 ) and Their Calculation Formulas

The paper gives description of regular elements of the semigroup BX(D) which are defined by semilattices of the class Σ2(X, 8), for which intersection the minimal elements is not empty. When X is a finite set, the formulas are derived, by means of which the number of regular elements of the semigroup is calculated. In this case the set of all regular elements is a subsemigroup of the semigroup BX(D) which is defined by semilattices of the class Σ2(X, 8).

The lemma is proved.
The lemma is proved.
, , , , , , , ,8 has a quasinormal representation of the form to be given below is a regular element of this semigroup iff there exist a complete α -isomorphism ϕ of the semilattice ( ) , V D α on some subsemilattice D' of the semilattice D that satisfies at least one of the following conditions: 1) ′ ∉ ∅ and satisfies the conditions: ′′ ∉ ∅ and satis- fies the conditions: ∉ ∅ and satisfies the conditions: ′′ ∉ ∅ and satisfies the conditions: ∉ ∅ and satisfies the conditions ∉ ∅ and satisfies the conditions Proof.In this case, when 7 6 Z Z ∩ ≠ ∅ , from the Lemma 1.2 it follows that diagrams 1-8 given in Figure 1 exhibit all diagrams of XI-subsemilattices of the semilattices D, a quasinormal representation of regular elements of the semigroup ( ) X B D , which are defined by these XI-semilattices, may have one of the forms listed above.Then the validity of the statements 1)-4) immediately follows from the Theorem 13.1.1 in [1], Theorem 13.1.1 in [2], the statements 5)-7) immediately follows from the Theorem 13.3.1 in [1], Theorem 13.3.1 in [2] and the statement 8) immediately follows from the Theorem 13.7.1 in [1], Theorem 13.7.1 in [2].
The theorem is proved.
Proof.According to the definition of the semilattice D we have From this and by the statement 1) of Lemma 1.1 we obtain ( ) The lemma is proved.
Of this we have that 3) is hold.Of this follows that ( ) ( ) is fulfilled.Now of the equalities (2.2) and (2.4) follows the following equality The lemma is proved.
( ) , then by statement 2) of the Lemma 1.1 and by Lemma 2.2 we obtain the validity of Lemma 2.3.
The lemma is proved.
So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: ≠ ∅ contradic- tion of the condition that representation of binary relation α is quazinormal.So, the equality ( ) ( ) = ∅ is true.The similar way we can show that the following equalities are hold: , , , , . So, the inclusion ( ) ( ) ( ) , then the conditions (3.4) and (3.5) are fulfilled, i.e.
The lemma is proved.
Of the last condition we have by assumption.Of the other hand, if the conditions of (3.9) are hold, then also hold the conditions of (3.8), i.e.
, , , , , , , ,8 If X is a finite set, then the following equalities are hold: ) and a quasinormal representation of a regular binary relation α has the form is a mapping of the set X in the semilattice D satisfying the conditions ( ) is empty set, and ( ) ( ) ( ) We are going to find properties of the maps 0 . Then by the properties (3.10) we have 1 1 by definition of the sets T Y α and T Y α ′ .Therefore by definition of the semilattice D. Therefore ( ) Then by properties (3.10) we have We have contradict of the equality t T α ′′ ′′ = .Therefore ( ) . Then by definition quasinormal representation binary relation α and by property (3.10) we ′ and 0 Y α .There- fore exist ordered system ( ) It is obvious that for disjoint binary relations exist disjoint ordered systems.Now, let are such mappings, which satisfying the conditions: 5) ( ) ( ) ( )  .Now we define a map f of a set X in the semilattice D, which satisfies the condition: and satisfying the conditions: ( ) , by lemma 2.5 we have that and ordered system ( ) exist one to one mapping.
By ([1], Theorem 1.18.2) the number of the mappings 0 1 2 3 , , , ) Note that the number ) ( ) does not depend on choice of chains

, , T T T D
′ ′′ ∈ of the semilattice D. Sins the number of such different chains of the semilattice D is equal to 18, for arbitrary , ,

T T T D ′ ′′ ∈
where T T T ′ ′′ ⊂ ⊂ , the number of regular elements of the set ) ( ) .
Note the number ( ) ( ) ( ) does not depend on choice of chains

, , T T T D
′ ′′ ∈ of the semilattice D. Since the number of such different chains of the semilattice D is equal to 18, for arbitrary , ,
So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: and a quasinormal representation of a regular binary relation α has the form Then by statement 4) of the Theorem 2.1, we have ≠ ∅ contradiction of the condition that representation of binary relation α is quazi- normal.So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: , By equalities (4.1) and (4.2) follows, that It is easy to see ( ) , , , , , , , .
Then from Theorem 1.1 we obtain ( ) ( ) . Then quasinormal representation binary relation α of the semigroup ( ) where , , ′′ ′ ≠ ∅ and by statement 5) of the Theorem 2.1 sa- tisfies the following conditions: Of this we have that the inclusions ( ) ( ) ( ) are fulfilled.Therefore, of the equality (5.1) follows, that ( ) ( ) (5.2) Now we show that the following equalities are hold: , ′′ ′ ≠ ∅ and by statement 5) of the Theorem 2.1 satisfies the following conditions: Of this conditions follows that But the inequality ( ) ≠ ∅ contradiction of the condition that representation of binary relation α is quazinormal.So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: ′′ ′ ≠ ∅ and by statement 5) of the Theorem 2.1 satisfies the following conditions: 2 ≠ ∅ contradiction of the condition that representation of binary relation α is qua- zinormal.So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: i.e., ( ) ( ) . So, the inclusion ( ) ( ) ( ) ( ) ( ) Of the other hand, if ( ) ( ) , then the conditions (5.4) and (5.5) are fulfilled, i.e., ( ) ( ) ( ) ( ) ( ) . Therefore, the equality ( ) ( ) ( ) ( ) ( ) The similar way we can show that the following equalities are hold: Now by equalities (5.2) and (5.3) we obtain the validity of Lemma 5.1.
The lemma is proved.
are arbitrary elements of the set { } Of the last condition we have , , , by supposition.Of the other hand, if the conditions of (5.7) are hold, then, also hold the conditions of (5.6) i.e.
Let f α is a mapping of the set X in the semilattice D satisfying the conditions ( ) for all t X ∈ .f 0α , f 1α , f 2α and f 3α are the restrictions of the mapping f α on the sets We are going to find properties of the maps f 0α , f 1α , f 2α and f 3α .1) t Z Z′ ∈ ∩ .Then by the properties (5.8) we have ( ) ( ) . Then by the properties (5.8) we have \ are such mappings, which satisfying the conditions: 5) ( ) . Now we define a map f of a set X in the semilattice D, which satisfies the condition: . Then binary relation β can be representation by form and satisfying the conditions: , , , .
), i.e., by Lemma 2.10 we have that and ordered system ( ) exist one to one mapping.
By ([1], Theorem 1.18.2) the number of the mappings 0 Note that the number For this we consider the following case.a) Let ( ) ( ) ≠ ∅ contradiction of the condition that representation of binary relation α is quazinormal.So, the equality ( ) ( ) The similar way we can show that the following equality is hold: ( ) ( ) and a quasinormal representation of a regular binary relation α has the form ( ) ( ) ( ) ( ) ( ) the condition that representation of binary relation α is quazinormal.So, the equality ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: ( ) ( ) ( ) ( ) ( ) ( ) By equalities (6.1) and (6.2) follows that ( ) ( ) ( ) ( ) ( ) It is easy to see .
Proof.First we show that the following equalities are hold: For this we consider the following case.a) Let ( ) ( ) The similar way we can show that the following equality is hold: ( ) ( ) = ∅ is hold.The similar way we can show that the following equalities are hold: Proof.First we show that the following equalities are hold: ( ) ( )  So, the equality ( ) ( ) = ∅ is hold.By equalities (8.1) and (8.2) follows that ( ) ( ) ( ) It is easy to see ( ) The lemma is proved.Let X be a finite set and 7 6 Z Z ∩ ≠ ∅ and us assume that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) exists an element β such that α β α α =   element Z of D (see ([1], Definition 1.14.2),([2], Definition 1.14.2)).

Theorem 1 . 1 .
Let R be the set of all regular elements of the semigroup

4 \ 1
Z Z ≠ ∅ and T Z ∩ = ∅ (see diagram 14 in Figure set of all XI-subsemilattices of the semilattice D isomorphic to i Q .Assume that and denote by the symbol ( ) R D′ the set of all regular elements α of the semigroup ( ) X B D′ , for which the semilattices ( ) , V D α and i Q are mutually α isomorphic and

2 )
Now let binary relation α of the semigroup ( ) X B D satisfying the condition 2) of the Theorem 2.1.In this case we have ⊂ .By definition of the semilattice D follows that

1 T
Then by the properties (3.10) we have the condition 3) of the Theorem 2.1, then

4 )
then by statement 3) of the Lemma 1.1, by Lemma 3.1 and by Lemma 3.3 we obtain the validity of Lemma 3.4.The lemma is proved.Now let binary relation α of the semigroup the condition 4) of the Theorem 2.1, then a quasinormal representation of a regular binary relation α has the form

(
the last equalities and by statement 4) of the Lemma 1.1 we obtain the validity of Lemma 4.1.The lemma is proved.5) Now let binary relation of the semigroup ( ) X B D satisfying the condition 5) of the Theorem 2.1.In this case we have ∅ .By definition of the semilattice D follows that

1 .
Let X be a finite set, the condition 5) of the Theorem 2.1, then , , , , , , D D D D D D D D D D ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ , where D D ′ ′′ ≠ , Z Y ⊇ and Z Y ′ ′ ⊇ .Then the following equalities are hold: by statement 6) of the Lemma 1.1 we obtain validity of Lemma 6.1.The lemma is proved.7)Letbinary relation α of the semigroup ( ) X B D satisfying the condition 7) of the Theorem 2.1.In this quasinormal representation of a regular binary relation α has the form - diction of the condition that representation of binary relation α is quazinormal.So, the equality ( ) ( ) quasinormal representation of a regular binary relation α has the form contradiction of the condition that representation of binary relation α is quazinor- mal.So, the equality ( ) ( ) quasinormal representation of a regular binary relation α has the form contradiction of the condition that representation of binary relation α is quazinormal.
by statement 8) of the Lemma 1.1 we obtain validity of Lemma 8.1.

1) Lemm 2.1. Let
By definition of the semilattice D we have and D Z′′ ⊇ .Of this and by the conditions T is contradiction of the equality t T . Since the number of such different elements of the semilattice D are equal to 7, the number of regular elements of the set