Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ1(X,10)

In this paper we give a full description of idempotent elements of the semigroup BX (D), which are defined by semilattices of the class Σ1 (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.


Introduction
Let X be an arbitrary nonempty set, D be an 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 α f on the set X that satisfies the condition The set of all such α f (f : X → D) is denoted by B X (D).It is easy to prove that B X (D) 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.
We denote by ∅ an empty binary relation or empty subset of the set X.The condition (x, y) ∈ α will be written in the form xαy. Further let x, y ∈ X, Y ⊆ X, α ∈ B X (D), T ∈ D, ∅ = D ′ ⊆ D, D = ∪D = Y ∈D Y and t ∈ D. Then by symbols we denote the following sets: By symbol ∧ (D, D ′ ) we mean an exact lower bound of the set D ′ in the semilattice D.
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 then it is obvious that any binary relation α of a semigroup B X (D) can always be written in the form α = T ∈V [α] (Y α T × T ) .In the sequel, such a representation of a binary relation α will be called quasinormal.
Note that for a quasinormal representation of a binary relation α, not all sets Y α T (T ∈ V [α]) can be different from the empty set.But for this representation the following conditions are always fulfilled: Equality 6.9] Let Y = {y 1 , y 2 , . . ., y k } and D j = {T 1 , T 2 , . . ., T j } be sets, where k ≥ 1 and j ≥ 1.Then the number s(k, j) of all possible mappings of the set Y on any such subset of the set D ′ j such that T j ∈ D ′ j can be calculated by the formula s(k, j) = j k − (j − 1) k .Lemma 1.5.Let D j = {T 1 , T 2 , . . ., T j }, X and Y be three such sets, that ∅ = Y ⊆ X.If f is such mapping of the set X, in the set D j , for which f (y) = T j for some y ∈ Y , then the number s of all those mappings f of the set X in the set D j is Proof.Let f 1 be a mappings of the set X\Y in the set D j , then the number of all such mappings is equal to j |X\Y | .Now let f 2 be all mappings of the set Y in the set D j , for which f (y) = T j for some y ∈ Y , then by Lemma 1.4 the number of all such mappings is equal to |Y | .We define the mapping f of the set X in the set D j by It is clear that the mapping f satisfies all the conditions of the given Lemma.
Thus the number s of all such maps is equal to all number of the pair (f 1 , f 2 ).
X (Q) and I D denote respectively the complete X-semilattice of unions, the set of all XIsubsemilattices of the semilattice D, the set of all right units of the semigroup B X (Q) and the set of all idempotents of the semigroup B X (D).Then for the sets E (r) X (Q) and I D the following statements are true: X (Q) is fulfilled for the finite set X. Theorem 1.9.[3] Let D = D, Z 1 , Z 2 , . . ., Z n−1 be some finite X-semilattice of unions and C(D) = {P 0 , P 1 , . . ., P n−1 } be the family of sets of pairwise nonintersecting subsets of the set X.If ϕ is a mapping of the semilattice D on the family of sets C(D) which satisfies the condition ϕ( D) = P 0 and ϕ(Z i ) = P i for any i = 1, 2, . . ., n − 1 and DZ = D\ {T ∈ D | Z ⊆ T }, then the following equalities are valid: 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 = 1, 2, . . ., n − 1) there exists a parameter that cannot be empty sets for D. Such sets P i (0 < i ≤ n − 1) are called basis sources, whereas sets P i (0 ≤ j ≤ n − 1) which can be empty sets too are called completeness sources.
It is proved that under the mapping ϕ the number of covering elements of the pre-image of a basis source is always equal to one, while under the mapping ϕ the number of covering elements of the pre-image of a completeness source either does not exist or is always greater than one (see, [[3]]).Therefore, ϕ(T )∩Z ′ = ∅ if and only if Z ′ ∈ DT .Of this follows that the inclusion ϕ(T ) = P ⊆ Z ′ is true iff D t = D\ DT for all t ∈ ϕ(T ) = P .

Results
Definition 2.1.Let D be complete X−semilattice of unions and Z be some fixed element of D. We say that a complete X−semilattice of unions D is Z−elementary if D satisfies the following conditions: Proof.The given Lemma immediately follows from the Z−elementary X−semilattice of unions.
Lemma 2.3.Let D be Z−elementary X−semilattice of unions.If subsemilattice D ′ of the semilattice D is not a chain, then D ′ is Z−elementary X−semilattice of unions.
Proof.Let D be Z−elementary X−semilattice of unions.Suppose that the subsemilattice D ′ of the semilattice D is not a chain.
1) It is clear, that the length of any chain of the semilattice D ′ is finite since Z is a chain.3) Further, let T and T ′ be such elements of the set D ′ such that T \T ′ = ∅ and T ′ \T = ∅ (i.e., the elements T and T ′ of D are incomparable).Then T ,T ′ ∈ D, since D ′ ⊆ D. From this we have T ∪ T ′ = Z by the definition of the Z−elementary X−semilattice union D.
From the conditions (1), ( 2) and (3) it follows, that D ′ is Z−elementary X−semilattice of unions.Definition 2.4.Let C and C ′ be finite different chains of the set 2 X and Z ∈ C∩C ′ .We say that the chains C and C ′ are Z−compatible if C and C ′ satisfy the following conditions:  Proof.Let D be Z−elementary X−semilattice of unions and C, C ′ be two different maximal subchains of the X−semilattice of unions D. a) 1) So, we can assumed that T \T ′ = ∅ and T ′ \T = ∅.Further, let the element . But, this process must stop, since the chains C ′ is finite.So, there exists a natural number s, such that Therefore, the chains are compatible.Let any two maximal subchains of the X−semilattice of unions D be Z−compatible.Then we have: 1) By supposition D is not a chain.
2) Every subchain of the semilattice D is finite, since all Z−comparable chains are finite.
3) By the definition of Z−comparable chains, the set 4) If T and T ′ are any incomparable elements of D, then there exist two maximal different chains C and C ′ such that T ∈ C\C ′ , T ′ ∈ C ′ \C and the chains C and Let D be Z−elementary X−semilattice of unions and be an XI−subsemilattice of the Z−elementary X−semilattice of unions D which satisfy the following conditions: (i.e., the number different elements covered by the element Z is two).Note that the diagram of the semilattice D is shown in Fig. 2.3.
Further, let ∪ {P j1 , . . ., P jk } ∪ {P 1j , . . ., P sj } be a family of sets, where every two elements are pairwise disjoint subsets of the set X, Then for the formal equalities of the semilattice Q we have : Here the elements P 0 , P 1 , . . ., P j−1 , P j1 , . . ., P jk , P 1j , . . ., P sj , P j+k+s , P j+k+s+1 , . . ., P m−1 are basis sources, the elements P j , P m are sources of completeness of the semilattice Q.
Then from the formal equalities and by Lemma 1.10 we have: . ., T j , T j1 , . . ., T 1j , T j+k+s+1 , . . ., T m−1 , T m } and Q ′ be the semilattice of unions generated by the set Theorem 2.8.Let D be a Z−elementary X−semilattice of unions and Q be any XI−subsemilattice of the X−semilattice of unions D. Then for the XI−semilattice Q we have: Proof.Let D be a Z−elementary X−semilattice of unions and Q be any XI−subsemilattice of the X−semilattice of unions D. (a ′ ) If Z / ∈ Q then by the definition of Z−elementary X−semilattice of unions it follows that Q is a finite X−chain.Now, let Z ∈ Q and T be the unique element of the semilattice Q which is covered by the element Z; If T 1 and T 2 are any incomparable elements of the semilattice Q satisfying the conditions T 1 ⊂ T and T 2 ⊂ T , then by the definition Z−elementary X−semilattice of unions it follows that Z = T 1 ∪ T 2 ⊆ T .The inclusion Z ⊆ T contradicts the condition T ⊂ Z. So, we have T 1 and T 2 are comparable elements of the semilattice The statement (a ′ ) is proved.
(b ′ ) Let T, T ′ and T ′′ be different elements of the semilattice Q which are covered by the element Z in the semilattice Q.Then But the equality T ′′ = Z contradict, that T ′′ is an element which is covered by the element Z in the semilattice Q.
2) Now suppose that the intersection any two different elements which are covered by the element Z in the semilattice Q is not empty.
It is clear that T = ∅ and T = t∈T ∧ (Q, Q t ), since Q is XI−semilattice of unions.From Lemma 2.3 it follows that Q is Z−elementary X−semilattice of unions.By the definition of the Z−elementary X−semilattice of unions D immediately follows that D\ {Z} is unique generated set of the semilattice D. It follows that T = ∧ (Q, Q t ′ ) for some t ′ ∈ T .On the other hand, The inclusion T ⊂ T ′ ⊂ Z contradicts the assumption that element T is covered by the element Z in the semilattice Q.This contradiction shows that number the elements which are covered by the element Z of the XI−semilattice Q are less or equal two.
For the elements T and T ′ of the semilattice Q we consider two case.
3) If T and T ′ are minimal elements of the X−semilattice unions c ′ ) Now suppose that the elements T ′ and T ′′ covered by the element Z in the semilattice Q are not minimal elements of the semilattice Q, i.e., T ⊂ T ′ and T ⊂ T ′′ for some T ∈ Q.Then by Lemma 2.7 we have the element T covered by the elements T ′ and T ′′ in the semilattice Q.It is clear, that the set {T, T ′ , T ′′ , Z} is a X−subsemilattice of the semilattice Q.
and is a subchain of the chain D Z .Now, let Z ′′ be any element of the set Q ′′ .Then Z ′′ ∈ Q, Z ′′ / ∈ Q ′ ∪{T, T ′ , T ′′ , Z} and Z ′′ ⊂ T ′ or Z ′′ ⊂ T ′′ since T ′ and T ′′ are maximal elements of the set Q\ (Q ′ ∪ {Z}).If Z ′′ and T are incomparable elements of the semilattice Q then Z = T ∪ Z ′′ ⊆ T ′ by the definition of Z−elementary X−semilattice unions and by the conditions T ⊂ T ′ and Z ′′ ⊂ T ′ .But the inclusion Z ⊆ T ′ contradicts the conditions T ′ ⊂ Z. So, Z ′′ and T are comparable elements of the semilattice Q.From this follows that Z ′′ ⊂ T .
In the case Z ′′ ⊂ T ′′ we can similarly prove that Z ′′ ⊂ T .Further let Z ′′ 1 and Z ′′ 2 are any incomparable elements of the set Q ′′ satisfying the conditions Z ′′ 1 ⊂ T and Z ′′ 2 ⊂ T .Then by the definition Z−elementary X−semilattice of unions it follows that It is easy to see, that: for any p = 1, 2, . . ., m − 1 and q = 1, 2, . . ., m. Proof.Let Q = {T 0 , T 1 , . . ., T m } be a subsemilattice of the semilattice D such that T 0 ⊂ T 1 ⊂ • • • ⊂ T m Then the given Theorem immediately follows from the Theorem 1.6 and Corollary 3 of [5].(see, also, Corollary 13.1.2 of [1]) .
Theorem 2.18.Let Q = {T 0 , T 1 , . . ., T j , . . ., T m } (m ≥ 3) be a semilattice and j be a fixed natural number such that 0 ≤ j ≤ m − 3 and X (Q) is the set of all right units of the semigroup B X (Q), then the following statements are true: Theorem 2.19.If D is Z−elementary X−semilattice of unions, then the following equalities are true: Proof.The given Theorem immediately follows from the Theorem 1.8.
Proof.The given Theorem immediately follows from the Theorem 2.10, 2.14 and 2.17.
Theorem 2.21.Let D and I D be any Z−elementary X−semilattice of unions and all idempotent elements of the Z−elementary X−semilattice of unions respectively.Then the following conditions are true.
Proof.The given Theorem immediately follows from the Theorem 2.19.Proof.Let D be any Z−elementary X−semilattice of unions and ε • ε = ε.As is known (see [1]) the group G X (D, ε) is anti-isomorphic to the group of all complete automorphisms of the semilattice V (D, ε).In this case the number of all complete automorphisms of the semilattice V (D, ε) is not greater than two.Therefore the order of maximal subgroup G X (D, ε) is not greater than two. (2.1) The semilattice satisfying the conditions (2.1) is shown in Fig. 2.8.Let C(D) = {P 0 , P 1 , P 2 , P 3 , P 4 } be a family sets, where P 0 , P 1 , P 2 , P 3 , P 4 are pairwise disjoint subsets of the set X and is a mapping of the semilattice D onto the family sets C(D).Then for the formal equalities of the semilattice D we have a form: Here the elements P 1 , P 2 , P 3 , P 4 are basic sources; the elements P 0 are sources of completeness of the Z 1 −elementary X−semilattice of unions D. Further, we have Z 4 ∩ Z 3 = (P 0 ∪ P 3 ) ∩ (P 0 ∪ P 2 ∪ P 4 ) = P 0 . ( and , where (2. 2) The semilattice satisfying the conditions (2.2) is shown in Fig. 2.9.Let C(D) = {P 0 , P 1 , P 2 , P 3 , P 4 , P 5 } be a family sets, where P 0 , P 1 , P 2 , P 3 , P 4 , P 5 are pairwise disjoint subsets of the set X and be a mapping of the semilattice D onto the family sets C(D).Then for the formal equalities of the semilattice D we have a form: Here the elements P 1 , P 2 , P 3 , P 4 are basic sources; the elements P 0 , P 5 are sources of completeness of the Z 1 −elementary X−semilattice of unions D. Further, we have Z 5 ∩ Z 3 = (P 0 ∪ P 3 ) ∩ (P 0 ∪ P 2 ∪ P 4 ∪ P 5 ) = P 0 . ( and , where , where

Definition 1 . 3 .
: a) ∧ (D, D t ) ∈ D for any t ∈ D; b) Z = t∈Z ∧ (D, D t ) for any nonempty Z element of D. Let D be an arbitrary complete X−semilattice of unions, α ∈ B X (D) and

Lemma 1 .
10. Let D = D, Z 1 , Z 2 , . . ., Z n−1 and C(D) = {P 0 , P 1 , . . ., P n−1 } be the finite semilattice of unions and the family of sets of pairwise nonintersecting subsets of the set X; ϕ = D Z 1 Z 2 . . .Z n−1 P 0 P 1 P 2 . . .P n−1 is a mapping of the semilattice D on the family of sets C(D).If ϕ(T ) = P ∈ C(D) for some D = T ∈ D, then D t = D\ DT for all t ∈ P .Proof.Let t and Z ′ be any elements of the set P (P = P 0 ) and of the semilattice D respectively.Then the equality P ∩ Z ′ = ∅ (i.e., Z ′ / ∈ D t for any t ∈ P ) is valid if and only if T / ∈ DZ ′ (if T ∈ DZ ′ , then ϕ(T ) ⊆ Z ′ by definition of the formal equalities of the semilattice D).Since DZ ′ = D\ {T ′ ∈ D | Z ′ ⊆ T ′ } by definition of the set DZ ′ .Thus the condition T / ∈ DZ ′ hold iff T ∈ {T ′ ∈ D | Z ′ ⊆ T ′ }.So, Z ′ ⊆ T and Z ′ ∈ DT by definition of the set DT .

Fig. 2 . 2 Fig. 2 . 3 Definition 2 . 5 .
Fig. 2.2 Fig. 2.3 Definition 2.5.The chain C of a X−semilattice of unions D is called maximal, if the inclusion C ⊆ C ′ implies that C = C ′ for any chain C ′ of the X−semilattice of unions D. Theorem 2.6.Suppose X−semilattice of unions D is not a chain.Then D is Z−elementary X−semilattice of unions iff any two maximal subchain of the X−semilattice of unions D are Z−compatible.
By assumption the sets C Z , C ′ Z and D Z are maximal chains of the X−semilattice of union D with smallest element Z. Then D Z = C Z = C ′ Z since by definition of the Z−elementary X−semilattice of unions D the maximal subchains D Z , C, C ′ of the X−semilattice D, with the smallest element Z are by definition unique.b) Let T ∈ C\C ′ and T ′ ∈ C ′ \C.Then T ⊂ Z, T ′ ⊂ Z and T = T ′ .If T \T ′ = ∅ and T ′ \T = ∅ then T ∪ T ′ = Z by definition of the semilattice D. Therefore, the chains C and C ′ are Z−compatible.

Definition 2 .
9. Let C(D) denote the set all chains of the Z−elementary X−semilattice unions D. N (D) = {|C| | C ∈ C(D)}, h(D) be the largest natural number of the set N (D),

Theorem 2 .
20.Let D be Z−elementary X−semilattice of unions and α ∈ B X (D).Binary relation α is an idempotent relation of the semigroup B X (D) iff binary relation α satisfies only one condition of the following conditions: a) α = (X × T ), where T ∈ D; b

Theorem 2 .
22. If D is any Z−elementary X−semilattice of unions, then for any idempotent binary relation ε from the semigroup B X (D) the order of maximal subgroup G X (D, ε) is not greater than two.