SOME REGULAR ELEMENTS, IDEMPOTENTS AND RIGHT UNITS OF COMPLETE SEMIGROUPS OF BINARY RELATIONS DEFINED BY SEMILATTICES OF THE CLASS LOWER INCOMPLETE NETS

Abstract: In this paper, we investigate such a regular elements α and idempotents of the complete semigroup of binary relations BX(D) defined by semilattices of the class lower incomplete nets, for which V (D,α) = Q. Also we investigate right units of the semigroup BX(Q). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup.


Introduction
Let X be an arbitrary nonempty set.Let D be some nonempty set of subsets of the set X, closed with respect to the operation of set-theoretic union of elements of the set D, i.e., ∪D ′ ∈ D for any nonempty subset D ′ of the set D. In that case, the set D is called complete X−semilettice of unions.The union of all elements of the set D is denoted by the symbol D. Clearly, D ∈ D is the largest element.
Recall that a binary relation on the set X is a subset of the cartesian product X×X.If α and β are binary relations on the set X with the elements x, y, z ∈ X the condition (x, y) ∈ α is denoted as xαy and xαyβz means the conditions xαy and yβz are satisfied simultaneously.The binary relation α −1 = {(x, y) : yαx} is usually called the binary relation inverse to α.The empty binary relation which is empty subset of X × X is denoted by ∅.The binary relation δ = α • β is called product of binary relations α and β.A pair (x, y) belongs to δ if only if there exists y ∈ X such that xαyβz.The binary operation • is associative.So, B X , the set of all binary relations on X, is therefore a semigroup with respect to the operation •.This semigroup is called the semigroup of all binary relations on the set X.
Let f be an arbitrary mapping from X into D. Then one can construct such a mapping f with a binary relation α f on X provided by the condition below, α f = x∈X ({x} × f (x)).The set of all such binary relation is denoted by B X (D).It is easy to prove that B X (D) is a semigroup with respect to the product operation of binary relations.This semigroup, B X (D), is called a complete semigroup of binary relations defined by an X−semilattice of unions D.
Further, let x, y ∈ X, Y ⊆ X, α ∈ B X (D), T ∈ D, ∅ = D ′ ⊆ D and t∈ D. Then we have the following notation, Now, let's take α ∈ B X (D).If β •α = β for any β ∈ B X (D), then α is called a right unit of semigroup B X (D).If α • α = α then α is called an idempotent element of semigroup B X (D).And if α • β • α = α for some β ∈ B X (D), then a binary relation α is called a regular element of semigroup B X (D).Let l(D ′ , T ) = ∪ (D ′ \D ′ T ).We say that a nonempty element T is a nonlimiting element of Now, we continue with some essential definitions and theorems given by the cited references.
Definition 1.2.[3, Definition 2] Let D and D ′ be some nonempty subsets of the complete X−semilattices of unions.We say that a subset D generates a set D ′ if any element from D ′ is a set-theoretic union of the elements from D.  Definition 1.5.[6, Definition 8] Let α be some binary relation of the semigroup B X (D).We say that a complete isomorphism ϕ between XI−semilattice of unions Q and Ω(D) is the set of all XI−subsemilattices of the complete X−semilat-tice of unions D such that D ′ ∈ Ω(D) iff there exists a complete isomorphism between the semilattices D ′ and D.

Let us denote
. ., T m−1 be some finite X−semilattice of unions and C(D) = {P 0 , P 1 , P 2 , . . ., P m−1 } be the family of sets of pairwise disjoint subsets of the set X.If ϕ is a mapping of the semilattice D to the family sets C(D) that satisfies the condition ϕ( D) = P 0 and ϕ(T i ) = P i (i = 1, 2, . . ., m − 1) and DZ = D\ {T ∈ D : Z ⊆ T } 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 P i (i = 0, 1, 2, . . ., m − 1) there exist such parameters that cannot be empty sets for D. Such sets P i (0 < i ≤ m − 1) are called basis sources, whereas sets P j (0 ≤ j ≤ m − 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.
Note that the set P 0 is always considered to be a source of completeness.
Lemma 2.1.Let Q be a lower incomplete net.Then Q is XI−semilattice of unions iff it satisfies the condition T 0k ∩ T s0 = ∅.
Then fromLemma 1.2 and from the formal equalities (2.1) we have: ∈ Q, when t / ∈ ∪ {P 0k , P 1k , . . ., P s−1k , P s0 , P s1 , . . ., P sk−1 }, i.e., If the equality T 0k ∩ T s0 = ∅ is true, then by ( * ) follows that Q is XIsemilattice of unions.Lemma 2.2.Let Q be a XI−lower incomplete net.Then following equalities are true: Proof.The given Lemma immediately follows from the formal equalities (2.1) of the semilattice Q.For the largest right unit ε of the semigroup B X (D) we have: Theorem 2.1.Let Q be a XI-lower incomplete net.Then a binary relation α of the semigroup B X (D) having a quasinormal representation of the form α = , is a regular element of the semigroup B X (D) iff for some α−isomorphism ϕ of the semilattice Q on some subsemilattice D ′ of the semilattice D the following conditions are fulfilled: Proof.It is easy to see that the set Qˆ= {T 10 , T 20 , . . ., T s0 , T 01 , . . ., T 0k } is an irreducible generating set of the semilattice Q.Moreover, all elements of the set Qˆ= Q 1 ∪ Q 2 are nonlimiting.Now by Theorem 1.4 we obtain a) From the condition a) of this theorem we immediately have the validity of the follo-wing inclusions: Let Q be a XI−lower incomplete net.Then a binary relation α of the semigroup B X (Q), which has a quasinormal representation of the form α = , is an idempotent element of the semigroup B X (D) iff the following conditions are fulfilled: Proof.This theorem immediately follows from Lemma 2.1, from the Theorem 2.1 and Theorem 1.5.
, is a right unit of the semigroup B X (Q) iff the following conditions are fulfilled: for any T ij ∈ Qˆ.
Proof.This theorem immediately follows from Theorem 1.3.
Proof.In the first place, we note that the given semilattice Q has one automorphisms (i.e., |Φ(Q, Next, assume that α ∈ R(Q, D ′ ) and a quasinormal representation of a regular binary relation α has the form Then, according to Theorem 2.1, we have for any Tij ∈ T01 , T02 , . . ., T0k , T10 , T20 , . . ., Ts0 .Further, let f α be a mapping the set X in the semilattice D satisfying the conditions f α (t) = tα for all t ∈ X.
By the Theorem 1.1 the number of the mappings  X (Q) be the set of all right units of the semigroup B x (Q).If X is a finite set, then the following formula is true Proof.By virtue of Theorem [4,Theorem 6.3.11]6.3.11(see [4]) we have E (r)

D
is partially ordered with respect to the set-theoretic inclusion.Let ∅ = D ′ ⊆ D and N (D, D ′ ) = {Z ∈ D : Z ⊆ Z ′ for any Z ′ ∈ D ′ }.It is clear that N (D, D ′ ) is the set of lower bounds of a nonempty subset D ′ included in D. If N (D, D ′ ) = ∅ then ∪N (D, D ′ ) belongs to D and it is the greatest lower bound of D ′ and is denoted by ∧(D, D ′ ) = ∪N (D, D ′ ).

Definition 1 . 3 .
[4, Definition 1.14.2]We say that a complete X−semilattice of unions D is an XI−semilattice of unions if it satisfies the following two conditions: a) ∧ (D, D t ) ∈ D for any t ∈ D, b) Z = t∈Z ∧ (D, D t ) for any nonempty element Z of D. Theorem 1.1.[4, Corollary 1.18.1]Let Y = {y 1 , y 2 , . . ., y k } and D j = {T 1 , . . ., T j } be some sets, where k ≥ 1 and j ≥ 1.Then the numbers s(k, j) of all possible mappings of the sets Y on any subset D ′ j of the set D j and T j ∈ D ′ j can be calculated by the formula s(k, j) = j k − (j − 1) k .Lemma 1.1.[1, Lemma 3.1] Let D complete X−semilattices of unions.If a binary relation ε having the form ε = ε(D, f ) = t∈ D ({t} × ∧ (D, D t )) ∪ X\ D × D is a right unit of the semigroup B x (D), then it is the largest right unit of this semigroup.Theorem 1.2.[1, Theorem 2.5] Let D ′ be a complete subsemilattice of the complete X−semilattice of unions D, D′ = ∪D ′ and f be an arbitrary mapping of the set X\ D′ in the semilattice D ′ .If D ′ is a complete XI−semilattice of unions then the binary relation

Definition 1 . 4 .
[6, Definition 7] A one-to-one mapping ϕ between the complete X−semilattices of unions D ′ and D ′′ is called a complete isomorphism if the condition ϕ (∪D 1 ) = T ′ ∈D 1 ϕ(T ′ ) is fulfilled for each nonempty subset D 1 of the semilattice D ′ .

Theorem 1 . 4 . 4 .
[4, Theorem 6.3.3]Let D be a finite X−semilattice of unions and α • σ • α = α for some elements α, σ ∈ B X (D); D(α) the set those elements T of the semilattice D = V (D, α)\ {∅} which are nonlimiting elements of the set D(α) T .Then binary relation α having a quasinormal representation of the form α = T ∈V (D,α)(Y α T × T ) is a reguler element of the semigroup B X (D)iff the set V (D, α) is a XI−semilattice of unions and for some α−isomorphism ϕ of the semilattice V (D, α) on some X−subsemilattice D ′ of the semilattice D the following conditions are fulfilled: i)T ∈ D(α) T Y α T ′ ⊇ ϕ(T ) for any T ∈ D(α); ii) Y α T ∩ ϕ(T ) = ∅ forany nonlimiting element T of the set D(α) T .Theorem 1.5.[4, Theorem 6.3.7]A regular element α of the semigroup B X (D) is idempotent iff the mapping ϕ satisfying the condition ϕ(T ) = T α for any T ∈ V (D, α) is an identity mapping of the semilattice V (D, α).Definition 1.6.[4, Definition 6.3.4]Let Q and D ′ be respectively some XI and X−subsemilattices of the complete X−semilattice of unions D. Then R ϕ (Q, D ′ ) is a subset of the semigroup B X (D) such that α ∈ R ϕ (Q, D ′ ) only if the following conditions are fulfilled for the elements of α and ϕ, a) The binary relation α be regular element of the semigroup B X (D), b) V (D, α) = Q, c) ϕ is a α−isomorphism between the complete semilattices of unions Q and D ′ satisfying the conditions i) and ii) of the Theorem 1.Definition 1.7.[4, Definition 6.3.4]Let Φ(D, D ′ ) be the set of all complete isomorphism ϕ between XI−semilattice of unions D and D ′ such that ϕ ∈ Φ(D, D ′ ) only if ϕ is a α−isomorphism for some α ∈ B X (D) and V (D, α) = D.

Lemma 1 . 2 .
Let D and C(D) = {P 0 , P 1 , . . ., P n−1 } are the finite semilattice of unions and the family of sets of pairwise nonintersecting subsets of the set X; ϕ is a mapping of the semilattice D on the family of sets C(D).If ϕ(T ) = P ∈ C(D)\ {P 0 } for some T ∈ D, then D t = D\ DT for all t ∈ P .Proof.Let t and Z ′ are 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 true 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

Corollary 2 . 1 .
Let Q be a XI−lower incomplete net and E (r)