Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class ( ) X n k 8 , 1 Σ + +

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the class ( ) 8 , 1 X n k Σ + + , and find uniquely irreducible generating set for the given semigroups.


Introduction
Let X be an arbitrary nonempty set, D is an X-semilattice of unions which is closed with respect to the set-theoretic union of elements from D, f be an arbitrary mapping of the set X in the set D. To each mapping f we put into correspondence a binary relation f α on the set X that satisfies the condition { } ( ) ( ) , , , , m C D P P P P − =  be the family of sets of pairwise nonintersecting subsets of the set X (the set ∅ can be repeated several times).If ϕ is a mapping of the semilattice D on the family of sets ( ) C D which satisfies the conditions 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 i P ( ) 0 1 i m < ≤ − there exist such pa- rameters that cannot be empty sets for D. Such sets i P are called bases sources, where sets j P ( ) 0 1 j m ≤ ≤ − , which can be empty sets too are called com- pleteness sources.
It is proved that under the mapping ϕ the number of covering elements of the pre-image of a bases 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 [1] [2] chapter 11).Definition 1.1.We say that an element α of the semigroup ( ) (see [1], chapter 1.3).

Result
Let ( ) be a class of all X-semilattices of unions whose every element is isomorphic to an X-semilattice of unions , , , , , which satisfies the condition: . 1).
Here the elements ( ) are bases sources, the element 0 P are sources of completeness of the semilattice D. Therefore X n k ≥ + (by symbol X we denoted the power of a set X), since ( ) In this paper we are learning irreducible generating sets of the semigroup ( ) Note, that it is well known, when 2 k = , then generated sets of the complete semigroup of binary relations defined by semilattices of the class , 2 2 1 ,5 In this paper we suppose, that 3 k n ≤ ≤ .
Remark, that in this case (i.e. 3 k ≥ ), from the formal equalities of a semilattice D follows, that the intersections of any two elements of a semilattice D is not empty.; From the formal equalities of the semilattise D immediately follows the following statements: q n q i i n q i i i q i q n q Z P Z Z P P P j n Z Z P P P q k The statements a), b) and c) of the lemma 2.0 are proved.
Lemma 2.0 is proved.
We denoted the following sets by symbols 1 D , 2 D and 3 D : . Then the following statements are true: 1) Let , From the formal equalities (2.0) of the semilattice D we obtain that: , .

T D D ′∈ ∪ and T T
′ ⊄ , then from the equalities 2.3 follows, that we consider the following cases:  , then we have , if .
But the equality T T′ = contradicts the inequality T T′ ≠ .Thus we have, that  , then we have, that , since T ′ is a minimal element of a semilattice D. On the other hand: , if 1 , ; , if ; n k n k i j j i i q n q i q n q j P P T P m n q q j β β β

 
The equality T T′ = contradicts the inequality T T′ ≠ .Also, the equality 3) If j n q = + ( ) , since T ′ is a minimal element of a semilattice D. On the other hand: , i q n q i q n q q n q P P T P P m q n P P P P P T P P The equality T T′ = contradicts the inequality T T′ ≠ .Also, the equality The statement 2) of the Lemma 2.1 is proved. Let , then from the formal equalities (2.0) of a semilattice D there exists such an element, that q j P Z ⊆ and q m P Z ⊆ , where 0 q m k ≤ ≤ + .So, from the equalities (2. ( ) . Then the following statements are true:


, where 1, 2, , ≥ , then by statement a) of the Lemma 2.1 follows that α is external element of the semigroup ( ) ≥ , then by statement a) of the Lemma 2.1 follows that α is external element of the semigroup ( )  where , 1, 2, , j q q k ≠ =  , then by the statement 2) of the Lemma 2.1 follows that α is external element of the semigroup ( ) , then from the statement 1) and 3) of the Lemma 2.1 follows that α is external element of the semigroup ( ) , .
We denoted the following sets by symbols 0 A and ( ) By definition of a set , , \ \ , . Then from the statements a), b) and c) of the Lemma 2.1 follows, that δ and β are generated by elements of the set ( ) follows that β is generated by elements of the set ( ) Lemma 2.5 is proved.
since representation of a binary relation δ is quasinormal.
The statement a) of the lemma 2.6 is proved.
2) Let quasinormal representation of a binary relation δ have a form where j q ≠ , then from the Lemma 2.4 follows that δ is generated by elements of the set ( ) since the representation of a binary relation δ is quasinormal.Thus, the element α is generated by elements of the set ( ) The statement a) of the lemma 2.7 is proved.

∈
a semigroup with respect to the op- eration 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 an empty subset of the set X.The condition ( ) , x y α ∈ will be written in the form x y α .Further, let , .We denote by the symbols yα , Yα , ( ) , V D α , X * and ( ) How to cite this paper: Diasamidze, Y., Givradze, O., Tsinaridze, N. and Tavdgiridze, G. (2018) Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class

Figure 1 .
Figure 1.Diagram of the semilattice D.
T and T ′ are minim- al elements of the semilattice D and 0 P ≠ ∅ by preposition.The equality T T′ = contradicts the inequality T T′ ≠ .The statement a) of the Lemma 2.1 is proved.
by preposition) by definition of a semilattice D.
then by the statement a) of the Lemma 2.1 follows that α is external element of the semigroup ( ) Then from the statement b) of the Lemma 2.1 follows that α is external element of the semi-, then by the statement a) of the Lemma 2.1 follows that α is external element of the semiproved.Now we learn the following subsemilattices of the semilattice D:

( ) 0 B
A follows that any element of the set is external element of the semigroup + .If quasinormal representation of a binary relation α has a form α is generated by elements of the elements of set ( ) 0 B A .Proof. 1).Let quasinormal representation of binary relations δ and β have a form then binary relation α is generated by elements of the elements of set ( ) 0 B A .Proof.Let quasinormal representation of the binary relations δ and β have a form:( then binary relation α is generated by elements of the elements of set ( ) 0 B A .Proof.Let quasinormal representation of a binary relations δ, β have a form ( )

0 B A . 2 )
+ .Then the following statements are true: 1) If quasinormal representation of a binary relation α has a form relation α is generated by elements of the set ( ) If quasinormal representation of a binary relation α has a form X α is generated by elements of the set ( ) representation of a binary relations δ, β have a form since T is a minimal element of the semilattice D. Now, let subquasinormal representations β of a binary relation β have a form { } ( ) ( )

2 β
is empty, since \ X D = ∅  , i.e. in the given case, subquasinormal representation β of a binary relation β is defined uniquely.So, we have that X T β β α = = × = (see property 2) in the case 1.1), which contradict the condi-

1 ,
generated by elements of the set 0 S .Indeed, let α be an arbitrary element of the semigroup ( ) X B D .Then quasi- normal representation of a binary relation α has a form V X α * ∈ A , then quasinormal representation of a binary relation α Lemma 2.3 follows that α is generated by elements of the elements of set ( ) Lemma 2.4 follows that α is generated by elements of the elements of set ( ) Lemma 2.5 follows that α is generated by elements of the elements of set ( ) then quasinormal representation of a binary rela- tion α has a form X D from the statement b) of the Lemma 2.6 follows that binary relation α is generated by elements of the set ( ) 0 B A .
since the representation of a binary relation β is quasinormal and by statement 3) of the Lemma 2.1 binary relations δ and β are external elements of the XB D .It is easy to see, that: equalities (2.0) and (2.1)), then from the Lemma 2.4 follows that δ is gener- since representation of a binary relation δ is quasinormal.