Generating Set of the Complete Semigroups of Binary Relations

Difficulties encountered in studying generators of semigroup ( ) X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup ( ) X B D , which are defined by semilattice D, can be generated by the set ( ) ( ) { } X B B D V X D α α ∗ = ∈ = , .


Introduction
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, then among the parameters there exist such parameters that cannot be empty sets for D. Such sets P i ( ) are called basis sources, whereas sets P i ( ) 0 1 j m ≤ ≤ − 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 com- pleteness source either does not exist or is always greater than one (see [1], Chapter 11).Some positive results in this direction can be found in [2]- [6].
Let 0 1 2 1 , , , , m P P P P −  be parameters in the formal equalities, ( ) ( ) ( ) The representation of the binary relation β of the form β and β  will be called subquasinormal and maximal subquasinormal.If β and β  are the subquasinormal and maximal subquasinormal representations of the binary relation β , then for the binary relations β and β  the following statements are true:


. For the last condition follows that z y β .We have x z y α β and ( ) The parameters P 1 , P 2 , P 3 are basis sources and the parameters 0 4 , P P are completeness sources, i.e.
a) Let ( ) , where T D ∈ .By element T we consider the following cases: 1. T D ≠  .In this case suppose that 0 1 2 3 4 1 where From the formal equality and equalities (6) and ( 5) we have: where From the formal equality and equalities (7) and ( 5) we have: is X-semilattice of unions.For the semilattice of unions ( ) , where, { } where From the formal equality and equalities (8) and ( 5) we have: ) ( ) .
. Then binary relation α has representation of the form ( ) ( ) ( ) where From the formal equality and equalities ( 11) and ( 5) we have: .
. Then binary relation α has representation of the form where From the formal equality and equalities (12) and ( 5) we have: ) .
. Then binary relation α has representation of the form ( ) ( ) ( ) where . From the formal equality and equalities (13) and ( 5) we have: .
. Then binary relation α has representation of the form ( ) ( ) ( ) where From the formal equality and equalities ( 14) and ( 5) we have: ) .
. Then binary relation α has representation of the form ( ) ( ) ( ) In this case suppose that where From the formal equality and equalities (15) and ( 5) we have:  .From the formal equality and equalities ( 16) and ( 5) we have:  ; .
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 ( )

β
subquasinormal representation of the binary relation β is quasinormal; is a mapping of the family of sets (

1 ,
, , , D Z Z Z Z D =  of unions given by the diagram of Figure 1.Formal equalities of the given semilattice have a form:

.
Then for the for- mal equalities of the semilattice D follows that

(
β is subquasinormal representation of a binary relation β ) is true.By assumption i.e. the quasinormal representation of a binary relation 1 α have a form

.
Then binary relation α has representation of the form