Applied Mathematics
Vol.09 No.04(2018), Article ID:84062,14 pages
10.4236/am.2018.94028
Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class
Yasha Diasamidze, Omari Givradze, Nino Tsinaridze, Giuli Tavdgiridze
Faculty of Mathematics, Physics and Computer Sciences, Shota Rustaveli State University, Batumi, Georgia
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Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 22, 2018; Accepted: April 24, 2018; Published: April 27, 2018
ABSTRACT
In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the class , and find uniquely irreducible generating set for the given semigroups.
Keywords:
Semigroup, Semilattice, Binary Relation
1. 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 on the set X that satisfies the condition . The set of all such ( ) is denoted by . It is easy to prove that 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 Æ an empty binary relation or an empty subset of the set X. The condition will be written in the form . Further, let , , , and . We denote by the symbols , , , and the following sets:
It is well known the following statements:
Theorem 1.1. Let be some finite X-semilattice of unions and be the family of sets of pairwise nonintersecting subsets of the set X (the set Æ can be repeated several times). If j is a mapping of the semilattice D on the family of sets which satisfies the conditions
and , 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 there exist such parameters that cannot be empty sets for D. Such sets are called bases sources, where sets , which can be empty sets too are called completeness sources.
It is proved that under the mapping j the number of covering elements of the pre-image of a
bases source is always equal to one, while under the mapping j 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 a of the semigroup is external if for all (see [1] [2] Definition 1.15.1).
It is well known, that if B is all external elements of the semigroup and is any generated set for the , then (see [1] [2] Lemma 1.15.1).
Definition 1.2. The representation of binary relation a is called quasinormal, if and for any ,
(see [1] [2] chapter 1.11).
Definition 1.3. Let . Their product is defined as follows: if there exists an element such that (see [1] , chapter 1.3).
2. 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:
(see Figure 1).
Figure 1. Diagram of the semilattice D.
It is easy to see that is irreducible generating set of the semilattice D.
Let be a family of sets, where are pairwise disjoint subsets of the set X and is a map
ping of the semilattice D onto the family of sets . Then the formal equalities of the semilattice D have a form:
(2.0)
Here the elements are bases sources, the element are sources of completeness of the semilattice D. Therefore (by symbol we denoted the power of a set X), since (see [1] [2] chapter 11).
In this paper we are learning irreducible generating sets of the semigroup defined by semilattices of the class .
Note, that it is well known, when , then generated sets of the complete semigroup of binary relations defined by semilattices of the class .
In this paper we suppose, that .
Remark, that in this case (i.e. ), from the formal equalities of a semilattice D follows, that the intersections of any two elements of a semilattice D is not empty.
Lemma 2.0 If , then the following statements are true:
a)
b)
c)
Proof. From the formal equalities of the semilattise D immediately follows the following statements:
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 , and :
Lemma 2.1. Let and . Then the following statements are true:
1) Let . If , then a is external element of the semigroup ;
2) Let , . If and , then a is external element of the semigroup .
3) Let and . If and , then a is external element of the semigroup ;
Proof. Let and for some . If quasinormal representation of binary relation d has a form
,
then
. (2.1)
From the formal equalities (2.0) of the semilattice D we obtain that:
(2.2)
where for any and by definition of a semilattice D from the class .
Now, let and for some , , then from the equalities (2.3) follows that since T and are minimal elements of the semilattice D and by preposition. The equality contradicts the inequality .
The statement a) of the Lemma 2.1 is proved.
Now, let and , for some , and , then from the equalities 2.3 follows, that
, if , or , , or
where . For the we consider the following cases:
1) If , then we have
,
since is a minimal element of a semilattice D. On the other hand,
But the equality contradicts the inequality . Thus we have, that .
2) Let , i.e. , then we have, that
,
since is a minimal element of a semilattice D. On the other hand:
The equality contradicts the inequality . Also, the equality contradicts the inequality for any and ( , by preposition) by definition of a semilattice D.
3) If , i.e. , then we have, that
,
since is a minimal element of a semilattice D. On the other hand:
The equality contradicts the inequality . Also, the equality , or contradicts the inequality for any and by definition of a semilattice D.
The statement 2) of the Lemma 2.1 is proved.
Let and . If and , , then from the formal equalities (2.0) of a semilattice D there exists such an element, that and , where . So, from the equalities (2.3) follows that and . Of from this and from the equalities (2.3) we obtain that there exists such an element , for which the equalities and , where . But such elements by definition of a semilattice D do not exist.
The statement c) of the Lemma 2.1 is proved.
Lemma 2.1 is proved.
Lemma 2.2. Let and . Then the following statements are true:
1) Let . If , then a is external element of the semigroup ;
2) Let . If , then a is external element of the semigroup ;
3) Let . If , then a is external element of the semigroup ;
4) Let , then a is external element of the semigroup ;
5) Let . If , , or , then a is external element of the semigroup ;
6) Let , then a is external element of the semigroup ;
7) Let , then a is external element of the semigroup .
Proof. Let a be any element of the semigroup . It is easy that . We consider the following cases:
Let , then since is subsemilattice of the semilattice D.
1) Let .
If , then , or , where , since is subsemilattice of the semilattice D.
If , then by statement c) of the Lemma 2.1 follows that a is external element of the semigroup .
2) Let .
If , then , or , where , since is a subsemilattice of the semilattice D.
If , then by statement a) of the Lemma 2.1 follows that a is external element of the semigroup .
3) Let .
If , then , or , , since is subsemilattice of the semilattice D.
If , then by statement a) of the Lemma 2.1 follows that a is external element of the semigroup .
4) Let , then by the statement a) of the Lemma 2.1 follows that a is external element of the semigroup .
5) Let .
If , then , or , or where and .
If where , then by the statement 2) of the Lemma 2.1 follows that a is external element of the semigroup ;
If , or , then from the statement 1) and 3) of the Lemma 2.1 follows that a is external element of the semigroup respectively.
6) Let . Then from the statement b) of the Lemma 2.1 follows that a is external element of the semigroup .
7) Let , then by the statement a) of the Lemma 2.1 follows that a is external element of the semigroup .
Lemma 2.2 is proved.
Now we learn the following subsemilattices of the semilattice D:
We denoted the following sets by symbols and :
By definition of a set follows that any element of the set is external element of the semigroup .
Lemma 2.3. Let . If quasinormal representation of a binary relation a has a form
where and , then a is generated by elements of the elements of set .
Proof. 1). Let quasinormal representation of binary relations d and b have a form
where .
,
since the representation of a binary relation b is quasinormal and by statement 3) of the Lemma 2.1 binary relations d and b are external elements of the semigroup . It is easy to see, that:
since (see equality (2.0))
if , and . Last equalities are possible since ( , by preposition).
Lemma 2.3 is proved.
Lemma 2.4. Let . If quasinormal representation of a binary relation a has a form , where , , then binary relation a is generated by elements of the elements of set .
Proof. Let quasinormal representation of the binary relations d and b have a form:
where and . Then from the statements a), b) and c) of the Lemma 2.1 follows, that d and b are generated by elements of the set and
, since
if , and . Last equalities are possible since ( by preposition).
Lemma 2.4 is proved.
Lemma 2.5. Let . If quasinormal representation of a binary relation a has a form , where , , then binary relation a is generated by elements of the elements of set .
Proof. Let quasinormal representation of a binary relations d, b have a form
where , and . Then from the Lemma 2.2 follows that b is generated by elements of the set , and
, since , (see equality(2.0))
since
if and . Last equalities are possible since ( by preposition).
Lemma 2.5 is proved.
Lemma 2.6. Let . Then the following statements are true:
1) If quasinormal representation of a binary relation a has a form , then binary relation a is generated by elements of the set .
2) If quasinormal representation of a binary relation a has a form , then binary relation a is generated by elements of the set .
Proof. 1) Let . If quasinormal representation of a binary relations d, b have a form
where ,
(see equalities (2.0) and (2.1)), then from the Lemma 2.4 follows that d is generated by elements of the set and from the Lemma 2.3 element b is generated by elements of the set and
, since
since representation of a binary relation d is quasinormal.
The statement a) of the lemma 2.6 is proved.
2) Let quasinormal representation of a binary relation d have a form
where , then from the Lemma 2.4 follows that d is generated by elements of the set and
, since and
,
since representation of a binary relation d is quasinormal.
The statement b) of the lemma 2.6 is proved.
Lemma 2.6 is proved.
Lemma 2.7. Let . Then the following statements are true:
a) If and , then binary relation is generated by elements of the elements of set ;
b) If and , then binary relation is external element for the semigroup .
Proof. 1) If quasinormal representation of a binary relation d has a form
,
where for all , then . Let quasinormal representation of a binary relations b have a form
, where f is any mapping of the set in the set . It is easy to see, that and two elements of the set belong to the semilattice , i.e. . In this case we have that for all .
since the representation of a binary relation d is quasinormal. Thus, the element a is generated by elements of the set .
The statement a) of the lemma 2.7 is proved.
2) Let , , for some and for some . Then we obtain that since T is a minimal element of the semilattice D.
Now, let subquasinormal representations of a binary relation b have a form
,
where is normal mapping. But complement mapping is empty, since , i.e. in the given case, subquasinormal representation of a binary relation b is defined uniquely. So, we have that (see property 2) in the case 1.1), which contradict the condition, that .
Therefore, if and , for some , then a is external element of the semigroup .
The statement 2) of the Lemma 2.7 is proved.
Lemma 2.7 is proved.
Theorem 2.1. Let , , and
Then the following statements are true:
1) If , then the is irreducible generating set for the semigroup ;
2) If , then the is irreducible generating set for the semigroup .
Proof. Let , and . First, we proved that every element of the semigroup is generated by elements of the set . Indeed, let a be an arbitrary element of the semigroup . Then quasinormal representation of a binary relation a has a form
,
where and . For the we consider the following cases:
1) If , then by definition of a set .
Now, let .
2) If , then quasinormal representation of a binary relation a has a form , where and from the Lemma 2.3 follows that a is generated by elements of the elements of set by definition of a set .
3) If , then quasinormal representation of a binary relation a has a form , where , and from the Lemma 2.4 follows that a is generated by elements of the elements of set by definition of a set .
4) If , then quasinormal representation of a binary relation a has a form , where , and from the Lemma 2.5 follows that a is generated by elements of the elements of set by definition of a set .
Now, let , then quasinormal representation of a binary relation a has a form , or , where .
5) If , then from the statement b) of the Lemma 2.6 follows that binary relation a is generated by elements of the set .
6) If , where , then from the statement a) of the Lemma 2.6 and 2.7 follows that binary relation a is generated by elements of the set .
Thus, we have that is a generating set for the semigroup .
If , then the set is an irreducible generating set for the semigroup since, is a set external elements of the semigroup .
The statement a) of the Theorem 2.1 is proved.
Now, let . First, we proved that every element of the semigroup is generated by elements of the set . The cases 1), 2), 3), 4) and 5) are proved analogously of the cases 1), 2), 3), 4) and 5 given above and consider case, when .
If , where , then from the statement a) of the Lemma 2.7 follows that binary relation a is generated by elements of the set .
If , where , then from the statement b) of the Lemma 2.6 follows that binary relation is external element for the semigroup .
Thus, we have that is a generating set for the semigroup .
If , then the set is an irreducible generating set for the semigroup since is a set external elements of the semigroup .
The statement b) of the Theorem 2.1 is proved.
Theorem 2.1 is proved.
Corollary 2.1. Let and
Then the following statements are true:
1) If , then is the uniquely defined generating set for the semigroup ;
2) If , then is the uniquely defined generating set for the semigroup .
Proof. It is well known, that if B is all external elements of the semigroup and is any generated set for the , then (see [1] [2] Lemma 1.15.1). From this follows that the sets and are defined uniquely, since they are sets external elements of the semigroup .
Corollary 2.1 is proved.
It is well-known, that if B is all external elements of the semigroup and is any generated set for the , then (Definition 1.1).
In this article, we find irredusible generating set for the complete semigroups of binary relations defined by X-semilattices of unions of the class . This generating set is uniquely defined, since they are defined by elements of the external elements of the semigroup .
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
References