Description of Some Right Unit Elements of
the Complete Semigroups of Binary Relations
of the Class
6
X
,8
.
Didem Yeşil Sungur
#1,GiuliTavdgiridze
*21
Canakkale Onsekiz Mart University, Faculty of Science and Art, Department of Mathematics,TURKEY
2
Shota Rustaveli BatumiState University, Faculty of Mathematics, Physics and Computer Sciences,GEORGİA
Abstract—Let
D
be anyX
semilattice of unions andQ
be a subsemilattice ofD
which satisfies the following conditions;Q
=
T
6,
T
5,
T
4,
T
3,
T
2,
T
1,
T
0
be aX
semilattice whereT
7
T
6
T
5
T
3
T
1
T
0,
0, 1 2 5 6
7
T
T
T
T
T
T
T
7
T
6
T
4
T
2
T
1
T
0,
T
2
T
3=
T
1,
T
4
T
3=
T
1,
,
=
25
4
T
T
T
T
2\
T
3
,
T
3\
T
2
,
T
5\
T
4
,
T
4\
T
5
,
T
4\
T
3
,
T
3\
T
4
.
In this paper, we investigate a binary relation
which is right unit element.Keywords—Semigroups, Binary relation, Right unit elements.
I.INTRODUCTION
It is well known that in the theory of semigroups, any semigroup is isomorphically embeddable in some semigroups of binary relations. Many studies related with this fact are available in literature. Among them, we refer the papers of Diasamidze [1 - 4]. In general he studied semigroups but, in particular, he investigates the semigroups of the binary relations. Moreover, in his papers he presents how the idempotents, one-sided units, regular, irreducible and externally adjoined elements of semigroups have exclusively important role in the investigation of the abstract properties of semigroups.
To construct all these work we first give some basic definitions.
A partially ordered set
A
is called a semilattice if there exists a greatest lower bound for every pair of elements ofA
.Let
X
be an arbitrary nonempty set,D
be some nonempty set of subsets of the setX
and also letD
closed under the union of its subsets
,
i.e.,
D
'
D
for any nonempty subsetD
' of the setD
.
In that case , the setD
is called a completeX
semilattice of unions. The union of all elements of the setD
is denoted by the symbolD
.
Clearly,D
D
is the largest element. We say that an element
Y
coversZ
in the semilatticeD
ifY
Z
and there is no other elementT
D
such thatY
T
Z
.
Using the cover relation, we can obtain a graphic representation of the semilattice of unionsD
. Each element ofD
is represented in the form of a small circle. IfY
Z
thenY
is located aboveZ
and alsoY
andZ
are connected by a rectilinear line.Recall that a binary relation on the set
X
is a subset of the cartesian productX
X
.
If
and
are binary relations on the setX
with the elementsx
,
y
,
z
X
the condition(
x
,
y
)
is denoted asx
y
and
x
y
z
means the conditionsx
y
andy
z
are satisfied simultaneously. The binary relation}
|
)
,
{(
=
1
x
y
y
x
is usually called the binary relation inverse to
.
The empty binary relation which is an empty subset ofX
X
is denoted by
.
The binary relation
=
is called the product of the binary relations
and
.
A pair(
x
,
z
)
belongs to
if and only if there existsy
X
such thatx
y
z
.
The binary operation
is associative. So,B
X,
the set of all binary relations onX
,
is therefore a semigroup withrespect to the operation
.
This semigroup is called the semigroup of all binary relations on the setX
.
,
=
|
.
\
2
=
,
|
=
,
}
|
{
=
},
|
{
=
)
,
(
,
}
|
{
=
2
..
T
Z
D
Z
D
X
Z
T
D
Z
D
Z
t
D
Z
D
D
Y
Y
D
V
X
Y
Y
' ' ' ' T X
' '
' ' T '
' t X
Let
f
be an arbitrary mapping fromX
intoD
.
Then one can construct such a mappingf
with a binary relation
f onX
provided by the condition below,
(
)
.
=
x
f
x
X x
f
The set of all such binary relations is denoted by
B
X(
D
).
It is easy to prove thatB
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 anX
semilattice of unionsD
.
Now, let’s take any
,
B
X(
D
).
If
=
then
is called a right unit element of semigroup).
(
D
B
X If
=
then
is called an idempotent element of semigroupB
X(
D
).
And if
=
for some
B
X(
D
)
then a binary relation
is called a regular element of semigroup).
(
D
B
XNote that the semilattice
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 thatN
(
D
,
D
')
is the set of all lower bounds of a nonempty subsetD
' included inD
.
IfN
(
D
,
D
')
then
N
(
D
,
D
')
belongs toD
and it is the greatest lower bound ofD
' and is denoted by
(
D
,
D
')
=
N
(
D
,
D
').
We say that a nonempty element
T
is a nonlimiting element ofD
' ifT
\
l
(
D
',
T
)
. A nonempty elementT
is limiting element ofD
' ifT
\
l
(
D
',
T
)
=
.
In this work, we use complete
X
semilattices of unions to define of all binary relations on a setX
,
which are called complete semigroups of binary relations. Let
Q
be a subsemilattice ofD
which satisfies the following conditions.Q
=
T
6,
T
5,
T
4,
T
3,
T
2,
T
1,
T
0
be aX
semilattice where,
0 1 3 5 6
7
T
T
T
T
T
T
T
7
T
6
T
5
T
2
T
1
T
0,T
7
T
6
T
4
T
2
T
1
T
0,
,
=
1 32
T
T
T
T
4
T
3=
T
1,
T
4
T
5=
T
2,
T
2\
T
3
,
T
3\
T
2
,
T
5\
T
4
,
T
4\
T
5
,
,
\
34
T
T
T
3\
T
4
.
Mainly, we investigate and determine right unit elements ofB
X(
Q
)
for a finite setX
and in particular, we give formulas for the calculation of the number of right unit elements. Now, we continue with some essential definitions and theorems given by the cited references.Definition 1.1 [2, Definition 1] Let
B
X(
D
),
T
V
(
X
,
),
Y
T=
{
y
X
|
y
=
T
}.
Then a
representation of a binary relation
of the form=
(
)
) , (
T
Y
T X V T
is called quasinormal.Note that, if
=
(
)
) , (
T
Y
T X V T
is a quasinormal representation of the binary relation
,
then thefollowing conditions are true,
1.
=
,
) , (
T X V T
Y
X
2.
'
T T
Y
Definition 1.2 [3, Definition 2] Let
D
andD
' be some nonempty subsets of the completeX
semilattices of unions. We say that a subset
D
generates a setD
' if any element fromD
' is a set-theoreticunion of the elements from
D
.
Definition 1.3 [4, Definition 1.14.2] We say that a complete
X
semilattice of unionsD
is anXI
semilattice of unions if it satisfies the following two conditions,
•
(
D
,
D
t)
D
for anyt
D
,
•
=
(
,
t)
Z t
D
D
Z
for any nonempty elementZ
ofD
.
Theorem 1.4 [4, Corollary 1.18.1] Let
Y
=
y
1,
y
2,
,
y
k
andD
j=
T
1,
,
T
j
be some sets, where1
k
andj
1.
Then the numberss
(
k
,
j
)
of all possible mappings of the setsY
on any subsetD
'j of the setD
j andT
j
D
'j can be calculated by the formula.
1)
(
=
)
,
(
k
j
j
kj
ks
Theorem 1.5 [4, Theorem 6.1.3]Let
D
be a completeX
semilattice of unions. The semigroupB
X(
D
)
possesses a right unit iff
D
is anXI
semilattice of unions.Lemma 1.6 [1, Lemma 3.1] Let
D
completeX
semilattices of unions. If a binary relation
having the form
(
,
))
((
\
)
)
(
=
)
,
(
=
D
f
x
D
D
tX
D
D
D t
is a right unit of the semigroup
B
X(
D
),
then it is the largest right unit of this semigroup.Definition 1.7 [5, Definition 7] A one-to-one mapping
between the completeX
semilattices of unions 'D
andD
'' is called a complete isomorphism if the condition)
(
=
)
(
1 1
' D ' T
T
D
is fulfilled for each nonempty subset
D
1 of the semilatticeD
'.
Definition 1.8 [5, Definition 8] Let
be some binary relation of the semigroupB
X(
D
).
We say that a complete isomorphism
betweenXI
semilattice of unionsQ
andD
is a complete
isomorphism if1.
Q
=
V
(
D
,
),
2.
(
)
=
for
V
(
D
,
)
and
(
T
)
=
T
for anyT
V
(
D
,
).
Theorem 1.9 [4, Theorem 6.3.3] Let
X
be a finite set,
B
X(
D
)
andD
be the set of thoseelements
T
of the semilatticeD
=
V
(
D
,
)
\
which are nonlimiting elements of the setD
T..
. A
binary relation
having a quasinormal representation of the form=
(
)
) , (
T
Y
TD V T
is a right unitelement of the semigroup
B
X(
D
)
iff) ( ..
T D ' T
3.
Y
T
T
for any nonlimiting elementT
of the set(
)
.
..
T
D
Definition 1.10 [4, Definition 6.3.4] Let
Q
,
D
'
be a set of all complete isomorphism of theXI
semilattice of unions
Q
on the semilatticeD
' such that
Q
,
D
'
only if
is a
isomorphism for some relation
B
X(
D
)
andV
D
,
=
Q
.
Q
is the set of allXI
subsemilattices of the completeX
semilattice of unionsD
such that
Q
Q
'
only if there exists some complete isomorphism between the semilatticesQ
' andQ
.
Theorem 1.11 [5, Theorem 22] Let
D
=
D
,
T
1,
T
2,
,
T
m
be some finiteX
- semilattice of unions and
P
P
P
P
mP
m
D
C
(
)
=
0,
1,
2,
,
1,
be the family of sets of pairwise disjoint subsets of the setX
,
is a mapping of the semilatticeD
on the family setsC
(
D
)
that satisfies the condition
(
D
)
=
P
0 andi i
P
T
)
=
(
for anyi
=
1,2,
,
m
andD
ˆ
Z=
D
\
T
D
|
Z
T
,
then the following equalities are valid:m
P
P
P
P
D
=
0
1
2
(1.1)and
.
,
1,2,
=
)
(
=
ˆ
0
T
foralli
m
P
Z
i Z D T
i
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 parametersP
i
i
=
0,1,2,
,
m
1
there exists such parameters thatcannot be empty sets forD
.
Such setsi
P
0
i
m
1
are called bases sources, whereas setsP
j
0
j
m
1
which can be empty sets too are called completeness sources.It is proved that under the mapping
the number of covering elements of the pre-image of a basis source is always equal to one, while under the mapping
the number of covering elements of the pre-image of a completeness source either does not exists or is always greater than one.Note that the set
P
0 is always considered to be a source of completeness.II. RESULTS
Xcvxcvxcv Let
X
be a finite set,D
=
T
7,
T
6,
T
5,
T
4,
T
3,
T
2,
T
1,
T
0
be aX
subsemilattice of unions of.
\
,
\
,
\
,
\
,
\
,
\
,
=
,
=
,
=
4 3 3 4 5 4 4 5 2 3 3 2 2 5 4 1 3 4 1 3 2 0 1 2 4 6 7 0 1 2 5 6 7 0 1 3 5 6 7
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
Let
7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0=
P
P
P
P
P
P
P
P
T
T
T
T
T
T
T
T
is a mapping of the semilattice
D
onto the family setsC
(
D
).
Then for the formal equalities of the semilatticeD
we have a form,0 7 7 0 6 7 6 4 0 5 7 6 5 3 0 4 7 6 5 4 2 0 3 7 6 5 4 3 0 2 7 6 5 4 3 2 0 1 7 6 5 4 3 2 1 0 0
=
=
=
=
=
=
=
=
P
T
P
P
T
P
P
P
P
T
P
P
P
P
P
T
P
P
P
P
P
P
T
P
P
P
P
P
P
T
P
P
P
P
P
P
P
T
P
P
P
P
P
P
P
P
T
(2.1)Here the elements
P
1,
P
2,
P
3,
P
4,
P
7 are basis sources, the elementsP
0,
P
5,
P
6 is sources of completeness of the semilatticeD
.
Theorem 2.1 The semigroup
B
X(
D
)
always has a right unit element.Proof. Let
t
D
,
D
t=
Z
D
|
t
Z
and
(
D
,
D
t)
is the exact lower bound of the setD
t inD
.
Then the formal equalities follows that,
)
(
D
B
X always has a right unit element.Lemma 2.2 For the semilattice
D
, the following equalities are true.
1 0 1 2 3 2 3 4 3 4 5 4 7 4 3 7 6 5 7 0\
=
\
=
\
=
\
=
\
=
=
T
T
P
T
T
P
T
T
P
T
T
P
T
T
T
P
P
P
T
P
Proof. The given Lemma immediately follows from the formal equalities 2.1 of the semilattice
D
.
Theorem 2.3 The binary relation
7 7
3 4
\
7
6
5\
4
5
4\
3
4
3\
2
3
0\
1
0
\
0
0
=
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
X
T
T
is the largest right unit of the semigroup
B
X(
D
).
Proof. Using Lemma 1.6 and Theorem 2.1 we have that
3 2 3
0 1
0
0
0
4 3 4 5 4 5 6 7 4 3 7 7 0 0 0 1 3 2 4 3 5 4 6 7 6 5 7 0 0 0
\
\
\
\
\
\
=
\
=
\
)
;
(
=
T
T
X
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
X
T
P
T
P
T
P
T
P
T
P
P
P
T
P
T
T
X
D
D
t
t D t
Corollary 2.4 A binary relation
having quasinormal representation as
,
=
0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
where
Y
7,
Y
6,
Y
5,
Y
4,
Y
3,
Y
0
,
is a right unit of the semigroupB
X(
D
)
iff the binary relation
satisfies the following conditions,
.
,
,
,
,
,
,
,
,
0 0 3 3 4 4 5 5 6 6 3 3 5 6 7 4 4 6 7 5 5 6 7 6 6 7 7 7
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
T
Y
Y
T
Y
Proof. It is easy to see, that the set
D
(
)
=
T
7,
T
6,
T
5,
T
4,
T
3,
T
2,
T
1
is a generating set of the semilatticeD
. Then the following equalities,
,
(
)
=
,
,
(
)
=
,
,
,
(
)
=
,
,
,
=
)
(
7 6 44 5 6 7 5 6 7 6 7
7
T
D
T
T
D
T
T
T
D
T
T
T
D
T
T
T
T
,
,
,
,
,
,
,
=
)
(
,
,
,
,
,
=
)
(
,
,
,
,
=
)
(
1 2 3 4 5 6 7 1 2 4 5 6 7 2 3 5 6 7 3T
T
T
T
T
T
T
D
T
T
T
T
T
D
T
T
T
T
D
T T T
,
,
,
,
,
,
,
,
=
)
(
7 6 5 4 3 2 1 01
T
T
T
T
T
T
T
T
D
T.
,
,
,
,
,
,
1 1 2 3 4 5 6 7 2 2 4 5 6 7 3 3 5 6 7 4 4 6 7 5 5 6 7 6 6 7 7 7T
Y
Y
Y
Y
Y
Y
Y
T
Y
Y
Y
Y
Y
T
Y
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
T
Y
Y
T
Y
For the last conditions we have
2 4 5 2 4 6 7 5 6 7 2 4 5 67
Y
Y
Y
Y
=
Y
Y
Y
Y
Y
Y
Y
T
T
Y
Y
2 2 2
=
T
Y
T
1 1 2 1 1 2 3 4 5 1 2 3 5 6 7 4 6 7 5 6 7 1 2 3 4 5 6 7=
=
T
Y
Y
T
Y
Y
T
T
T
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
is always satisfied.
Now for using limiting element definition we find,
(
)
,
=
(
)
\
=
,
\
(
)
,
7
=
7\
=
7,
7 7
7 7 7
7
T
D
T
T
l
D
T
T
T
D
l
T
T
T
(
)
,
=
(
)
\
=
,
\
(
)
,
6
=
6\
7,
6 6 7 6 6 6
6
T
D
T
T
T
l
D
T
T
T
D
l
T
T
T
D
(
)
5,
T
5
=
D
(
)
5\
T
5
=
T
6,
T
5\
l
D
(
)
5,
T
5
=
T
5\
T
6
,
l
T
T
T
(
)
,
=
(
)
\
=
,
\
(
)
,
4
=
4\
6,
4 4 6 4 4 4
4
T
D
T
T
T
l
D
T
T
T
D
l
T
T
T
D
(
)
3,
T
3
=
D
(
)
3\
T
3
=
T
5,
T
3\
l
D
(
)
3,
T
3
=
T
3\
T
5
,
l
T
T
T
D
(
)
2,
T
2
=
D
(
)
2\
T
2
=
T
2,
T
2\
l
D
(
)
2,
T
2
=
T
2\
T
2=
,
l
T
T
T
(
)
,
=
(
)
\
=
,
\
(
)
,
1
=
1\
1=
,
1 1 1 1 1 1
1
T
D
T
T
T
l
D
T
T
T
D
l
T
T
T
D
(
)
0,
T
0
=
D
(
)
0\
T
0
=
T
1,
T
0\
l
D
(
)
0,
T
0
=
T
0\
T
1
.
l
T
T
TTherefore,
T
7,
T
6,
T
5,
T
4,
T
3 andT
0 are nonlimiting elements of the sets 34 5
6
7
,
(
)
,
(
)
,
(
)
,
(
)
)
(
TD
TD
TD
TD
TD
and0
)
(
TD
respectively. By the statementc
)
Theorem 1.9it follows, that the conditions
6 5 5 4 4 3 3 0 06
T
,
Y
T
,
Y
T
,
Y
T
,
Y
T
Y
are hold. As a result of these we have,
.
,
,
,
,
,
,
,
,
,
0 0 3 3 4 4 5 5 6 6 3 3 5 6 7 4 4 6 7 5 5 6 7 6 6 7 7 7
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
T
Y
Y
T
Y
(2.2)Theorem 2.5
E
X r
D
are the set of all right unit elements of the semigroupB
X
D
.
IfX
be a finite set, then we calculate the number of right unit elements through this formula
4\7 6\ 7 5\ 4 5\4 4\3 4\3
3
2
1
3
2
3
2
2
=
T T T T T T T T T T T T Tr X
D
E
.
8
3
8
3
4
T3\T2 T3\T2 T0\T1 T0\T1
X\T0
Proof. Assume that
E
Xr
D
and from Corollary 2.4, we have that
has a quasinormal representation as=
Y
TT
,
D T
.
,
,
,
,
,
0 0 3
3
4 4 5
5 6
6 3 3 5 6 7
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
Y
Y
Y
Further, let
f
be a mapping from the setX
in the semilatticeD
satisfying the conditionf
(
t
)
=
t
for allt
X
.
We can definef
0,
f
1,
f
2,
f
3,
f
4,
f
5 andf
6 which are the restrictions of the mapping
f
on the setsT
7,
T
3
T
4
\
T
7,
T
5\
T
4,
T
4\
T
3,
T
3\
T
2,
T
0\
T
1 andX
\
T
0,
respectively. It is clear, that the intersection disjoint sets of the set
T
7,T
3
T
4\
T
7,
T
5\
T
4,
T
4\
T
3,
T
3\
T
2,
T
0\
T
1,
X
\
T
0
are empty set and
T
T
T
T
T
T
T
T
T
T
T
X
T
X
T
3
4\
7
5\
4
4\
3
3\
2
0\
1
\
0=
Now, we are going to find properties of maps
f
0,
f
1,
f
2,
f
3,
f
4,
f
5 andf
6.
1. Let
t
T
7 . Then, by using Equation 2.2 and the definition of the setY
7 , we have 70 7
7
Y
f
(
t
)
=
T
T
t
for allt
T
7.
2. Let
t
T
3
T
4
\
T
7.
Then, by using Equation 2.2 and definition of the sets
7
Y
andY
6, we have
6 7 3 5 6 7 4 6 7 4 3 7 4
3
T
\
T
T
T
Y
Y
Y
Y
Y
Y
Y
=
Y
Y
T
t
7 6
1
(
t
)
T
,
T
f
for allt
T
3
T
4
\
T
7.Suppose that
Y
6
T
6
andt
1
Y
6
T
6
t
1
Y
6 andt
1
T
6
t
1
=
T
6 andt
1
T
6.
If 71
T
t
thent
1
T
7
Y
7
f
1(
t
1)
=
T
7 for somet
1
T
7.
Which contradicts witht
1
=
T
6(
T
6 is not equal toT
7 inD
). SoY
6
T
6
T
3
T
4
\
T
7 andf
1(
t
1)
=
T
6 for somet
1
T
6\
T
7.
3. Let
t
T
5\
T
4.
Then, by using Equation 2.2 and definition of the sets
6 7
,
Y
Y
andY
5 , we have
5 6 7 5 4
5
\
T
T
Y
Y
Y
T
t
f
2(
t
)
=
t
T
7,
T
6,
T
5
for allt
T
5\
T
4.Suppose that
Y
5
T
5
andt
2
Y
5
T
5
t
2
Y
5 andt
2
T
5
t
2
=
T
5.
Ift
2
T
4 then.
4 6 7 4 2
Y
Y
Y
T
t
Therefore,t
2
T
7,
T
6,
T
4
.
Which contradicts of the equalityt
2
=
T
5,
since
7,
6,
4
.
5
T
T
T
T
Sof
2(
t
2)
=
T
5 for somet
2
T
5\
T
4.
4. Let
t
T
4\
T
3.
Then, by using Equation 2.2 and definition of the setsY
7,
Y
6 andY
4, we have
T
4\
T
3T
4t
Y
7
Y
6
Y
4
t
T
7,
T
6,
T
4
.
Thereforef
3(
t
)
T
7,
T
6,
T
4
for all.
\
34
T
T
t
Suppose that
Y
4
T
4
andt
3
Y
4
T
4
t
3
Y
4 andt
3
T
4
t
3
=
T
4.
Ift
3
T
3 then.
3 5 6 7 3 3
Y
Y
Y
Y
T
t
Therefore,t
3
T
7,
T
6,
T
5,
T
3
.
Which contradicts of the equality,
=
43
T
t
sinceT
4
T
7,
T
6,
T
5,
T
3
.
Sof
3(
t
3)
=
T
4 for somet
3
T
4\
T
3.
5. Let
