Regular Elements of the Complete
Semigroups of Binary Relations
DidemYeşil Sungur
#1,GiuliTavdgiridze
*2, Barış Albayrak
#3, Nino Tsinaridze
*41
Canakkale Onsekiz Mart University, Faculty of Science and Art, Department of Mathematics,TURKEY 2,4
Shota Rustaveli BatumiState University, Faculty of Mathematics, Physics and Computer Sciences,GEORGİA 3
Canakkale Onsekiz Mart University, Biga School of Applied Sciences, TURKEY
Abstract—In this paper, we investigate regular elements properties in given semilattice
T
T
T
mT
mT
mT
m
Q
=
1,
2,
,
3,
2,
1,
. Additionally, we will calculate the number of regular elements of)
(
D
B
X for a finite setX
.
Keywords—Semigroups, Binary relation, Regular elements.
I.INTRODUCTION
Let
X
be an arbitrary nonempty set andB
X be semigroup of all binary relations on the setX
.
IfD
is a nonempty set of subsets ofX
which is closed under the union thenD
is called a completeX
semilattice of unions.Let
x
,
y
X
,
Y
X
,
B
X,
T
D
,
D
'
D
andt
D
.
Then we have the following notations,
)
,
(
=
)
,
(
},
forany
|
{
=
)
,
(
|
=
,
|
=
},
|
{
=
}
|
{
=
)
,
(
,
=
},
)
,
(
|
{
=
..
' '
' ' '
'
' ' ' ' T ' '
' ' T ' '
t
Y y
D
D
N
D
D
D
Z
Z
Z
D
Z
D
D
N
T
Z
D
Z
D
Z
T
D
Z
D
Z
t
D
Z
D
D
Y
Y
D
V
y
Y
x
y
X
x
y
Let
f
be an arbitrary mapping fromX
intoD
.
Then one can construct a binary relation
f onX
by
(
)
.
=
x
f
x
X x
f
The set of all such binary relations is denoted byB
X(
D
)
and called a complete semigroup of binary relations defined by anX
semilattice of unionsD
.
This structure was comprehensively investigated in Diasamidze [1].A complete
X
semilattice of unionsD
is anXI
semilattice of unions if
(
D
,
D
t)
D
for anyD
t
and=
(
,
t)
Z t
D
D
Z
for any nonempty elementZ
ofD
.
)
(
D
B
X
is idempotent if
=
and
B
X(
D
)
is regular if
=
for some)
(
D
B
X
.Let
D
' be an arbitrary nonempty subset of the completeX
semilattice of unionsD
.
Set).
\
(
=
)
,
(
T'' '
D
D
T
D
l
We say that a nonempty elementT
is a nonlimiting element ofD
' if
)
,
(
\
l
D
T
T
' . Also, a nonempty elementT
said to be limiting element ofD
' ifT
\
l
(
D
',
T
)
=
.
Let
D
=
D
,
Z
1,
Z
2,
,
Z
m1
be finiteX
semilattice of unions andC
D
=
P
0,
P
1,
P
2
,
P
m1
be the family of pairwise nonintersecting subsets ofX
.
If
1 1
0
1 1
=
m m
P
P
P
Z
Z
D
is a mapping fromD
on
D
C
thenD
=
P
0
P
1
P
2
P
m1
and
Z
P
T
Z D D T
i
\ 0
=
satisfy.In this paper, we take in particular
Q
=
T
1,
T
2,
,
T
m3,
T
m2,
T
m1,
T
m
subsemilattice ofX
semilattice of unionsD
which the elements are satisfyingm m
m
T
T
T
T
T
T
1
3
5
3
1
,T
1
T
3
T
5
T
m3
T
m2
T
m ,m m
m
T
T
T
T
T
T
2
4
5
3
1
,T
2
T
4
T
5
T
m3
T
m2
T
m ,m m
m
T
T
T
T
T
T
2
3
5
3
1
,T
2
T
3
T
5
T
m3
T
m2
T
m , ,3 1 2
T
=
T
T
T
4
T
3=
T
5 ,T
m2
T
m1=
T
m,T
2\
T
1
,T
1\
T
2
,T
4\
T
3
andT
3\
T
4
,T
m2\
T
m1
,T
m1\
T
m2
.
We will investigate the properties of regular element
B
X(
D
)
satisfyingV
(
D
,
)
=
Q
. Moreover, we will calculate the number of regular elements ofB
X(
D
)
for a finite setX
.
Theorem 1.1 [2, Theorem 10] Let
and
be binary relations of the semigroupB
X(
D
)
such that.
=
IfD
(
)
is some generating set of the semilatticeV
(
D
,
)
\
and)
(
=
) , (
T
Y
T D V T
is a quasinormal representation of the relation
,
thenV
(
D
,
)
is a complete
XI
semilattice of unions. Moreover, there exists a complete isomorphism
between the semilattice)
,
(
D
V
andD
'=
T
|
T
V
(
D
,
)
,
that satisfies the following conditions:1.
T
=
T
and
T
=
T
for allT
V
(
D
,
)
2.
(
)
) ( ..
T
Y
'T
T D ' T
for anyT
D
(
),
3.
Y
T
(
T
)
for all nonlimiting element
T
of the setD
T,
4. If
T
is a limiting element of the setD
T,
then the equality
B
T
=
T
is always holds for the set
T
=
Z
D
|
Y
(
T
)
B
T Z
.On the other hand, if
B
X(
D
)
such thatV
(
D
,
)
is a completeXI
semilattice of unions and if some complete
isomorphism
fromV
(
D
,
)
to a subsemilatticeD
' ofD
satisfies the conditions)
)
d
b
of the theorem, then
is a regular element ofB
X(
D
).
Theorem 1.2 [1, Theorem 6.3.5] Let
X
be a finite set. If
is a fixed element of the set
(
D
,
D
')
and0
=
)
(
D
m
andq
is a number of all automorphisms of the semilatticeD
then.
)
,
(
=
)
(
0' '
D
D
R
q
m
D
R
II. RESULTS
Let
X
be a finite set,D
be a completeX
semilattice of unions and
T
T
T
T
mT
mT
mT
m
. 1 2
5, 3 4
3, 1 2 2
1 1
2
4 3 3
4 2
1 1
2
2 3
6 5 3 2
1 3
6 5 3 2
2 3
6 5 4 2
1 3
6 5 4 2
2 3
6 5 3 1
1 3
6 5 3 1
=
,
=
=
,
\
,
\
,
\
,
\
,
\
,
\
,
,
,
,
,
,
m m
m m m m
m
m m
m
m m
m
m m
m
m m
m
m m
m
m m
m
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
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
C
Q
=
P
i|
i
=
1,2,
,
m
. Then 1 21
=
P
P
P
P
T
m m
m
m
,1 2
1
=
P
P
P
T
m m
m
1 1
2
=
P
P
P
T
m m
m
,...,1 2 3
4
=
P
P
P
P
T
m
,=
3
T
P
m
P
4
P
2
P
1,1
2
=
P
P
T
m
,2 4
1
=
P
P
P
T
m
are obtained.
First, we investigate that in which conditions
Q
isXI
semilattice of unions. We determine the greatest lower bounds of the each semilatticeQ
t inQ
fort
T
m.
We get,
1 2
3
2 1
3
3 4
4 1
3 5
4 3
2 1
3 2
1
2 1
1 2
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
=
P
t
T
T
T
P
t
T
T
T
P
t
T
T
P
t
T
T
T
T
P
t
T
T
T
T
P
t
T
T
T
P
t
T
T
P
t
T
T
P
t
Q
Q
m m m m
m m
m m m
m m
m m
m m
m
m m
m
m
t
(2.1)From the Equation (2.1) the greatest lower bounds for each semilattice
Q
t1
T
T
23
T
T
45
T
6
T
3
m
T
2
m
T
T
m1m
T
2 21 2 4 1 4 3 1 1 4 3 1 3 4 3 1 3 3 1 1 3 1 2 2 1 3 2 1
=
)
,
(
=
)
,
(
=
)
,
(
=
)
,
(
=
)
,
(
,
,
=
)
,
(
=
)
,
(
=
)
,
(
=
)
,
(
,
,
=
)
,
(
=
)
,
(
,
,
=
)
,
(
=
)
,
(
,
,
,
=
)
,
(
=
)
,
(
,
,
,
=
)
,
(
=
)
,
(
=
)
,
(
T
Q
Q
T
Q
Q
N
P
t
Q
Q
Q
Q
N
P
t
T
Q
Q
T
T
Q
Q
N
P
t
T
Q
Q
T
Q
Q
N
P
t
T
Q
Q
T
T
Q
Q
N
P
t
T
Q
Q
T
T
Q
Q
N
P
t
T
Q
Q
T
T
T
Q
Q
N
P
t
T
Q
Q
T
T
T
Q
Q
N
P
t
Q
Q
Q
Q
N
P
t
t t t t t t t t m t m t m m t m t m m t m m t m m t m m t m t t m
(2.2)are obtained. If
t
P
m ort
P
2,
then
(
D
,
D
t)
=
D
.
So,P
m
P
2=
. Also using the Equation (2.2), we have seen easily(
Q
,
Q
t)
D
.
i T t
Lemma 2.1
Q
isXI
semilattice of unions if and only ifT
1
T
4=
Proof.
:
LetQ
be aXI
semilattice of unions. ThenP
m
P
2=
andT
1=
P
2,T
4=
P
1
P
3 by Equation (2.1). ThereforeT
1
T
4=
sinceP
1,
P
2 andP
3 are pairwise disjoint sets.:
IfT
1
T
4=
,
thenP
m
P
2=
. Using the Equation (2.2), we see that(
,
t)
=
i.
i T tT
Q
Q
So, wehave
Q
isXI
semilattice of unions.Lemma 2.2 Let
G
=
T
1,
T
2,
,
T
m1
be a generating set ofQ
. Then the elementsT
1,
T
2,
T
4,
T
6,
,
T
m1are nonlimiting elements of the sets
,
1
T
G
,
2
T
G
,
4
T
G
1 6
,
,
TmT
G
G
respectively andT
3,
T
5 is limiting element sof the sets5 3
,
TT
G
G
respectively. Proof. Definition of' T
D
..
and
(
,
)
=
(
\
i),
iT i
i
T
T
G
T
G
l
i
1,2,
,
m
1
, we find nonlimiting and limiting elements ofi T
G
.Now, we determine properties of a regular element
ofB
X(
Q
)
whereV
(
D
,
)
=
Q
and).
(
=
1 =
i i m
i
T
Y
Theorem 2.3 Let
B
X(
Q
)
with a quasinormal representation of the form
=
(
)
1 =
i i m
i
T
Y
such thatQ
D
V
(
,
)
=
. Then
B
X(
D
)
is a regular iffT
1
T
4=
and for some complete
-isomorphism 'D
Q
:
D
,
the following conditions are satisfied:,
)
(
,
)
(
,
)
(
,
)
(
,
)
(
,
)
(
),
(
),
(
),
(
),
(
),
(
),
(
),
(
),
(
1 1
2 2
3 3
4 4
6 6
4 4
1 1
3 2
1
2 2
3 2
1
3 3
2 1
4 4
2 1
6 6
5 4 3 2 1
4 4
2 2 2
1 1
m m
m m
m m
m m
m m
m
m m
m
m m
m m
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
Y
Y
Y
T
Y
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
Y
Y
Y
T
Y
Y
T
Y
T
Y
(2.3)
Proof. Let
G
=
T
1,
T
2,
,
T
m1
be a generating set ofQ
.:
Since
B
X(
D
)
is regular andV
(
D
,
)
=
Q
isXI
semilattice of unions, by Theorem 1.1, there exits a complete
isomorphism
:
Q
D
'. By Theorem 1.1(
a
),
T
=
T
for allT
V
(
D
,
)
. Applying the Theorem 1.1(
b
)
and)
(
),
(
),
(
),
(
),
(
),
(
),
(
),
(
1 1
3 2
1
2 2
3 2
1
3 3
2 1
4 4
2 1
6 6
5 4 3 2 1
4 4
2 2 2
1 1
m m
m
m m
m
m m
m m
T
Y
Y
Y
Y
T
Y
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
T
Y
Y
Y
Y
Y
Y
T
Y
Y
T
Y
T
Y
Moreover, considering that the elements
T
1,
T
2,
T
4,
T
6,
,
T
m1 are nonlimiting elements of the setsG
T1,
,
2
T
G
,
4
T
G
1 6
,
,
TmT
G
G
respectively and using the Theorem 1.1(
c
)
, following properties
(
4)
,
6(
6)
,
4(
4)
,
,
1(
1)
4
T
Y
T
Y
mT
mY
mT
mY
are obtained. Therefore there exists a complete
isomorphism
which holds given conditions.:
SinceV
(
D
,
)
=
Q
,V
(
D
,
)
isXI
semilattice of unions. Let
:
Q
D
'
D
be complete
isomorphism which holds given conditions. So, considering Equation (2.3), satisfying Theorem 1.1).
(
)
(
a
c
Remembering thatT
3 andT
5 are limiting elements of the sets 3T
G
and 5T
G
, we constitute the set
=
|
(
3)
3
3
Z
G
Y
T
T
B
T Z
and
=
|
(
5)
.
5
5
Z
G
Y
T
T
B
T Z
It has been proved that
T
3=
T
3B
and
B
T
5=
T
5 in [?, Theorem 3.4]. By Theorem 1.1, we conclude that
is the regularNow we calculate the number of regular elements
, satisfying the hyphothesis of Theorem 2.3. Let)
(
D
B
X
be a regular element which is quasinormal representation of the form=
(
)
1 =
i i m
i
T
Y
and.
=
)
,
(
D
Q
V
Then there exist a complete
isomorphism
:
Q
D
'=
(
T
1),
(
T
2),
,
(
T
m)
satisfying the hyphothesis of Theorem 2.3. So,
R
(
Q
,
D
').
We will denote
(
T
i)
=
T
i,
i
=
1,2,
m
.
Diagram of the
D
'=
T
1,
T
2,
,
T
m
is shown in the following figure. Then the Equation (2.3) reduced tobelow equation.
)
(
,
)
(
,
)
(
,
)
(
,
)
(
,
)
(
),
(
),
(
),
(
),
(
),
(
),
(
1 1
2 2
3 3
4 4
6 6
4 4
1 1
3 2
1
2 2
3 2
1
6 6
5 4 3 2 1
4 4
2 2 2
1 1
m m
m m
m m
m m
m m
m
m m
m
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
T
Y
Y
Y
Y
T
Y
Y
Y
Y
T
Y
Y
Y
Y
Y
Y
T
Y
Y
T
Y
T
Y
(2.4)
On the other hand,
T
1,
T
2,
T
3\
T
4,
,
T
k1\
T
k
k
=
3,5,6,7,
,
m
5,
m
2
,
,
(
T
m1
52
)
\
m
m
T
T
,T
m1\
T
m2 ,T
m2\
T
m1,X
\
T
mare also pairwise disjoint sets and union of these setsequals
X
.
Lemma 2.4 For every
R
(
Q
,
D
'),
there exists an ordered system of disjoint mappings
=
3,5,6,7,
,
5,
2
\
,
,
\
,
,
{
T
1T
2T
3T
4
T
k1T
kk
m
m
,
,
}.
\
,
\
,
\
)
(
T
m1
T
m2T
m5T
m2T
m1X
T
mProof. Let
f
:
X
D
be a mapping satisfying the conditionf
(
t
)
=
t
for allt
X
.
We consider the restrictions of the mappingf
asf
1,
f
2,
f
4,
,
f
k,
,
f
(m3),
f
(m2),
f
(m1),
f
m on the setsk
k
T
T
T
T
T
T
1,
2,
3\
4,
,
1\
k
=
3,5,6,7,
,
m
5,
m
2
,
,
m m
m m m
m
T
T
T
T
X
T
T
)
\
,
\
,
\
(
1
2 5 2 1 respectively.Now, considering the definition of the sets
Y
i,
i
=
1,2,
,
m
1
together with the Equation (2.4) we have,1 1
1 1
1
t
Y
f
(
t
)
=
T
,
t
T
T
t
2 2
2 2
2
t
Y
f
(
t
)
=
T
,
t
T
T
t
1 2 3
3 4 43 2 1 4
3
\
T
t
Y
Y
Y
f
(
t
)
=
T
,
T
,
T
,
t
T
\
T
T
t
k
k kk
k k
k k k
k
T
T
t
T
T
T
t
f
Y
Y
Y
T
T
T
t
T
T
t
\
,
,
,
,
)
(
\
\
1 1
2 1
1 2
1 1 1
1
1
T T2
3
T T4
5
T
6
T
3 m
T
2 m
T Tm1
Tm4
m
5 2
1
3 1
3) (
3 2
1 2 1
5 2
1
\
)
(
,
,
,
)
(
\
)
(
m m
m
m m
m m
m
m m
m
T
T
T
t
T
T
t
f
Y
Y
Y
T
T
t
T
T
T
t
1 3 2
2 11) (
2 3 1
2 1
2
\
,
,
,
,
)
(
\
m m m
m m
m m m
m m
T
T
t
T
T
T
t
f
Y
Y
Y
T
t
T
T
t
m m
i m
i m
m
t
X
T
X
Y
f
t
Q
t
X
T
T
X
t
\
\
=
(
)
,
\
1 =
Besides,
Y
k1
T
k1
so there is an elementt
k1
Y
k1
T
k1.
Thent
k1
=
T
k1 andt
k1
T
k1.
Ift
k1
T
k thent
k1
T
k
Y
1
Y
k.
Thust
k1
T
1,
,
T
k
which is in contradiction with theequality
t
k1
=
T
k1.
So, there is an elementt
k1
T
k1\
T
k such thatf
k(
t
k1)
=
T
k1.Similarly,
f
(m3)(
t
m3)
=
T
m3 for somet
m3
(
T
m1
T
m2)
\
T
m5 ,f
(m1)(
t
m2)
=
T
m2 for some 12
2
\
m mm
T
T
t
. Therefore, for every
R
(
Q
,
D
')
there exists an ordered system)
,
,
,
(
f
1f
2
f
m .On the other hand, suppose that for
,
R
(
Q
,
D
')
which
,
be obtained)
,
,
,
(
=
1 2
f
f
f
mf
andf
=
(
f
1,
f
2,
,
f
m)
.
I
ff
=
f
, we get
=
f
f
(
t
)
=
f
(
t
),
t
X
t
=
t
,
t
X
=
f
which contradicts to
. Therefore different binary relations’s ordered systems are different.Lemma 2.5 Let
Q
be anXI
semilattice of unions andf
=
(
f
1,
f
2,
,
f
m)
be ordered system fromX
in the semilattice
D
such that
.
)
(
,
\
:
,
\
=
)
(
and
,
,
,
)
(
,
,
,
,
\
:
,
\
)
(
=
)
(
and
,
,
)
(
,
,
,
\
)
(
:
,
\
=
)
(
and
,
,
)
(
,
,
,
\
:
,
,
,
)
(
,
,
,
\
:
,
=
)
(
,
:
,
=
)
(
,
:
1
1 2 2
2 2
1
2 3 1
1 2 3 1
1 2 1
5 2
1 3
3 3
3
3 1
3 3 1
5 2
1 3
1 1 1 1
1 1
1 1
1
3 2 1 4
3 2 1 4 3 4
2 2
2 2 2
1 1
1 1 1
Q
t
f
Q
T
X
f
T
T
t
T
t
f
T
T
T
t
f
T
T
T
T
T
f
T
T
T
t
T
t
f
T
T
t
f
T
T
T
T
T
f
T
T
t
T
t
f
T
T
t
f
T
T
T
T
f
T
T
T
t
f
T
T
T
T
T
f
T
t
f
T
T
f
T
t
f
T
T
f
m m
m
m m m
m m
m
m m m
m m m
m m
m m
m m
m m
m
m m
m m
m m
m
k k k k k
k
k k
k k
k k
Then
=
(
x
f
(
x
))
B
X(
D
)
X x
