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Volume 4, Issue 3, March – 2019 International Journal of Innovative Science and Research Technology

ISSN No:-2456-2165

Structure of an Idempotent M-Normal


Commutative Semigroups
[1]
R.Rajeswari [2]D.Deva Margaret Helen [3]G.Soundharya
[1]
Assistant Professor of Mathematics [2][3]Students of I.M.Sc. Mathematics
Thassim Beevi Abdul Kader College For Women
Kilakarai,Ramanathapuram
Tamilnadu,India

Abstract:- This paper concerned with basic concepts G. Definition: A Semigroup S is said to be right quasi
and some results on idempotent m-normal normal if it satisfy the identity abc=abac for all a,b,c ϵ
commutative semigroup satisfying the identities of the S.
three variables.This is used to frame some structure for
m normal commutative semigroups,here we consider III. MAIN RESULTS
the semigroup satisfying some properties of m normal
commutative semigroup,left quasi normal and right A. Definition:
quasi normal.If it is right and left rectangular then it is An idempotent commutative semigroup (S, .) is said
semilattice. to be m-power left normal if it satisfy the identity
abmcm=acmbm..
Keywords:- Lattice, RegularSemigroup, Idempotent,
Normal, Quasi. B. Definition:
An idempotent commutative semigroup (S, .) is said
I. INTRODUCTION to be m-power left quasi normal if it satisfy the identity
abmcm=acmbmcm..
The concept of a semigroup is very simple and plays
an important role in the development of mathematics. The C. Definition:
theory of semigroup is similar to group theory and ring An idempotent commutative semigroup (S, .) is said
theory. The earliest major contribution to the theory of to be m-power regular if it satisfy the identity
semigroups are strongly motivated by comparisons with abmcma=abmacma.
group and rings. In this paper the results of ring theory
were adopted for semigroups. D. Definition:
An idempotent commutative semigroup (S, .) is said
II. PRELIMINARIES to be m-power left semi regular if it satisfy the identity
abmcma=abmacmabmcma.
In this section we present some basic concept of
idempotent m normal semigroups and some basic definition E. Definition:
of semigroups. An idempotent commutative semigroup (S, .) is said
to be m-power left semi normal if it satisfy the identity
A. Definition: A semigroup (S, .) is said to be singular if abmcma=acmbmcma.
it satisfy the identity ab=a(ab=b) for all a,b in S.
F. Definition:
B. Definition: A semigroup (S, .) is said to be rectangular An idempotent commutative semigroup (S, .) is said
if it satisfy the identity aba=a for all a,b in S. to be total if it satisfy the identity S2=S

C. Definition: A semigroup (S, .) is said to be regular if it G. Note:


satisfy the identity aba=ab(aba=ba) for all a,b in S. An idempotent m power commutative semigroup is
singular then it satisfy the semi lattice condition.
D. Definition: A semigroup (S, .) is said to be total if it
satisfy the identity S2=S. H. Lemma:
An Idempotent m-power commutative semigroup S is
E. Definition: A Semigroup S is said to be normal if it left(right) normal if and only if it is left(right) quasi normal.
satisfy the identity abca=acba for all a,b,c ϵ S.
 Proof:
F. Definition: A Semigroup S is said to be left quasi Let (S, .) be an idempotent m power commutative
normal if it satisfy the identity abc=acbc for all a,b,c ϵ semigroup and (S, .) be left(right) normal.
S. We have to prove (S, .) be left(right) quasi normal,

IJISRT19MA618 www.ijisrt.com 758


Volume 4, Issue 3, March – 2019 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
Then ,  Proof :
abmcm =acmbm Let (S, .) be an idempotent m power commutative
abmcmcm= acmbmcm semigroup,assume that (S, .) is left quasi normal.
abmcm=acmbmcm Then,
abmcm =acmbmcm
abmcm =acmbmcm
Therefore (S, .) is a left quasi normal. abmcma=acmbmcma(post multiply by a)
Conversely, =abmcmcma
=abmcma
Let (S, .) is a left quasi normal. =abmbmcma
We have to prove that (S, .) be a normal. =abmcmbma.

abmcm=acmbmcm Hence (S, .) is right semi normal.


=acmcmbm Conversely,
=acmbm. Suppose that (S, .) is right semi normal
ab c =acmbm.
m m
Then,
Hence proves the lemma.
abmcma =abmcmbma
I. Theorem: a.abmcm =acmbmbma
m m m m
An idempotent m power commutative semigroup S is ab c =ac b a
a left quasi normal if and only if it is left semi regular. =acmcmbm
= acmbmcm.
 Proof: ab c =acmbmcm
m m

Let us consider (S, .) is a left quasi normal,


We have to prove it is left semi regular hence (S, .) is left quasi normal.
Then This proves the theorem.

abmcm a=abmcmcma K. Theorem:


=a.acmbmcmbma (a2=a) An idempotent m-power commutative semigroup (S,
=abmacmbmcma .) is left(right) quasi normal if and only if it is right semi
=abma.acmbmcma regular.
=abmacmabmcma  Proof:
abmcma= abmacmabmc ma.
Hence (S, .) is left semi regular. Let (S, .) be an idempotent commutative semigroup
and assume that (S, .) is left quasi normal.
Conversely, We have to prove (S, .) is right semi-regular.
Suppose that (S, .) is left semi regular. Then,
We have to prove that it is left quasi normal
abmcm=acmbmcm
m m m m m m
ab c a = ab ac ab c a abmcma=aacmbmbmcma
=aabmcmabmcma =aabmcmbmcmaa
=abmacmabmcma =abmcmabmacma
=abmcmaabmcma . abmcma=abmcmabmacma.
ab c =abmcmabmcma
m m

=abmcmbmacma Hence (S,.) is right semi regular.


=abmbmcmcmaa Conversely,
=abmcmcma Let (S, .) is right semi regular
m m m
=ac b c a Then,
=acmbmcm (a2=a idempotent condition)
abmcm =acmbmcm. abmcma=abmcmabmacma.
aabmcm=acmbmbmabmcmaa
hence (S, .) is a left quasi normal abmcm =acmbmbmacma
This proves the theorem. =acmbmcmaa
=acmbmcma
J. Theorem: ab c =acmbmcm.
m m

An idempotent m-power commutative semigroup (S,


.) is left(right) quasi normal if and only if it is right semi Hence it is left quasi normal.
normal. This proves the theorem.

IJISRT19MA618 www.ijisrt.com 759


Volume 4, Issue 3, March – 2019 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
L. Lemma:
An idempotent commutative m power semigroup is REFERENCES
right and left singular then it is semi lattice
[1]. Hall T.E(1973) “ On regular semigroup” J.Algebra 24.
 Proof : [2]. Grillet,P.A(1974) “The structure of regular
Let (S, .) be a semigroup with left and right regular semigroups 1”,semigroup forum 8 ,177-187.
Then, [3]. Hall,T.E “On regular semigroups whose structure
forms subsemigroup”.
abma=abm→① [4]. Howie,J.M. “Fundamentals of semigroup theory”.
abma=bma→② [5]. P.Sreenivasalu Reddy and Mulugeta dawud “Structure
from ①and②we get of Regular Semigroups”.
abm=bma [6]. D.Radha, P.Meenakshi “Some Structures of
so (S, .) is commutative. idempotent commutative semigroups”.
Now to prove (S, .)is a band [7]. Special classes of semigroups by Attila Nagy.
abma=abm [8]. On (m,n) Regular Semigroups Dragica N.Krogvic.
a(bma)=abm
abm(bma)=abm
a(bm)2a=abm
a2(bm)2=abm
(abm)2=abm.
So (S, .) is a band.

Hence it is semi lattice since it is a commutative band.


This proves the lemma.

M. Corollary 1:
An idempotent m power commutative semigroup
satisfies the singular property then it is semi lattice.

N. Corollary 2:
An idempotent m power commutative semigroup is
right(left) singular then it is regular.

O. Corollary 3:
An idempotent m power commutative semigroup is
rectangular then it is semi regular.

P. Lemma:
An idempotent commutative semigroup(S, .) is m
power left singular then it is rectangular

 Proof:
Let an idempotent commutative semigroup is m power
right singular
We have to prove it is rectangular.
abm=a
post multiply by a on both sides
abma=a.a
=a2
abma=a(idempotent condition a2=a)
This proves the lemma

IV. CONCLUSION

The structure of m normal commutative semigroup


gives the proper structure to the semigroup.This structure is
the basement for further works in m normal
semigroups.This paper is very useful for the scholars who
will do their work in normal semigroup and this will be
applied in many chemical industries.

IJISRT19MA618 www.ijisrt.com 760

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