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ISSN No:-2456-2165
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
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