The Fuzzy Regularity of Bilinear Form Semigroups

Proceedings of ”The 6th SEAMS-UGM Conference 2011”
Topic, pp. xx–xx.

THE FUZZY REGULARITY OF
BILINEAR FORM SEMIGROUPS
1ST KARYATI, 2ND SRI WAHYUNI, 3RD BUDI SURODJO, 4TH SETIADJI
Abstract. A bilinear form semigroup is a semigroup constructed based on a bilinear form.
In this paper, it is established the properties of a bilinear form semigroup as a regular
semigroup. The bilinear form semigroup will be denoted by
. We get some properties
of the fuzzy regular bilinear form subsemigroup of
, i.e: For every is a fuzzy regular
bilinear form subsemigroup of
if and only if
,
is a regular subsemigroup
of
, provided
. If
is a nonempty subset of
, then

is a regular
subsemigroup of
if and only if , the characteristics function of , is a fuzzy regular
bilinear form subsemigroup of
. If is a fuzzy regular bilinear form subsemigroup of
, then
.
Keywor ds a nd Phr a ses : fuzzy subsemigr oup, fuzzy r egula r bilinea r for m subsemigr oup

1. INTRODUCTION
The basic concept of fuzzy subset is established by Zadeh since 1965
(Aktas; 2004). The concept has been developed in many areas of algebra
structures, such as fuzzy subgroup , fuzzy subring, fuzzy subsemigroup, etc.
Refers to the articles of Asaad (1991), Kandasamy (2003), Mordeson & Malik
(1998), Ajmal (1994), Shabir (2005) , we define the fuzzy subset
of a set
that is a mapping from into the interval
, i.e.
. Let
be a

fuzzy subset of
and
, then the level subset
is defined as set of all
elements in such that their images of the fuzzy subset are more than equal to
, i.e.
. Moreover, let be a semigroup, then a mapping
is called a fuzzy subsemigroup of a semigroup if
(1)
The property of a level subset that is a fuzzy subsemigroup of a semigroup
if and only if for every nonempty level subset is a subsemigroup of .
Let
and be vector spaces over field , with the characteristics of
is zero. A function
is called a bilinear form if is linear with
respect to each variable. Every bilinear form determines two
linear
, which is defined as
and
transformation,

, which is defined as
. In this case,
and
are
the dual spaces of
and , respectively. In this paper, we denote
and
as a set of all linear operator of and , respectively. For
, we
can construct the following sets:

1

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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

The element
the bilinear form
if

Further, we denote the following sets:

is called an adjoin pair with respect to
for all
and
.

Based on these sets, we denote a set as follow:

is a semigroup with respect to the binary operation
The structure of the set
define as
. The semigroup
is called a bilinear
form semigroup related to the bilinear form .
Let
be a semigroup and
, an element
is called a regular
element if there exists

such that
. A semigroup is called
regular if every element is regular. The element is called a completely regular
such that
and
. A semigroup is
if there exists
called a completely regular if every element is a completely regular element.
The purpose of this paper is to define the fuzzy regular bilinear form
subsemigroup and investigates the characteristics of it.

2. RESULTS
form

Let
and

be bilinear form semigroups, with respect to the bilinear
, respectively. For any
, we define a set as follow:

and

We define a fuzzy regular bilinear form subsemigroup of
Definition 2.1. If
there exists

is a fuzzy subsemigroup of
such that:

, as follow:

and every

(2)
then

is called a fuzzy regular bilinear form subsemigroup .

Based on the property of a fuzzy subsemigroup, we can investigate the

following proposition:
Proposition 2.1. The fuzzy subsemigroup
subsemigroup of
if and only if
of bilinear form semigroup
.

is a fuzzy regular bilinear form
,
is a regular subsemigroup

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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

is a
Proof: It is always satisfy that is a fuzzy subsemigroup if and only if
subsemigroup of In fact, if is a fuzzy regular bilinear form subsemigroup
of
for every

, then there exists
such that (2)
is hold, i.e. :
.
Furthermore,
,
Hence, we obtain
semigroup
.

i.e.

.
is a regular subsemigroup of bilinear form

Conversely, suppose
,the level subset
is a regular
subsemigroup of bilinear form semigroup
, provided

. On the other
such that
and
,
hand there is
. Set
. Clearly that
and for every
,
. This contradicts the fact that
is a regular
. So, the equation (2) is hold.
subsemigroup of bilinear form semigroup

The following proposition give the relation between the nonempty subset
of bilinear form semigroup
and the characteristic function of this subset.
Proposition 2.2. If is a nonempty subset of
, then
is a regular

subsemigroup of bilinear form semigroup
if and only if
the
characteristic function of , is a fuzzy regular bilinear form subsemigoup of
.
Proof: If
is a regular subsemigroup of bilinear form semigroup
the following equation is hold

and if

implies

i.e.
regular subsemigroup of
there exists
Therefore,
From this point we have
semigroup
.

Conversely, if
for every

, then

. For every
, if
then
. Furthermore, since is a
, then for every
bilinear form semigroup
such that
or
.

is a fuzzy regular bilinear form subsemigroup of

is a fuzzy regular bilinear form subsemigroup of
, we get
.

Hence,
.

, then

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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

Thus,

and implies

.

In addition, if
Finally we conclude that
form
.

. Thus

then we obtain
.
is a regular subsemigrup bilinear


The following proposition give the condition of a semigroup
homomorphism, such that the image of a fuzzy regular bilinear form of
is
a fuzzy regular bilinear form subsemigroup of
too, and vise versa.
Proposition 2.3. Let
.
onto

be a semigroup surjective homomorphism from

1. If is fuzzy regular bilinear form subsemigroup of
regular bilinear form subsemigroup of
2. If is a fuzzy regular bilinear form subsemigroup of
a fuzzy regular bilinear form subsemigroup of

, then
, then

is fuzzy
is

Proof:
is a regular subsemigroup bilinear form
1. For every
, provided
. In the fact, if there is
such that
is not a nonempty set and is not regular semigroup, then there is
such that:

,

,i.e.

and
,

,

and

Let

:

and

Now there is

with
i.e.

For any

,
with

and
.
, we have

or

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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

Clearly,

,
and so
i.e.
is not regular. This is contradicts that
regular bilinear form subsemigroup of semigroup

2. If is a fuzzy regular bilinear form subsemigroup of semigroup
for every
,
, there exists
that
. Since
is surjective, for every
there exists
such that
.
For every
,
, there exists

,
is a fuzzy

, then
such

with

Based on the previous property, we can conclude that for every
, there exists
Furthermore, we have:

Finally, based on (2),
.
of semigroup

Proposition 2.4. If
semigroup
, then

such that

.

is a fuzzy regular bilinear form subsemigroup


is a fuzzy regular bilinear form subsemigroup of

Proof: It is always hold
then

. For every

, if

that implies

If
, then there exists
bilinear form subsemigrup, so we have

It is proved that

, so

. Since

.

is a fuzzy regular
. Hence



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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

3. CONCLUDING REMARK
Based on the discussion, by defining a fuzzy regular bilinear form
subsemigroup, we have investigate and prove the following results.
1. The fuzzy semigroup
if and only if
form semigroup

is a fuzzy regular bilinear form subsemigroup of
,
is a regular subsemigroup of bilinear

2. If is a nonempty subset of
, then
is a regular subsemigroup of
bilinear form semigroup
if and only if
, the characteristic function
of , is a fuzzy regular bilinear form subsemigroup of
3. Let be a semigroup surjective homomorphism from
onto
a. If
is fuzzy regular bilinear form subsemigroup of
, then
is
fuzzy regular bilinear form subsemigroup of
b. If
is a fuzzy regular bilinear form subsemigroup of
, then
is a fuzzy regular bilinear form subsemigroup of
4. If

is a fuzzy regular bilinear form subsemigroup of

, then

References
[1] AJMAL, NASEEM., Homomorphism of Fuzzy groups, Corrrespondence Theorm and
Fuzzy Quotient Groups. Fuzzy Sets and Systems 61, p:329-339. North-Holland. 1994
[2] AKTAŞ, HACI, On Fuzzy Relation and Fuzzy Quotient Groups. International Journal
of Computational Cognition Vol 2 ,No 2, p: 71-79. 2004
[3] ASAAD, MOHAMED, Group and Fuzzy Subgroup. Fuzzy Sets and Systems 39, p:323328. North-Holland. 1991
[4] HOWIE, J.M., An Introduction to Semigroup Theory. Academic Press, Ltd, London,
1976
[5] KANDASAMY, W.B.V., Smarandache Fuzzy Algebra . American Research Press and
W.B. Vasantha Kandasamy Rehoboth. USA. 2003
[6] KARYATI, S.WAHYUNI, B. SURODJO, SETIADJI , Ideal Fuzzy Semigrup. Seminar
Nasional MIPA dan Pendidikan MIPA di FMIPA, Universitas Negeri Yogyakarta,
tanggal 30 Mei 2008. Yogyakarta, 2008.
[7] KARYATI, S. WAHYUNI , B. SURODJO, SETIADJI, The Fuzzy Version Of The
Fundamental Theorem Of Semigroup Homomorphism. The 3rd International Conference
on Mathematics and Statistics (ICoMS-3)Institut Pertanian Bogor, Indonesia, 5-6
August 2008. Bogor, 2008.
[8] KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI, Beberapa Sifat Ideal Fuzzy
Semigrup yang Dibangun oleh Subhimpunan Fuzzy. Seminar Nasional Matematika,

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KARYATI, S. WAHYUNI, B. SURODJO, SETIADJI

FMIPA, UNEJ . Jember, 2009.

[9] MORDESON, J.N AND MALIK, D.S. Fuzzy Commutative Algebra. World Scientifics
Publishing Co. Pte. Ltd. Singapore, 1998.

KARYATI: Ph. D student, Mathematics Department, Gadjah Mada University
E-mails: yatiuny@yahoo.com

SRI WAHYUNI : Mathematics Department, Gadjah Mada University
E-mails: swahyuni@ugm.ac.id

BUDI SURODJO: Mathematics Department, Gadjah Mada University
E-mails: surodjo_b@ugm.ac.id

SETIADJI: Mathematics Department, Gadjah Mada University