Partial Bi Semimodules over Partial Semirings

(1)

Partial Bi-Semimodules over Partial Semirings

V. Amarendra Babu

1

_{, M. Srinivasa Reddy}

2,*

_{, P.V.Srinivasa Rao}

2

1_{Department of Mathematics, Acharya Nagarjunana University, Guntur, 522510, Andhra Pradesh, India} 2_{D. V. R & Dr. H. S. MIC College of Technology, Kanchikacherla, 521180, Krishna(dt), Andhra Pradesh, India}

*Corresponding Author: [email protected]

Abstract

A partial semiring is a structure possessing an infinitary partial addition and a binary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a partial semiring. In this paper we introduce the notions of ( R, S ) - partial bi-semimodule and ( R, S ) - homomorphism of ( R, S ) - partial bi-semimodules and extended the results on partial semimodules over partial semirings by P. V. Srinivasa Rao [8] to ( R, S ) - partial bi-semimodules.

Keywords

( R, S ) - Partial Bi-Semimodule, (N : M ), ( R, S ) - Homomorphism, Bourne Relation, Steady ( R, S )- Homomorphism And Absorbing Subbi-Semimodule.

1. Introduction

Partially defined infinitary operations occur in the contexts ranging from integration theory to programming lanugage

semantics. The general cardinal algebras studied by Tarski in 1949, ∑- structures studied by Higgs in 1980, Housdorff topoligical commutative groups studied by Bourbaki in 1966, sum-ordered partial monoids and sum-ordered partial semirings studied by Arbib, Manes, Benson and Streenstrup are some of the algebraic structures of the above type.

The study of

pfn

(

D

,

D

)

(the set of all partial functions of a set D to itself),Mfn(D,D)(the set of all multi functions of a set

D

to itself) and Mset(D,D)(the set of all total functions of a set D to the set of all finite multi sets of D) play an important role in the theory of computer science, and to abstract these structures Manes and Benson[6] introduced the notion of sum ordered partial semirings(so-rings). Motivated by the work done in partially-additive semantics by Arbib, Manes[3] and in the development of matrix theory of so-rings by Martha E. Streenstrup[7], G. V. S. Acharyulu[1] in 1992 studied the conditions under which an arbitrary so-ring becomes a pfn(D,D), Mfn(D,D)and Mset(D,D). Continuing this study, P. V. Srinivasa Rao[9] in 2011 developed the ideal theory for so-rings and partial semimodules over partial semirings.

In this paper we introduce the notions of ( R, S ) - partial bi-semimodule, ( R, S ) - homomorphism and absorbing subbi-semimodules and we generalise the results of semirings (Jonathan S. Golan [4] ) and results of partial semirings (Srinivasa Rao. P.V [9]) to the class of (R, S ) – partial bi-semimodules .

2. Preliminaries

In this section we collect important definitions, results and examples which were already proved for our use in the next sections.

2.1 Definition. [6] A partial monoid is a pair

(

M

,

Σ

)

where

M

is a non empty set and ∑ is a partial addition defined on

some, but not necessarily all families

(

x

_i

:

i

∈

I

)

in

M

subject to the following axioms:

Unary Sum Axiom: If

(

x

_i

:

i

∈

I

)

is a one element family in

M

and I = { j }, then

∑

(

x

_i

:

i

∈

I

)

is defined and equals j

x

.

Partition - Associativity Axiom: If

(

x

i

:

i

∈

I

)

is a family in

M

and If

(

I

j

:

j

∈

J

)

is a partition of

I

, then

)

:

(2)

summable, and

∑

(

x

_i

:

i

∈

I

)

=

∑

(

∑

(

x

_i

:

i

∈

I

_j

)

:

j

∈

J

)

.

2.2 Definition. [6] The sum ordering≤ on a partial monoid

(

M

,

Σ

)

is the binary relation ≤ such thatx ≤ y if and only if there exists a h in

M

such that y = x + h, for x, y∈

M

.

2.3 Definition. [6] A partial semiring is a quadruple

(

R

,

Σ

,

⋅

,

1 )

, Where

(

R

,

Σ

)

is a partial monoid with partial addition ∑,

)

1 ,

,

(

R

⋅

is a monoid with multiplicative operation ‘

⋅

’ and unit ‘1’, and the additive and multiplicative structures obey the following distributive laws:

If

∑

(

x

i

:

i

∈

I

)

is defined in R, then for all y in R,

∑

(

y

⋅

x

i

:

i

∈

I

)

and

∑

(

x

i

⋅

y

:

i

∈

I

)

are defined and

∑

=

⋅

=

⋅

i i i i i i

i i

y

x

y

x

y

x

y

[

]

(

),

[

]

(

).

2.4 Definition. [6] A sum-ordered partial semiring (or so-ring, in short), is a partial semiring in which the sum ordering is a partial ordering.

2.5 Definition. [1] Let R be a so-ring. A subset N of R is said to be an ideal of R if the following are satisfied: (I1) if

(

x

i

:

i

∈

I

)

is a summable family in R and xi

∈N

for every i∈I then

∑

i i

x

_∈

_N_{, (I2) if}_{x ≤ y}_and_y∈_N_then_x∈N_,

and

(I3) if x

∈

N and r

∈

R then xr, rx

∈

N.

2.6 Definition. [2] A subset N of a so-ring R is said to be a bi-ideal of R if the following are satisfied: (B1) if

(

x

i

:

i

∈

I

)

∈

N for every i

∈

I then

∑

i i

x

_∈

_N_,

(B2)if x ≤ y and y

∈

N then x

∈

N, and (B3)if x, y

∈

N and r

∈

R then xry

∈

N.

2.7 Definition. [2]A subset N of a so-ring R is said to be a partial bi-ideal of R if the following are satisfied: i. if

(

x

_i

:

i

∈

I

)

∈

N for every i

∈

I then

∑

i i

x

_∈

_N_{, and}

ii. if x, y

∈

N and r

∈

R then xry

∈

N.

2.8 Definition. [3]Let

M

₁

,

M

₂be

(

R

,

S

)

- partial bi-semimodules. Then a mapping

φ

:

M

₁

→

M

₂ is said to be an

addativemapping if

φ

(

Σ

_i

m

i

)

=

Σ

_i

φ

(

m

i

)

for any summable family (mi :

i

∈

I

) in M1.

2.9 Definition. [7] Let

(

R

,

Σ

,

⋅

,

1 )

be a partial semiring and

(

M

,

Σ

)

be a partial monoid. Then

M

is said to be a left partial semimodule over

R

if there exists a function

∗

:

R

×

M

→

M

:

(

r

,

x

)



r

∗

x

which satisfies the following axioms for

x

,

(

x

_i

:

i

∈

I

)

in

M

and

r

1

,

r

2

,

(

r

j

:

j

∈

J

)

in

R

i. if

Σ

_i

x

i exists then

r

∗

(

Σ

_i

x

i

)

=

Σ

_i

(

r

∗

x

i

),

ii. if

∑

j j

r

_{exists then}

(

r

)

x

(

r

_j

x

),

j

j j

∗

Σ

=

∗

∑

iii.

r

1

∗

(

r

2

∗

x

)

=

(

r

1

⋅

r

2

)

∗

x

,

and

iv.

1

_R

∗

x

=

x

.Anlogously, one can define right partial semimodules over R. Throughout this paper R, S donote partial semirings.

3. (

R

,

S

) - Partial Bi-Semimodules

In this section we prove that (0 : M )R = (0 : m)Rfor every nonzero m in M where M is a bi-austere

(

R

,

S

)

- partial

bi-semimodule.

3.1. Definition. Let

R

,

S

be partial semirings and

(

M

,

Σ

)

be a partial monoid. Then

M

is said to be an

(

R

,

S

)

- partial bi-semimodule if it satisfies the following axioms:

i.

M

is a left partial semimodule over

R

,

ii.

M

is a right partial semimodule over

S

, and (iii) for any

r

∈

R

,

x

∈

M

,

s

∈

S

,

r

∗

x

∗

s

∈

M

.

(3)

subbi-semimodule of

M

if

N

is closed under

_Σ

and

∗

.

N

be a subbi-semimodule of an

(

R

,

R

)

- partial bi-semimodule M. Then define ( N : M ) as

}

|

{

)

:

(

N

M

=

r

∈

R

rMR

⊆

N

.

N

(

R

,

S

)

- partial bi-semimodule

M

.Then define

(

N

:

M

)

_R and S

M

N

:

)

(

as

(

N

:

M

)

_R

=

{

r

∈

R

|

rMS

⊆

N

}

and

(

N

:

M

)

_S

=

{

s

∈

S

|

RMs

⊆

N

}.

3.5. Remark. Let N be a subbi-semimodule of an

(

R

,

S

)

- partial bi-semimodule M. Then

(

N

:

M

)

_R and

(

N

:

M

)

_S are partial bi-ideals of R and S respectively.

Proof: Note that

(

N

:

M

)

_R

=

{

r

∈

R

|

rMS

⊆

N

}

. First we prove that

(

N

:

M

)

_R is a partial bi-ideal of R. Let

)

:

(

x

i

∈

I

be a summable family in

R

and each

x

i

∈

(

N

:

M

)

R

∀

i

∈

I

. Then

∑

i i

x

_{exists and}

_x

_i

_MS

_⊆

_N

I

i

∈

∀

.

⇒

∑

⊆

i i

N

MS

x

_.

⇒

∑

⊆

i i

N

MS

x

)

(

_{and hence}

∑

∈

i i R

M

N

x

(

:

)

_{. Let}

R

M

N

y

x

,

∈

(

:

)

and

R

r

∈

. Then

xMS

⊆

N

and

yMS

⊆

N

. Now

(

xry

)

MS

=

xr

(

yMS

)

⊆

xrN

⊆

N

.

⇒

xry

∈

(

N

:

M

)

_R . Hence

(

N

:

M

)

_R is a partial bi-ideal of R. Similarly we can prove that

(

N

:

M

)

_S is a partial bi-ideal of S.

3.6. Remark. Let

N

(

R

,

R

)

-partial bi-semimodule

M

. Then

(

N

:

M

)

is a partial bi-ideal of

R

.

3.7. Theorem. If

N

and

N

′

are subbi-semimodules of an

(

R

,

S

)

M

and if

A

,

B

are non empty subsets of

M

then ( i )

A

⊆

B

⇒

(

N

:

B

)

_R

⊆

(

N

:

A

)

_R

,

( ii )

(

N



N

′

:

A

)

_R

=

(

N

:

A

)

_R



(

N

′

:

A

)

_R

,

and ( iii ) if

Σ

(

a

,

b

)

exists for all

a

∈

A

,

b

∈

B

then

(

N

:

A

)

_R

_

(

N

:

B

)

_R

⊆

(

N

:

A

+

B

)

_R with equality holding if

B

A

M

∈



0

.

Proof: (i). Suppose

A

⊆

B

and let

x

∈

(

N

:

B

)

_R . Then

xBS

⊆

N

.

⇒

xbS

⊆

N

∀

b

∈

B

.

⇒

xaS

⊆

N

A

a

∈

∀

.

⇒

xAS

⊆

N

.

⇒

x

∈

(

N

:

A

)

_R. Hence

(

N

:

B

)

_R

⊆

(

N

:

A

)

_R.

(ii). Note that

x

∈

(

N

_

N

′

:

A

)

_R

⇔

xAS

⊆

N



N

′

⇔

xAS

⊆

N

and

xAS

⊆

N

′

⇔

x

∈

(

N

:

A

)

_R and R

A

N

x

∈

(

′

:

)

⇔

x

∈

(

N

:

A

)

_R

_

(

N

′

:

A

)

_R.

(iii). Suppose

Σ

(

a

,

b

)

exists for all

a

∈

A

,

b

∈

B

. Then

A

+

B

=

{

Σ

(

a

,

b

)

|

a

∈

A

,

b

∈

B

}

exists and nonempty. Let R

R

N

B

A

N

x

∈

(

:

)



(

:

)

. Then

xAS

⊆

N

and

xBS

⊆

N

.

⇒

x

(

A

+

B

)

S

⊆

N

.

⇒

x

∈

(

N

:

A

+

B

)

_R . R

R

N

B

N

A

B

A

N

:

)

(

:

)

(

:

)

(

⊆

+

⇒

_

. Suppose

0

_M

∈

A

_

B

and let

x

∈

(

N

:

A

+

B

)

_R .

N

S

B

A

x

+

⊆

⇒

(

)

.

⇒

xAS

+

xBS

⊆

N

. Since

0

_M

∈

A

and

0

_M

∈

B

,

0 +

xBS

⊆

N

and

xAS

+

0 ⊆

N

.

N

xAS

⊆

⇒

and

xBS

⊆

N

.

⇒

x

∈

(

N

:

A

)

_R and

x

∈

(

N

:

B

)

_R .

⇒

x

∈

(

N

:

A

)

_R

_

(

N

:

B

)

_R . R

R

N

A

N

B

A

N

:

)

(

:

)

(

:

)

(

+

⊆



⇒

. Hence

(

N

:

A

+

B

)

_R

=

(

N

:

A

)

_R



(

N

:

B

)

_R.

A

be a non empty subset of an

(

R

,

S

)

M

. Then the subbi-semimodule generated by

A

is the intersection of all subbi-semimodules of

M

containing

A

and is denoted by

RAS

.

Here we are able to generalise the results of partial semimodules over the partial semirings.

3.9. Theorem. Let M be an (R, S)-partial bi-semimodule. Then for any non empty subset A of M,

,

|

)

(

{

r

a

s

r

R

s

S

RAS

=

Σ

_i i

∗

i

∗

i i

∈

i

∈

a

i

∈

A

,

i

∈

I

}

Proof: Take T =

{

Σ

_i

(

r

i

∗

a

i

∗

s

i

)

/

r

i

∈

R

,

s

i

∈

S

,

a

i

∈

A

}

. First we prove that T is a subbi-semimodule of

M

. Let

)

:

(

x

i

∈

I

M

such that

x

i

∈

T,

i

∈

I

. Then

Σ

_i

x

i exists and each

S

s

A

a

R

r

s

a

r

x

i

=

Σ

_j

(

i_j

∗

i_j

∗

i_j

),

i_j

∈

,

i_j

∈

,

i_j

∈

.

⇒

Σ

_i

x

i

=

Σ

_i

Σ

_j

(

r

ij

∗

a

ij

∗

s

ij

)

exists and is in T. Let

r

∈

R

,

s

∈

S

and

x

∈

T. Then

r

∈

R

,

s

∈

S

and

x

=

Σ

_i

(

r

i

∗

a

i

∗

s

i

)

.

⇒

r

∗

x

∗

s

=

r

∗

Σ

_i

(

r

i

∗

a

i

∗

s

i

)

∗

s

=

s

a

r

i i i i

∗

(4)

A

a

=

1

R

∗

1

S

∀

∈

and hence

A

⊆

T

. To prove that T is smallest, let

N

be a subbi-semimodule of

M

containing

A

and let

x

∈

T. Then

x

=

Σ

_i

(

r

i

∗

a

i

∗

s

i

)

,

r

i

∈

R

,

a

i

∈

A

,

s

i

∈

S

.

⇒

r

i

∈

R

,

a

i

∈

N

,

s

i

∈

S

∀

i

∈

I

.

⇒

x

=

Σ

_i

(

r

i

∗

a

i

∗

s

i

)

∈

N

. Hence T is the smallest subbi-semimodule of

M

containing

A

.

M

be an

(

R

,

S

)

- partial bi-semimodule. Then the set

∑

∈I i i

N

_{of all possible sums of elements of}



_i _I

N

i

∈ , is the smallest subbi-semimodule of

M

containing each

N

i.

3.11. Definition. A non empty subset

N

of an

(

R

,

S

)

M

is said to be subtractive if and only if for any

m

,

m

′

∈

M

,

m

+

m

′

∈

N

and

m

∈

N

implies

m

′

∈

N

.

3.12. Definition. An

(

R

,

S

)

M

is said to be bi-austere if and only if

{

0 }

and

M

are the only subtractive subbi-semimodules of

M

.

N

(

R

,

S

)

M

and

A

be a non empty subset of

M

. Then

(

N

:

A

)

_R=

_

{(

N

:

a

)

_R

|

a

∈

A

}

.

3.14. Theorem. If

M

is a bi-austere

(

R

,

S

)

-partial bi-semimodule then

(

0 :

M

)

_R

=

(

0 :

m

)

_Rfor every non zero

m

in

M

.

Proof: Since

(

0 :

M

)

_R

=

_

{(

0 :

m

)

_R

|

m

∈

M

}

, we have

(

0 :

M

)

_R

⊆

(

0 :

m

)

_R for every nonzero

m

in

M

. Suppose

(

0 :

m

)

_R

⊄

(

0 :

M

)

_R for some nonzero

m

in

M

.Then

(

0 :

m

)

_R

⊄

(

0 :

m

′

)

_R for some non zero

M

m

′

∈

. Take

N

=

{

x

∈

M

|

(

0 :

m

)

R

⊆

(

0 :

x

)

R

}

. Then

0 ≠

m

∈

N

and

0 ≠

m

′

∉

N

and hence

M

N

⊂

}

0 {

. Now we prove that N is a subtractive subbi-semimodule of

M

. Let

(

x

i

:

i

∈

I

)

M

and

x

_i

∈

N

,

i

∈

I

. Then

Σ

_i

x

i exists and

(

0 :

m

)

R

⊆

(

0 :

x

i

)

R

∀

i

∈

I

.

)

:

0 (

)

:

0 (

)

:

0 (

_i

i

i i

R

x

m

⊆

=

Σ

⇒

∑

_{and hence}

_x

_i

_N

i

∈

Σ

_{. Let}

r

∈

R

,

s

∈

S

and

x

∈

N

. Then

r

∈

R

,

s

∈

S

and

R

x

m

)

(

0 :

)

:

0 (

⊆

.

⇒

(

0 :

m

)

_R

⊆

(

0 :

r

∗

x

∗

s

)

_R and hence

r

∗

x

∗

s

∈

N

. Hence N is a subbi-semimodule of

M

. To prove that

N

is subtractive, let

x

,

y

∈

M

such that

x

∈

N

and

Σ

(

x

,

y

)

∈

N

. Then

(

0 :

m

)

_R

⊆

(

0 :

x

)

_R and

(

0 :

m

)

R

⊆

(

0 :

Σ

(

x

,

y

)

R

)

. Now let

r

∈

(

0 :

m

)

R . Then

r

∈

(

0 :

x

)

R and

r

∈

(

0 :

Σ

(

x

,

y

))

R.

⇒

rxS

=

0

and

r

(

Σ

(

x

,

y

))

S

=

0

.

⇒

rxS

=

0

and

Σ

(

rxS

,

ryS

)

=

0

.

⇒

Σ

(

0 ,

ryS

)

=

0

.

⇒

ryS

=

0 .

⇒

r

∈

(

0 :

y

)

_R.

R

y

m

)

(

0 :

)

:

0 (

⊆

⇒

and hence

y

∈

N

.

⇒

N

is a non trivial subtractive subbi-semimodule of

M

, a contradiction. Hence

(

0 :

M

)

_R

=

(

0 :

m

)

_Rfor every non zero

m

in

M

.

3.15. Remark. If

N

is a subtractive subbi-semimodule of an

(

R

,

S

)

M

and

A

is a non empty subset of

M

then

(

N

:

A

)

_R is a subtractive partial bi-ideal of

R

.

Proof: By the remark 3.5,

(

N

:

A

)

_R is a partial bi-ideal of

R

. Let

x

,

y

∈

R

∋

x

+

y

∈

(

N

:

A

)

_Rand

x

∈

(

N

:

A

)

_R . Then

x

+

y

∈

(

N

:

a

)

_R and

x

∈

(

N

:

a

)

_R

∀

a

∈

A

.

⇒

(

x

+

y

)

aS

⊆

N

and

xaS

⊆

N

.

⇒

(

x

+

y

)

as

∈

N

and

N

xas

∈

∀

s

∈

S

.

⇒

xas

+

yas

∈

N

and

xas

∈

N

∀

s

∈

S

.

⇒

yas

∈

N

∀

s

∈

S

.

⇒

yAS

⊆

N

.

R

A

N

y

∈

(

:

)

⇒

and hence

(

N

:

A

)

_R is a subtractive.

(

R

,

S

)

M

is said to be entire if and only if

r

∗

m

∗

s

≠

0

_M whenever

,

0

R

≠

r

∈

R

0

S

≠

s

∈

S

and

0

M

≠

m

∈

M

.

3.17. Theorem. A partial semiring

R

is entire if and only if there exists a non trivial entire

(

R

,

R

)

-partial bi-semimodule.

Proof: If

R

is entire then

R

is a non trivial

(

R

,

R

)

-partial bi-semimodule. Suppose

∃

a non trivial entire

(

R

,

R

)

M

. Then

∃

0

_M

≠

m

∈

M

. Let

r

,

r

′

∈

R

∋

r

′

=

0

.

⇒

(

r

′

)

∗

m

∗

(

r

′

)

=

0

_M .

M

r

m

r

∗

(

′

∗

)

∗

′

=

0

(5)

3.18 Remark. Let

M

be an

(

R

,

S

)

-partial bi-semimodule. If

N

,

N

′

,

N

′′

are subbi-semimodules of

M

such that

N

is subtractive and

N

′

⊆

N

, then

N

∩

(

N

′

+

N

′′

)

=

N

′

+

(

N

∩

N

′′

)

.

Proof: Clearly

N

∩

(

N

′

+

N

′′

)

⊇

N

′

+

(

N

∩

N

′′

)

. Let

x

∈

N

∩

(

N

′

+

N

′′

)

. Then

x

∈

N

and

x

∈

N

′

+

N

′′

.

z

y

x

N

x

∈

=

+

⇒

,

for some

y

∈

N

′

,

z

∈

N

′′

.

⇒

x

=

y

+

z

∈

N

and

y

∈

N

. Since

N

is subtractive,

z

∈

N

)

(

N

z

y

x

=

+

∈

′

+

∩

′′

⇒

. Hence the remark.

4. (

R

,

S

) - Homomorphisms and Absorbing Subbi-Semimodules

In this section we introduce the notions of

(

R

,

S

)

- homomorphism, Bourne relation, steady

(

R

,

S

)

-homomorphism and absorbing subbi-semimodules and study various characteristics of them.

M

be an

(

R

,

S

)

- partial bi-semimodule and

θ

be an equivalence relation on

M

. Then

θ

is said to be an

(

R

,

S

)

- congruence relation on

M

if and only if it satisfies the following: ( i )

θ

is closed under the additive operation of the product (R, S)- partial bi-semimodule

M

×

M

. i.e., if

(

x

_i

:

i

∈

I

)

and

(

y

_i

:

i

∈

I

)

are summable families in M such that

(

x

i

,

y

i

)

∈

θ

then

∑

_∈

(

,

i

)

∈

θ

,

I i i

y

x

_{( ii ) if}

_r

_∈

_R

_,

_s

_∈

_S

_,

₍

_x

_,

_y

₎

_∈

_θ

_then

θ

∈

∗

,

)

(

r

x

s

r

y

s

.

(

M

,

Σ

,

∗

)

be an

(

R

,

S

)

- partial bi-semimodule and

θ

be an

(

R

,

S

)

- congruence relation on

M

. Then their quotient is the structure

₍

_M

_/

_θ

_,

_Σ

′

_,

_⋅

₎

where

M

/

θ

=

{[

x

]

θ

/

x

∈

M

}

(

[

x

]

θ is the equivalence class containing

x

with respect to

θ

),

_Σ

′

and

⋅

are defined as follows:

A family

([

x

i

]

θ

:

i

∈

I

)

is summable in

M

/

θ

if and only if

(

x

i

:

i

∈

I

)

is summable in

M

. Then we write

θ θ

[

]

[

i _i i i

x

=

Σ

x

′

Σ

. And ‘

⋅

’ is a function

R

×

M

/

θ

×

S

→

M

/

θ

:

(

r

[,

x

]

θ

,

s

)



r

⋅

[

x

]

θ

⋅

s

where

θ θ

[

]

[

x

s

r

x

s

r

⋅

=

∗

∀

r

∈

R

,

s

∈

S

and

x

∈

M

.

The following example shows that

M

/

θ

need not be an

(

R

,

S

)

- partial bi-semimodule.

4.3. Example. We know that

(

P

(

D

),

Σ

,

⋅

)

is a partial semiring, where

∑







∪

∩

=

∀

≠

=

i

j i i

i

_undefined

_otherwise

j

i

A

if

A

.

,

φ

and

A

⋅

B

=

A

∩

B

. Take

R

:

=

S

:

=

M

:

=

P

(

D

)

. Then

M

is an

(

R

,

S

)

- partial bi-semimodule.

Let

D

=

{

x

,

y

}

. Then

θ

=

{(

φ

,

φ

),

({

x

},

{

x

}),

({

y

},

{

y

}),

(

D

,

D

),

({

x

},

D

),

(

D

,

{

x

}),

(

φ

,

{

y

}),

({

y

},

φ

)}

is an

)

,

(

R

P

(

D

)

. Now

P

(

D

)

/

θ

=

{

φ

,

{

x

}

},

where

φ

=

{

φ

,

{

y

}}

=

{

y

}

,

{

x

}

=

{{

x

},

D

}

=

D

. Here

{

x

}

+

{

y

}

is defined. But

{

x

,

y

}

+

{

y

}

is not defined and hence

{

x

}

+

{

y

}

is not well defined. Hence

P

(

D

)

/

θ

is not a partial bi-semimodule.

θ

be an

(

R

,

S

)

- congruence relation on an

(

R

,

S

)

M

. Then a necessary and sufficient condition for

M

/

θ

to be a partial semiring is that the family

(

y

_i

:

i

∈

I

)

is summable whenever

(

x

_i

:

i

∈

I

)

is summable and

x

_i

θ

y

_i

,

i

∈

I

.

N

(

R

,

S

)

M

. Then the Bourne relation

≡

_N

on

M

is defined as

m

≡

_N

m

′

⇔

there exists

n

,

n

′

∈

N

such that

m

+

n

=

m

′

+

n

′

.

(

R

,

S

)

M

is said to be a complete

(

R

,

S

)

-partial bi-semimodule if for any family

(

m

_i

:

i

∈

I

)

in M,

Σ

_i

m

i is in M.

4.7. Remark. TheBourne relation

≡

_N is an

(

R

,

S

)

- congruence relation on a complete

(

R

,

S

)

(6)

Proof: Clearly

≡

_N is closed under the additive operation of product (R, S) - partial bi-semimodule

M

.

Let

r

∈

R

,

s

∈

S

and

(

m

,

m

′

)

∈≡

_N

.

⇒

∃

n

,

n

′

∈

N

∋

m

+

n

=

m

′

+

n

′

.

⇒

r

∗

(

m

+

n

)

∗

s

=

r

∗

(

m

′

+

n

′

)

∗

s

.

s

n

r

s

m

r

s

n

r

s

m

r

∗

+

∗

=

∗

′

∗

+

∗

′

∗

⇒

Since

n

,

n

′

∈

N

and

N

is a subbi-semimodule of M,

N

s

n

r

s

n

r

∗

,

∗

′

∗

∈

.

⇒

r

∗

m

∗

s

≡

N

r

∗

m

′

∗

s

. Hence

≡

N is an

(

R

,

S

)

M

. We denote the equivalence class of

m

as

m

/

N

and the quotient

M

/

≡

Nby

M

/

N

.

N

(

R

,

S

)

M

. Then the subtractive closure of

N

is the intersection of all subtractive subbi-semimodules of

M

containing

N

.

4.9. Remark. If

N

is a subbi-semimodule of a complete

(

R

,

S

)

M

. Then

0 /

N

is the subtractive closure of

N

.

Proof: Note that

0 /

N

=

{

m

∈

M

|

m

≡

N

0 }

=

{

m

∈

M

|

∃

n

∈

N

∋

m

+

n

∈

N

}

. First we prove that

0 /

N

is a subtractive subbi-semimodule of

M

. Let

(

m

_i

:

i

∈

I

)

M

such that

m

_i

∈

0 /

N

,

i

∈

I

. Then

0

N i

m

≡

∀

_i

∈

_I

.

⇒

Σ

_i

m

i

≡

N

0

and hence

Σ

_i

m

i

∈

0 /

N

. Let

r

∈

R

,

s

∈

S

and

m

∈

0 /

N

. Then

r

∈

R

,

s

∈

S

and

m

≡

_N

0

_M .

⇒

r

∗

m

∗

s

≡

_N

r

∗

0

_M

∗

s

.

⇒

r

∗

m

∗

s

≡

_N

0

_M and hence

r

∗

m

∗

s

∈

0 /

N

. Let

N

m

M

m

,

′

∈

∋

+

′

∈

0 /

and

m

∈

0 /

N

.

⇒

(

m

+

m

′

)

≡

_N

0

and

m

≡

_N

0

.

N

n

m

N

n

′

∈

∋

+

′

+

∈

∃

⇒

,

(

)

and

m

+

n

′

∈

N

.

⇒

∃

m

+

n

+

n

′

∈

N

∋

m

′

+

(

m

+

n

+

n

′

)

∈

N

.

0

N

m

′

≡

⇒

and hence

m

′

∈

0 /

N

. Therefore

0 /

N

M

. For any

0

0 ,

+

=

+

∈

N

a

and hence

a

≡

N

0

. Hence

a

∈

0 /

N

.

⇒

N

⊆

0 /

N

. Now let

N

′

be a subtractive subbi-semimodule of

M

∋

N

⊆

N

′

. Let

m

∈

0 /

N

. Then

∃

n

∈

N

∋

m

+

n

∈

N

.

⇒

∃

n

∈

N

′

∋

m

+

n

∈

N

′

. Since

N

′

is subtractive,

m

∈

N

′

and hence

0 /

N

⊆

N

′

. Hence the remark.

4.10. Remark. If

N

is a subbi-semimodule of a complete

(

R

,

S

)

M

. Then the congruence relations

≡

N and

≡

0/N on

M

coincide.

Proof: Since

N

⊆

0 /

N

. Clearly

≡

N ⊆

≡

0/N . Now let

m

≡

0/N

m

′

. Then

∃

x

,

y

∈

0 /

N

∋

m

+

x

=

m

′

+

y

.

N

y

m

N

x

m

)

/

(

)

/

(

+

=

′

+

⇒

.

⇒

m

/

N

+

x

/

N

=

m

′

/

N

+

y

/

N

. Since

x

,

y

∈

0 /

N

.

⇒

x

/

N

=

0 /

N

and

N

y

/

=

0 /

.

⇒

m

/

N

=

m

′

/

N

.

⇒

m

≡

_N

m

′

and hence

≡

_N ⊆

≡

0/N.

M

₁

,

M

₂ be

(

R

,

S

)

- partial bi-semimodules. Then a mapping

φ

:

M

₁

→

M

₂ is said to be an

)

,

(

R

S

-mapping if

φ

(

r

∗

x

∗

s

)

=

r

∗

φ

(

x

)

∗

s

∀

x

∈

M

,

r

∈

R

,

s

∈

S

.

4.12. Definition. A mapping

φ

:

M

₁

→

M

₂ is called an

(

R

,

S

)

-homomorphism of

(

R

,

S

)

- partial bi-semimodules M1,

M2 if ( i )

φ

is an additive mapping, and ( ii )

φ

is an

(

R

,

S

)

- mapping.

φ

:

M

₁

→

M

₂ be an

(

R

,

S

)

-homomorphism of

(

R

,

S

)

- partial bi-semimodules . Then the kernel

of

φ

is

ker

φ

=

{

x

∈

M

₁

|

φ

(

x

)

=

0 }

, for any subset

M

of

M

₁ ,

φ

M

=

{

φ

(

m

)

|

m

∈

M

}

and for any

}

)

(

|

{

)

(

,

1 1

2

y

x

M

x

y

M

y

∈

_φ

−

=

∈

_φ

=

_.

4.14. Theorem. Let

M

be a complete

(

R

,

S

)

- partial bi-semimodule. Then a subset

N

of

M

is subtractive subbi-semimodule if and only if there exists an

(

R

,

S

)

- homomorphism

α

:

M

→

M

′

satisfying

N

=

ker(

α

)

.

Proof: Suppose there exists an

(

R

,

S

)

- homomorphism

α

:

M

→

M

′

such that

N

=

ker(

α

)

. To prove that

ker(

α

)

is a subbi-semimodule of

M

, let

(

x

_i

:

i

∈

I

)

M

∋

each

x

_i

∈

ker(

α

)

.

⇒

α

(

x

_i

)

=

0

.

0 )

(

=

Σ

⇒

_i

i

α

x

.

⇒

α

(

Σ

i

x

i

)

=

0

.

⇒

Σ

i

x

i

∈

ker(

α

)

. Let

r

∈

R

,

s

∈

S

and

x

∈

ker(

α

)

.Then M

M

s

r

s

x

r

s

x

r

)

(

)

0

0 (

∗

=

∗

α

∗

=

∗

=

α

.

⇒

r

∗

x

∗

s

∈

ker(

α

)

. Hence

ker(

α

)

is a subbi-semimodule of

M

. To prove that

N

is subtractive, let

x

,

y

∈

M

∋

x

+

y

∈

N

and

x

∈

N

. Then

α

(

x

+

y

)

=

0

and

α

(

x

)

=

0

.

0 )

(

=

⇒

α

y

.

⇒

y

∈

ker(

α

)

=

N

. Hence

N

M

.