1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 48-51
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www.iosrjournals.org 48 | Page
Parallel Summable Range Symmetric Incline Matrices
Shakila Banu
Department of Mathematics, Karpagam University,Coimbatore-641 021.
Abstract: The sum and parallel sum of parallel summable range symmetric matrices are range symmetric in £
and explore some additional properties of parallel summable range symmetric matrices are determined.
I. Introduction:
Incline is an algebraic structure and is a special type of a semiring. Inclines are additively idempotent
semirings in which products are less than or equal to either factor. The notion of inclines and their applications
are described comprehensively in Cao, Kim and Roush [2]. In [4], the equivalent conditions for EP-elements
over an incline are determined. In [6], the equivalent conditions for EP matrix are discussed and in [7],the
equivalent conditions of k-range symmetric matrices over an incline are determined.
II. Preliminaries:
In this section, some basic definition and required results are given.
Definition 2.1: A non empty set £ with two binary operations ‘+’ and ‘·‘ is called an incline if it satisfy the
following conditions.
1) (£,+) is a semilattice.
2) (£,·) is a semigroup.
3) x(y+z) = xy+xz for all x,y,zє£
4) x+xy = x and y+xy=y for x,y є£
Lemma2.2[5]: For A,Bє£mn We have the following:
(i) R(B)  R(A)B=XA for some Xє£m
(ii) C(B) C(A)B=AY for some Yє£n
Lemma 2.3[5]: For Aє£mn,Bє£np,we have the following
(i)R(AB) R(A)
(ii)C(AB) C(B)
Definition 2.4: A matrix A ε £m is said to be regular if there exists a matrix X ε £n such that AXA =A. Then
X is called a g-inverse of A and A{1} denotes the set of all g-inverses of A. That is, A{1}={X/ AXA =A}.
Lemma 2.5[5]: For A,Bє£mn, if A is regular, then
(i)R(B)  R(A)B=BA-
A for each A-
єA{1}
(ii)C(B)  CA)B=AA-
B for each A-
єA{1}
Definition 2.6[6]: A є £n is range symmetric incline matrix if and only if R(A)=R(AT
).
III. Parallel Summable Range Symmetric Incline Matrices
Definition 3.1: A pair of matrices 𝐴, 𝐵 £ n are said to be parallel summable (p.s) if 𝐴 + 𝐵 is regular and
𝐴(𝐴 + 𝐵)−
𝐵 is invariant under the choice of g-inverse (𝐴 + 𝐵)−
of (𝐴 + 𝐵). If 𝐴 and 𝐵 are parallel summable
then the parallel sum of 𝐴 and 𝐵 denoted as 𝐴: 𝐵 is defined as 𝐴: 𝐵=𝐴(𝐴 + 𝐵)−
𝐵.
Theorem 3.1: For incline matrices 𝐴, 𝐵£n, if 𝐴 + 𝐵 is regular then the following are equivalent:
(i) 𝑅 (𝐴)  𝑅 (𝐴 + 𝐵) and 𝐶 (𝐵)  𝐶 (𝐴 + 𝐵)
(ii) 𝐴 = 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵) and 𝐵 = (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵 for all (𝐴 + 𝐵)−
of 𝐴 + 𝐵.
(iii) 𝐴(𝐴 + 𝐵)−
𝐵 isinvariant, 𝐴 = 𝐴 + 𝐴 𝐴 + 𝐵 −
(𝐴 + 𝐵)and 𝐵 = 𝐵 + (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵
(iv) 𝐴 and 𝐵 are parallel summable
(v) 𝐴 𝑇
and 𝐵 𝑇
are parallel summable
Proof
(i)  (ii)
𝑅 𝐴  𝑅 𝐴 + 𝐵  𝐴 = 𝑋(𝐴 + 𝐵) [By Lemma 2.2]

2. Parallel Summable Range Symmetric Incline Matrices
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 𝐴 = 𝑋(𝐴 + 𝐵)(𝐴 + 𝐵)−
(𝐴 + 𝐵)
[By Definition 2.5]
 𝐴 = 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵)
and 𝐶 (𝐵)  𝐶 (𝐴 + 𝐵) 𝐵 = (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵 for all (𝐴 + 𝐵)−
of 𝐴 + 𝐵 .
(ii)  (iii)
𝐴(𝐴 + 𝐵)−
𝐵 = 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵)(𝐴 + 𝐵)−
(𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵
= 𝐴 𝐴 + 𝐵 −
𝐴 + 𝐵 𝐴 + 𝐵 −
𝐴 + 𝐵 (𝐴 + 𝐵)−
𝐵
= 𝐴 𝐴 + 𝐵 −
𝐴 + 𝐵 𝐴 + 𝐵 −
𝐵 = 𝐴𝑌𝐵
where 𝑌 = 𝐴 + 𝐵 −
𝐴 + 𝐵 𝐴 + 𝐵 −
 𝐴 + 𝐵 1 .
Hence 𝐴(𝐴 + 𝐵)−
𝐵 is invariant.
By definition, 𝐴 + 𝐴 = 𝐴 we get 𝐴 = 𝐴 + 𝐴 (𝐴 + 𝐵)−
(𝐴 + 𝐵). Similarly 𝐵 = 𝐵 + (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵.
(iii)  (ii)
Since 𝐴 = 𝐴 + 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵)  𝐴  𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵).
Suppose 𝐴 > 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵), then
𝐴(𝐴 + 𝐵)−
𝐵 > 𝐴 𝐴 + 𝐵 −
𝐴 + 𝐵 𝐴 + 𝐵 −
𝐵 > 𝐴𝑌𝐵
where 𝑌 = 𝐴 + 𝐵 −
(𝐴 + 𝐵) 𝐴 + 𝐵 −
 (𝐴 + 𝐵){1}.
which contradicts the invariance of 𝐴(𝐴 + 𝐵)−
𝐵.
Therefore 𝐴 = 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵). Similarly 𝐵 = (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵
(iv)  (v)
𝐴 and 𝐵 are parallel summable. Then 𝐴 𝑇
and 𝐵 𝑇
are parallel summable.
(iv)  (i)
𝐴 and 𝐵 are p.s by Definition (3.1) it follows that 𝑅 (𝐴)  𝑅 (𝐴 + 𝐵) and 𝐶 (𝐵)  𝐶 (𝐴 + 𝐵).
Hence the Theorem.
Lemma 3.3: For 𝐴, 𝐵 £ n with 𝑅(𝐴) = 𝑅(𝐵) and 𝐶(𝐴) = 𝐶(𝐵), 𝐴 is range symmetric if and only if 𝐵 is
range symmetric.
Proof
𝐴 is range symmetric  𝑅 (𝐴) = 𝑅 (𝐴 𝑇
) = 𝐶 (𝐴)
[By Definition 2.6]
 𝑅(𝐵) = 𝐶 (𝐵) = 𝑅(𝐵 𝑇
)
 𝐵 is range symmetric
Hence the Theorem.
Both the conditions in Lemma (3.3) are essential. This is illustrated in the following example.
Example 3.4
Consider the incline {£,[0,1],+,·}
Let 𝐴 =
0.5 0.5
1 0.3
and 𝐵 =
1 0.5
0.5 0.3
.
Any element (𝑥 , 𝑦) in 𝑅(𝐴) is of the form
(𝑥, 𝑦) = (0.5 , 0.5) + (1 , 0.3) for ,  F .
(𝑥, 𝑦) = ((0.5), (0.5)) + ( , (0.3)).
𝑥 = sup {(0.5) , }
𝑦 = sup {(0.5) , (0.3)}
Thus
𝑅 (𝐴) = {(𝑥, 𝑦): 0 𝑥 1, 0 𝑦 0.5}
𝑅 (𝐵) = {(𝑥, 𝑦): 0 𝑥 1, 0 𝑦 0.5}
𝐶 (𝐴) = {(𝑥, 𝑦): 0 𝑥 0.5, 0 𝑦 1}
𝐶 (𝐵) = {(𝑥, 𝑦): 0 𝑥 1, 0 𝑦 0.5}
Here 𝑅(𝐴) = 𝑅(𝐵) but 𝐶(𝐴) ≠ 𝐶(𝐵). B is symmetric hence range symmetric.
𝑅 (𝐴 𝑇
) = {(𝑥, 𝑦): 0 𝑥  0.5, 0 𝑦 1} ≠ 𝑅(𝐴). 𝐴 is not range symmetric. Hence the Theorem
fails.
Theorem 3.5: For 𝐴, 𝐵£n if 𝑅(𝐵)  𝑅(𝐴)  𝑅(𝐴 + 𝐵) and 𝐶(𝐵)  𝐶(𝐴)  𝐶(𝐴 + 𝐵). Then 𝐴 is range
symmetric if and only if (𝐴 + 𝐵) is range symmetric.
Proof
𝑅(𝐵)  𝑅(𝐴)  𝐵 = 𝑋𝐴 for some 𝑋£n [By Lemma 2.2]
Then, 𝐴 + 𝐵 = 𝐴 + 𝑋𝐴
= (𝐼 + 𝑋)𝐴
This implies 𝑅(𝐴 + 𝐵)  𝑅(𝐴). Together with the given condition,

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𝑅(𝐴)  𝑅(𝐴 + 𝐵) yields 𝑅(𝐴) = 𝑅(𝐴 + 𝐵) → (3.3)
Similarly,𝐶(𝐵)  𝐶(𝐴)  𝐵 = 𝐴𝑌 for some Y£n [By Lemma 2.2]
Then, 𝐴 + 𝐵 = 𝐴 + 𝐴𝑌
= 𝐴(𝐼 + 𝑌)
This implies 𝐶(𝐴 + 𝐵)  𝐶(𝐴)
Since 𝐶(𝐴)  𝐶(𝐴 + 𝐵), we get 𝐶(𝐴) = 𝐶(𝐴 + 𝐵). → (3.4)
By using (3.3) and (3.4) in Lemma (3.3) we get 𝐴 is range symmetric if and only if (𝐴 + 𝐵) is range symmetric.
Theorem 3.6: Let 𝐴 and 𝐵 be range symmetric matrices with 𝑅(𝐴) = 𝑅(𝐵). If 𝐴 and 𝐵 are parallel summable
then 𝐴 + 𝐵 is range symmetric.
Proof
𝐴 and 𝐵 are range symmetric  𝑅(𝐴) = 𝑅(𝐴 𝑇
) and
𝑅(𝐵) = 𝑅(𝐵 𝑇
) → (3.5)
Since 𝑅(𝐴) = 𝑅(𝐵) along with (3.5) we get 𝐶(𝐴) = 𝐶(𝐵). → (3.6)
𝐴 and 𝐵 are p.s  𝑅 (𝐴)  𝑅 (𝐴 + 𝐵) and 𝐶 (𝐵)  𝐶(𝐴 + 𝐵) follows from Definition (3.1).
→ (3.7)
By using (3.6) and (3.7) we get 𝑅(𝐵)  𝑅(𝐴)  𝑅(𝐴 + 𝐵) and 𝐶(𝐵)  𝐶(𝐴)  𝐶(𝐴 + 𝐵).
Thus 𝐴 + 𝐵 is range symmetric follows from Theorem (3.5).
Theorem 3.7: Let 𝐴 and 𝐵 be p.s range symmetric matrices with 𝑅(𝐴: 𝐵)  𝑅(𝐴) and C(𝐴: 𝐵)  𝐶(𝐵) . Then
(𝐴: 𝐵) is range symmetric.
Proof
𝐴 and 𝐵 are p.s fuzzy matrices by Definition 𝑅(𝐴: 𝐵)  𝑅(𝐵) and 𝐶(𝐴: 𝐵)  𝐶(𝐴) and along
with the given conditions
𝑅 𝐴: 𝐵 = 𝑅(𝐴)  𝑅(𝐵)
= 𝑅(𝐴 𝑇
)  𝑅(𝐵 𝑇
) [𝐴 and 𝐵 are range symmetric]
= 𝑅 𝐴 𝑇
: 𝐵 𝑇
.
Hence A:B is range symmetric.
Lemma 3.8: Let 𝐴 and 𝐵 be p.s range symmetric matrices such that (𝐴 + 𝐵) is symmetric. If there exist incline
matrices 𝐻 and 𝐾 such that 𝐻−
𝐻 = 𝐾𝐾−
= 𝐼 then(𝐻𝐴: 𝐻𝐵) 𝑇
= 𝐾 𝐵: 𝐴 and (𝐴𝐾: 𝐵𝐾) 𝑇
= (𝐵: 𝐴)𝐻.
Proof
Since 𝐴 and 𝐵 are range symmetric matrices , 𝐴 𝑇
= 𝐻𝐴 and 𝐵 𝑇
= 𝐾𝐵.→ (3.8)
(𝐻𝐴: 𝐻𝐵) 𝑇
= [𝐻(𝐴: 𝐵)] 𝑇
= (𝐴: 𝐵) 𝑇
𝐻 𝑇
= (𝐵 𝑇
: 𝐴 𝑇
) 𝐻 𝑇
= 𝐵 𝑇
(𝐵 𝑇
+ 𝐴 𝑇
)−
𝐴 𝑇
𝐻 𝑇
[By Definition 3.1]
= 𝐾𝐵((𝐵 + 𝐴) 𝑇
)−
𝐴 [(𝐴 + 𝐵) is symmetric]
= 𝐾(𝐵: 𝐴)
Similarly (𝐴𝐾: 𝐵𝐾) 𝑇
= (𝐵: 𝐴)𝐻 can be proved in the same manner.
Theorem 3.9: Let 𝐴, 𝐵 Fn be p.s range symmetric matrices such that 𝐴 + 𝐵 is symmetric. If there exist 𝐻,
𝐾Fn such 𝐻−
𝐻 = 𝐾𝐾−
= 𝐼n then the following hold.
(1) 𝐴  𝐾(𝐵: 𝐴) and 𝐵  (𝐵: 𝐴)𝐻 for some 𝐻, 𝐾 Fn
(2) 𝐴 + 𝐵  𝐾 𝐵: 𝐴 + (𝐵: 𝐴)𝐻 for some 𝐻, 𝐾 Fn
Proof
Since 𝐴 and 𝐵 are range symmetric matrices, 𝐴 𝑇
= 𝐻𝐴 and 𝐵 𝑇
= 𝐵𝐾.
𝐴 and 𝐵 are p.s  𝑅(𝐴)  𝑅(𝐴 + 𝐵)
 𝐴 = 𝐴(𝐴 + 𝐵)−
(𝐴 + 𝐵)
 𝐴 = 𝐴(𝐴 + 𝐵)−
𝐴 + 𝐴(𝐴 + 𝐵)−
𝐵
 𝐴  𝐴(𝐴 + 𝐵)−
𝐵
 𝐴  𝐴: 𝐵
and 𝐶 𝐵  𝐶 𝐴 + 𝐵  𝐵 = (𝐴 + 𝐵)(𝐴 + 𝐵)−
𝐵
 𝐵 = 𝐴(𝐴 + 𝐵)−
𝐵 + 𝐵(𝐴 + 𝐵)−
𝐵
 𝐵  𝐴(𝐴 + 𝐵)−
𝐵
 𝐵  𝐴: 𝐵
Thus 𝐴  (𝐴: 𝐵)
Premultiply by 𝐻 and using 𝐴 𝑇
= 𝐻𝐴, we get

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 𝐻𝐴  𝐻(𝐴: 𝐵)
 𝐴 𝑇
 𝐻𝐴: 𝐻𝐵
 𝐴  (𝐻𝐴: 𝐻𝐵) 𝑇
 𝐴  𝐾𝐵 ∶ 𝐾𝐴
 𝐴  𝐾(𝐵: 𝐴)
Since 𝐵  𝐴: 𝐵
Postmultiply by 𝐻 and using 𝐵 𝑇
= 𝐵𝐾 we get 𝐵  (𝐵: 𝐴)𝐻.
On addition we get,
𝐴 + 𝐵  𝐾(𝐵: 𝐴) + (𝐵: 𝐴)𝐻 for some 𝐻, 𝐾 £n.
Acknowledgement
This work was supported by Grants-in-Aid of Women’s of Scientists-A, Department of Science and
Technology, New Delhi.
References
[1] Z. Q. Cao, K. H. Kim, F. W. Roush (1984) Incline algebra and Applications, John Wiley and Sons, New York.
[2] K. H. Kim, F. W. Roush (1980) Generalized Fuzzy Matrices, Fuzzy Sets Sys.,4, 293-315.
[3] AR.Meenakshi(2008) Fuzzy Matrix Theory and its Applications, MJP Publishers, Chennai.
[4] AR. Meenakshi, S. Anbalagan (2010) EP Elements in an incline, International Journal of Algebra,4,541-550.
[5] AR. Meenakshi, P. Shakila Banu (2010) g-inverses of matrices over a regular incline, Advances in Algebra,3,33-42.
[6] AR. Meenakshi, P. Shakila Banu (2012) EP Matrices over an Incline, International Journal of Mathematical Science,11, 175-182.
[7] P. Shakila Banu, k-Range Symmetric Incline Matrices (Communicated)