BASIC RESULTS ON INTERVAL - INCLINE MATRICES

International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 115 BASIC RESULTS ON INTERVAL - INCLINE MATRICES P. Ramasamy* & S. Anbalagan** * Research Scholar, Department of Mathematics, Raja Serfoji Government College (Affiliated to Bharathidasan University), Thanjavur, Tamil Nadu, India ** Assistant Professor, Department of Mathematics, Raja Serfoji Government College (Affiliated to Bharathidasan University), Thanjavur, Tamil Nadu, India Cite This Article: P. Ramasamy & S. Anbalagan, “Basic Results on Interval - Incline Matrices”, International Journal of Multidisciplinary Research and Modern Education, Volume 11, Issue 2, July - December, Page Number 115-127, 2025. Copy Right: © Crystal Pen Publication, 2025 (All Rights Reserved). This is an Open Access Article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DOI: Abstract: Inclines are special type of semirings and also the generalization of Boolean fuzzy, Distributive Lattice and fuzzy algebra. Here we generalized the incline as interval incline. Interval valued fuzzy is best example for interval incline. We have discussed some basic properties of interval incline and interval incline matrices. Key Words: Incline, Interval Incline, Fuzzy 1. Introduction: Inclines are special type of semiring, that is additively idempotent semiring which product are less than or equal to each factors Cao gave the notion of incline in 1983 and He co - authored with Kim and Roush have wrote the book incline incline algebra and its applications which is the first book in incline algebra [ 3 ]. Kim and Roush surveyed and outline properties of incline and incline matrices in [ 8 ]. Some basic properties of incline elements as well as regular incline elements has investigated by Meenakshi and Anbalagan in [ 11 ]and have an example which deals that incline not only contain distributive lattice which also contain non - distributive lattice A several authors studied the incline matrices like invertibility [ 5 ] indices periods [ 3 ] ect. A number of authors [ 6, 7, 9, 11, 16, 17 ] investigated the fuzzy matrices, boolean matrices and distributive matrices. The operators for interval matrices were discussed in [14] Ganesan [ 2 ] examined the properties of interval matrices and for interval valued fuzzy sets by Turksen [ 19 ] in 1986.Shyamal and M. pal [ 15 ] studied and outlined about interval valued fuzzy matrix and then many authors studied the same in various concepts. In this paper, we extend the concept of incline matrices to interval incline matrices as the generalization of interval valued fuzzy matrices. In section 2, we give some definitions and results needed to this paper. In section 3, we have defined interval in incline and using the set of all intervals we generalized the incline as interval incline. In section 4, we investigate the invertibility conditions and discussed the properties. 2. Preliminaries: In this section, we gave some needed definitions and results. Definition 2.1: ( [ 9 ] ). An incline is a nonempty set H with binary operations ,. ( H × H → H ) are given , denoted as addition and multiplication, such that for all ,,x y z        , , ,x y y x x y z x y z x y z xy xz                 , , , ,y z x yx zx x yz xy z x x x x xy x y xy y          Definition 2.2: ( [ 9 ] ). The incline  is said to be commutative, if ,,xy yx x y   Definition 2.3: ( [ 9 ] ).   , is an incline with order relation "" defined on  such that for ,,x y x y  if and only if x y y . If xy then y is said to be dominate x . Property 2.1: ( [ 12 ] ). Incline H with order relation "" for ,x y H , xy x and xy y . Property 2.2: ( [ 12 ] ). Incline H with order relation "" for ,x y H x y x and x y y    . Throughout this paper LI denotes the interval incline. 3. Basic Properties: In this section, we define the interval incline and examine some properties and operations. To define an interval in an incline. we need the usual ordering , here we have consider '' as in definition 2.3, which implies, the properties 2.1 and 2.2 also. Definition 3.1: In an incline L for , LU xx L interval is defined by   , LU xx  { x  L / LU x x x } and the set of all intervals denoted by LI .The binary operations addition and multiplication for any two elements in interval incline are given in the following has an extension of the operations on interval valued fuzzy sets [ 8 ] and incline algebra [1,9 ]. For any two elements     , , , L U L U x x y y  LI         , , , L U L U L L U U i x x y y x y x y    .         , . , . , . L U L U L L U U ii x x y y x y x y . International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 116 Proposition 3.1: The conditions in the definition 2.1 , which are examine for interval in an incline . Proof: For any three elements       , , , , , L U L U L U x x y y z z  LI           , , , ,. L U L U L L U U L L U U i x x y y x y x y y x y x                                , , , , , , , , , . L U L U L U L U L L U U L L L U U U L U L U L U ii x x y y z z x x y z y z x y z x y z x x y y z z                 iii In a similar manner of   ii we can obtain.                , , , , , , . L U L U L U L U L U L U x x y y z z x x y y z z .                                , , , ,, , , ,, , , , , . L U L U L U L U L L U U L L L U U U L L L L U U U U L L U U L L U U L U L U L U L U iv x x y y z z x x y z y z x y z x y z x y x z x y x z x y x y x z x z x x y y x x z z                 v In a similar manner of   iv , we can obtain               , , , , , , , . L U L U L U L U L U L U L U y y z z x x y y x x z z x x             ,, , ,. L U L U L L U U LU vi x x x x x x x x xx                 , , , , ,. L U L U L U L L L U U U LU vii x x x x y y x x y x x y xx        viii In similar manner of   vii , we can obtain        , , , , . L U L U L U L U y y x x y y y y From Proposition 3.1, we have set of intervals in an incline from an incline so we called which is interval incline. Throughout this paper LI denotes the interval incline. Definition 3.2: An interval incline is a non-empty set of elements LI with binary operations addition and multiplication denoted as ,. defined on LI  LI  LI . For all       , , , , , L U L U L U x x y y z z  LI           , , , , , L U L U L U L U i x x y y y y x x                     , , , , , , , L U L U L U L U L U L U ii x x y y z z x x y y z z                      , , , , , , , , L U L U L U L U L U L U L U iii x x y y z z x x y y x x z z                   , , , , , , , , L U L U L U L U L U L U L U iv y y z z x x y y x x z z x x                    , , , , , , , L U L U L U L U L U L U v x x y y z z x x y y z z         , , , , L U L U L U vi x x x x x x           , , , , , L U L U L U L U vii x x x x y y x x          , , , , . L U L U L U L U viii y y x x y y y y International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 117 For example, interval values fuzzy sets, Interval distributive lattices are the best example for interval inclines. Moreover, as in ( [ 9 ] page. 459 ), if we take any interval elements       , , , 0,1 L U L U x x y y  , then under the operations of 12 7 ,7 we can obtain the interval incline LI. Here we define the order relation '' for interval incline as a extension of definition 2.3. Definition 3.3: For any two elements     , , , L U L U x x y y  LI           , , , , , . L U L U L U L U L U x x y y if and only if x x y y y y   From the definition 2.3, we have the following properties as a generalization of [ 12 ]. Property 3.1: For any two elements     , , , L U L U x x y y  LI ,   i For addition             , , , , , . L U L U L U L U L U x x y y x x x x y y                , , , , , . L U L U L U L U L U x x y y x x y y y y    Using definition 3.2, we have the following       , , , L U L U L U x x y y x x and       , , , . L U L U L U x x y y y y   ii For multiplication Now, from the last two conditions in definition 3.1           , , . , , L U L U L U L U x x x x y y x x           , , . , , L U L U L U L U y y x x y y y y Using definition 3.2, we have the following      , , , L U L U L U x x x x y y and      , , , . L U L U L U y y x x y y Definition 3.4: An element   , LU aa of LI is said to be idempotent ,if     2 ,, L U L U a a a a , the set of all idempotents elements in LI is denoted by I ( LI ). . .,ie I ( LI) = {   , LU aa  LI      2 ,, L U L U a a a a }. Example 3.1: Incline 2 7 will be forms Interval Incline if we take the elements all are sub - intervals of   0,1 , then with the some binary operations in   10 . In this Interval Incline only two idempotent elements are   0,0 and   1,1 . i.e., I ( LI )=       0,0 , 1,1 . In this interval valued Fuzzy set I ( LI )= I V F S. Remark 3.1: In this interval incline elements having the lower limit and the upper limit are same then the interval incline coincides with the incline. An Interval incline matrix ( IIM ) is a matrix with the entries are the ( interval ) elements belongs to the Interval incline by ( LI ). In particular the interval incline matrix have the lower incline matrix and upper incline matrix, that is 'A IIM s which is defined as   ,, L U ijL ijU A A A a a    . Let LI mn ( LI n ) denotes the set of all mn interval incline matrices (IIM ) ( nn interval incline square matrix ). Definition 3.5: An interval incline matrices of order mn is defined as   ,, L U ijL ijU A A A a a    where , ij ijL ijU a a a   th is ij element of A interval incline matrix containing all the elements as intervals L A is the lower matrix of A and U A is the upper matrix of A . The following definition deals with the basic operations on IIM. Definition 3.6:   i For   , LU AA and   , LU BB  LI mn , the sum is defined as       , , , , L U L U L L U U i jL i jL i jU i jU A A B B A B A B a b a b          .   ii For   ,, L U i jL i jU m n A A a a    and   ,, L U i jL i jU n p B B b b    ,      . . ., , , , L U L U L U C AB ie C C A A B B is defined as follows with order mp .        , , , , L U L U L U L L U U C C A A B B A B A B   ,, ikL jkL ikU jkU ikL ikU kk a b a b c c      International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 118   iii A LI n then the transpose of A is defined as , , , T T T T T L U ijL ijU jiL jiU A A A a a a a              .   iv For ,AB LI mn ,     ,, L U L U L L U U ijL ijL ijU ijU A A B B iff A B and A B iff a b and a b     , 1, 2, ...,im and 1, 2, ..., .jn Remark 3.2: Like remark 3.1, each entries of L A and u A are same then the internal incline matrix coincides with incline matrix. Definition 3.7: An interval Incline matrix is said to be null matrix, if all its elements are zero i.e all elements are   0,0 and the matrix is denoted by O Definition 3.8: A matrix P LI mn is called permutation matrix if only one entry of its every row and every column is   1,1 and other entries are   0,0 . Definition 3.9: In an Interval Incline Matrix   ,, L U ijL ijU A A a a   of order nn is called Identity IIM, if all the diagonal entries are   1,1 and all other entries are   0,0 . Which is denoted by   , LU II . That is identity matrix of IIM’s special of permutation matrix. Remark 3.3: We conclude that by the above remark 3.2, Definition 3.7 and Definition ( 3.9 ) coincides with Null matrix and Identity matrix of incline respectively. We present some basic properties of IIM’s. The commutative, associative and distributive laws are valid for IIM’s under the operations    and   . respectively. Property 3.2: For any three interval incline matrices   , LU mn AA  ,   , LU np BB  , and   , LU mp CC  .       , , , L U L U ijL ijL ijU ijU i A A B B a b a b       , ijL ijL ijU ijU b a b a        , L L U U B A B A       , , . L U L U B B A A           , , , L U L U L U ii A A B B C C =         , , , L U L U L U A A B B C C Proof:           , , , , , L U L U L U ijL ijU ijL ijL ijU ijU A A B B C C a a b c b c               , ijL ijL ijL ijU ijU ijU a b c a b c          , , , ijL ijU ijL ijU ijL ijU a a b b c c                        , , , . L U L U L U A A B B C C     iii (a )         , . , . , . , . , L U L U L U ikL kjL ikU kjU jiL jiU jk A A B B C C a b a b c c                 . , . ikL kjL jiL ikU kjU jiU j k k a b c a b c       , ijL jiL ijU jiU j d c d c    Where,     . ijL ikL kjL ijU ikU kjU kk d a b and d a b  ( b )         , . , . , , , L U L U L U ijL ijU ikL kjL ikU kjU jk A A B B C C a a b c b c                  ,. ijL ikL kjL ijU ikU kjU j k k a b c a b c       From     a and b , we can conclude International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 119                 , . , . , , . , . , . L U L U L U L U L U L U A A B B C C A A B B C C   iv       , , , T T L U L U ijL ijL ijU ijU A A B B a b a b       , jiL jjL jiU jiU a b a b      ,, jiL jiU jjL jiU a a b b          ,, TT ijL ijU ijL ijU a a b b              , , . TT L U L U A A B B   v           , . , , . , TTT L U L U L U L U A A B B B B A A Proof:         , . , , , T T L U L U ijL ijU ijL ijU A A B B a a b b             , T ijL ijL ijU ijU kk a b a b         , kjL jkL kjU jkU kk b a b a         , , . T T L U L U B B A A Property 3.3: Let   ,, L U ijL ijU mn A A a a    be the interval incline matrix then         , , , L U L U L U i A A A A A A . Proof:     , , , , L U L U ijL ijuU ijL ijuU A A A A a a a a            , ijL ijL ijuU ijU a a a a      , ijL ijU aa   =   ,. LU AA       sin , 0,0 , L U L U ce A A A A   , 0,0 , . ijL ijU ijL ijU a a a a            Therefore, the interval incline matrix is special types of semigroup with respect to addition as well as multiplication. That is, additively idempotent, commutative over additive semigroup. 3.1 Convergence and Periodicity of Interval Incline Matrix (IIM): Let   ,, L U ijL ijU A A a a   be square IIM of order n . Thus the power of   , LU AA is defined as         , , , .... , mm L U L U L U L U A A A A A A A A m times    i.e.,   , , , .... , mm ijL ijU ijL ijU ijL ijU ijL ijU a a a a a a a a m times               and naturally   ,th ij element of , , . m m m m L U ijL ijU A A is a a         Definition 3.10: Let   ,, L U ijL ijU A A a a   be a square interval incline matrix of order n . If there exists an integer m such that 11 ,, m m m m L U L U A A A A           11 ,, m m m m ijL ijU ijL ijU a a a a           holds, then the power of an interval incline matrix is said to be converge. Generally, an Interval Incline Matrix is said converge when its power converge. International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 120 Definition 3.11: Let   ,, L U ijL ijU A A a a   be a square IIM of order n . If there exists two integers m and s such that ,, m s m s s s ijL ijU ijL ijU a a a a           holds, then m is said to be the periodicity of   ,, L U ijL ijU A A a a   and s is the starting point of   , LU AA corresponding to m . Let ns be any positive integer then 0ns and multiplying , n s n s LU AA    on both sides in ,, s m s m s s L U L U A A A A           , we get ,, n s s m n s s m n s s n s s L U L U A A A A                    , , , n m n m n n L U L U A A A A           Which means that every ns be also a starting point corresponding to m . Remark 3.4: Every periodicity m of an IIM   , LU AA is a multiple of the least periodicity d . We have generalized the following properties from [ 10,13,18 ]. Property 3.4: Let       , , , , , L U L U L U A A B B C C IIM with suitable order, such that     ,, L U L U A A B B ,implies         , , , , L U L U L U L U A A C C B B C C . Proof: Let us take any there IIM’s namely,     , , , , , L U ijL ijU L U ijL ijU m n m n A A a a B B b b           and   ,, L U ijL ijU np C C c c    such that     ,, L U L U A A B B . Let ( I )       , , , , L U L U ikL ikU kjL kjU A A C C a a c c   , ikL kjL ikU kjU k a c a c    ,, ijL ijU mp dd    Where ijL ikL kjL k d a c and ijU ikU kjU k d a c and ( II )       , , , , L U L U ikL ikU kjL kjU B B C C b b c c   , ikL kjL ikU kjU k b c b c    , ijL ijU mp ee    Where ijL ikL kjL k e b c and ijU ikU kjU k e b c Since,     ,, L U L U A A B B ijL ijL ijU ijU a b and a b   Again,   ,, L U ijL ijU C C c c   , the elements of ijL ijU c and c  LI np ijL ikL kjL ikL kjL ijL kk d a c b c e    and ijU ikU kjU ikU kjU ijU kk d a c b c e    Hence,         , , , , L U L U L U L U A A C C B B C C . Property 3.5: For A LI n if , , , , p p q q p p q q L U L U L U L U A A A A or A A A A                  holds for pq , then   , LU AA converges. Proof: If , , , , p p q q p p q q L U L U ijL ijU ijL ijU A A A A a a a a                    holds then         p q p q ijL ijL ijU ijU a a and a a we have             11 ... ... p q q p q q ijL ijL ijL ijU ijU ijU a a a and a a a        International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 121 Since,     pq ijL ijL aa ,     pq ijU ijU aa , therefore a finite number of distinct lower IIM and upper IIM in the corresponding interval ( position ) occur in the power of   , LU A A A . So that,           11 ... . . p q q s s ijL ijL ijL ijL ijL a a a a a        . and           11 . .. ... p q q t t ijU ijU ijU ijU ijU a a a a a        for some finite s and t , where st or st . Simply a finite number of distinct interval incline matrices occur in the power of   , LU A A A . Hence,   , LU AA converges. Similarly, it can be shown when ,, p p q q L U L U A A A A          . Property 3.6:   , LU AA be LI nn for all ,i j n if there exists kn such that ijL ikL kjL a a a and ijU ikU kjU a a a then   , LU AA converge to , pp LU AA   , for some 1pn . 4. Invertible Interval Incline Matrices: In this section we have investigated the invertible properties of IIM’s as the extension of [ 3,4 ] . For any positive integer n, always stands for any one of   1,2,3,... ) .   n denotes the least common multiple of the integers 1,2, ... ,n . The set M ( LI n ) constitute a partially ordered monoid with respect to the matrix multiplication. Definition 4.1: Let S be a semigroup and   , LU a a S .Assume that ,, k k k d k d L U L U a a a a           for some positive integers k and d are called an index and period   , LU aa respectively. In this case, we say that   , LU a a a has index. Definition 4.2: An interval incline LI with     0,0 1,1and are called an integral interval incline if it has no non-zero elements     ,, L U L U x x and y y such that     , 0,0 L L U U x y x y  and       , , 1,1 L U L U x x y y . Definition 4.3: An interval incline matrix   , LU AA M (LI n ) is said to be right invertible (left invertible), if     ,, L L U U A B A B I I ,       ,, L L U U B A B A I I for some   , LU BB M ( LI n ). The matrix   , LU BB is called a right inverse (left inverse) of   , LU AA . If   , LU AA is both right and left invertible then it is said to be invertible and   , LU BB , denoted by 11 , LU AA    is called an inverse of it. Remark 4.1: A matrix of IIM’s is invertible , then its right and left inverse coincide with the inverse it. Lemma 4.1: I ( LI ) is interval distributive lattice. Proof: Since     0,0 , 1,1  I ( LI ) we see that I ( LI ) is nonempty. For any     , , , L U L U x x y y  I ( LI ) , we have           22 , , , L U L U L L U U x x y y x y x y        22 , L L U U x y x y      2 2 2 2 2 , 2 L L L L U U U U x x y y x x y y          , L L L L L L U U U U U U x x y x y y x x y x y y       Suppose that 2 2 2 2 ,, L L U U L L U U x x x x and y y y y            , L L L L L L U U U U U U x x y x y y x x y x y y               , 3.1 L L U U x y x y ByDefinition        ,, L U L U x x y y          22 , , , L U L U L L U U x x y y x y x y     ,, L L U U L L U U x y x y x y x y International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 122 2 2 2 2 , L L U U x y x y     , L L U U x y x y     ,, L U L U x x y y Hence,     ,, L U L U x x y y I ( LI ) ,   , L L U U x y x y  I ( LI ) and             , , , , , , L U L U L L U U L U L U x x y y x y x y x x y y      I ( LI ) . If         , , , , L U L U L U L U x x z z and y y z z for some   , LU zz I ( LI ) , then     22 , , , L L U U L U L U x y x y z z z z    , which means that         , , , , L L U U L U L U x y x y x x y y  I ( LI ) Thus, I ( LI ) is interval distributive lattice. Lemma 4.2: If   , LU AA M ( LI n ) then             2 2 2 2 ,, n n n n n n n n n n L U L U A A A A           Proof: Let   ,, L U ijL ijU A A A a a    and   2 k n n . Denote     ,, pp pp L U ijL ijU A A a a      . Then for any ,i j n , we have     1 1 1 2 1 2 1 1 1 2 1 , ,..., , , , ... , kk k kk ijL ijU ii L iiU ii L ii U i jL i jU i i i n a a a a a a a a              Consider any term 1 1 1 2 1 2 1 1 , , ... , kk ii L iiU ii L ii U i jL i jU a a a a a a          of     , kk ijL ijU aa   Divide it into   nn products of u elements in order, 1 1 1 2 1 2 1 1 1 1 1 1 1 1 , , ... , , ... , . . . , n n n n n n n n k n k n k n k n k k ii L iiU i i L i i U i i L i i U i i L i i U i i L i i U i jL i jU a a a a a a a a a a a a                                 For every product of n interval elements, the number of the subscripts is equal to 1n their values are in n . Hence there exists two subscripts such that their value are same. Therefore, every product of n interval elements contains a sub product of type 11 , ... , rr pp L ppU p pL p pU a a a a         from the every product of n interval elements above, select exactly one subproduct of such type. Then there exists a integer rn such that   n subproduct of r elements are chosen atleast. In fact or else for every rn , the number of selected subproduct of r elements is smaller than   n , and so the total number of the selected subproduct is smaller than   nn . Now, select exactly     nr subproduct of r interval elements and leave them from 1 1 1 2 1 2 1 1 , , ... , kk ii L iiU ii L ii U i jL i jU a a a a a a          . Then we obtain a term of     , k n k n ijL ijU aa    . Hence we have,     1 1 1 2 1 2 1 1 , , ... , , kk k n k n ii L iiU i i L i i U i jL i jU ijL ijU a a a a a a a a                Since, the inequality above holds for any term 1 1 1 2 1 2 1 1 , , ... , , , kk kk ii L iiU i i L ii U i jL i jU ijL ijU a a a a a a of a a              It follows that     ,, k n k n kk ijL ijU ijL ijU a a a a       . This completes the proof. Lemma 4.3: If D is a interval distributive lattice and   , LU A A A D( LI n ) , where D( LI n ) - matrices over interval distributive lattice of order n , then     ,, n n n n nn L U L U A A A A       . Proof: Let   ,, L U ijL ijU A A A a a    then for any ,i j n we have that International Journal of Multidisciplinary Research and Modern Education (IJMRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.315, ISSN (Online): 2454 - 6119, Volume 11, Issue 2, July - December, 2025 123     1 1 1 2 1 2 1 1 1 1 1 2 1 2 1 1 1 2 1 , ,..., , , . .. , , , , . .. , . kk nn n nn ii L iiU i i L i i U i jL i jU ijL ijU ii L iiU i i L i i U i jL i jU i i i n a a a a a a a a a a a a a a                       Consider any term,     1 1 1 2 1 2 1 1 , , ... , , . nn nn ii L iiU i i L i i U i jL i jU ijL ijU a a a a a a of a a            Put 0n i i and i j ,since u in for all   0u u n there exists p and t such that 0p t n   and pt ii . Then we obtain that, 0 1 0 1 1 2 1 2 1 1 , , .. . , n n n n i i L i i U i i L i i U i i L i i U a a a a a a              0 1 0 1 1 1 1 1 1 1 , ... , , , .. . , p p p p t t t t n n n n i i L i i U i i L i i U L U i i L i i U i i L i i U a a a a b b a a a a                       0 1 0 1 1 1 1 1 1 1 , ... , , ... , .. . , p p p p t t t t n n n n i i L i i U i i L i i U L U i i L i i U i i L i i U a a a a b b a a a a                             00 ,. nn n n n n i i L i i U aa    Where   1 1 1 1 , , ... , p p p p t t t t L U i i L i i U i i L i i U b b a a a a         and     1 n ltp  . Since the inequality above holds for any term of     , nn ijL ijU aa   , we see that             ,, n n n n nn ijL ijU ijL ijU a a a a       . This completes the proof. Lemma 4.4: If D is an interval distributive lattice and   , LU A A A D( LI n ) then             2 2 2 2 ,, n n n n n n n n n n L U L U A A A A           . Proof: By Lemma 4.2, we have that                 2 2 2 2 ,, n n n n n n n n n n n n L U L L U U A A A A A A                           2 2 2 2 ,, n n n n n n n n n n n n L L U U L U A A A A A A           By Lemma 4.3, we obtain that                 2 2 2 2 ,, n n n n n n n n n n n n n n n n L U L L U U A A A A A A                        2 2 2 2 , , . n n n n n n n n n n nn L L U U L U A A A A A A           This completes the proof. 4.1 Some Conditions for Invertibility of IIM’s: In this section, we take the interval incline LI with     0,0 1,1and where   1,1 is the multiplicative identity. Here we give some necessary and sufficient conditions fot the IIM’s to be invertible [ 4 ]. Theorem 4.1: If   , LU A A A M( LI n ) is right invertible , then         ,, n n n L U L U A A A I I I      . Proof: Since   , LU A A A is right invertible.     ,, L L U U L U A X A X I I for some   , LU XX M( LI n ) . Let   2 k n n and   dn then ,, k d k d K K L U L U A A A A           by Lemma 4.8 we have that   ,, d d d L U L U A A A I I                   , , . .. , , , . .. , dd L U L U L U L L U U L U L U A A A A A A A X A X X X X X   , , , , , d d k k k k k d k k d k k k k k L U L U L U L L U U L L U U A A A A X X A X A X A X A X                    