Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1745-1761 © Research India Publications http://www.ripublication.com The Determinant and Rank of a Lattice Matrix Geena Joy Department of Mathematics, Union Christian College, Aluva, Kochi, Kerala, India - 683102 K. V. Thomas Department of Mathematics, Bharata Mata College, Thrikkakara, Kochi, Kerala, India - 682021 Abstract This paper deals with the determinant of a matrix over a dually Browerian, distributive lattice L with the greatest element 1 and the least element 0 , and proves that determinant of product of matrices over L is less than equal to the product of determinants of the matrices. Also, this paper introduces the concept of rank of a matrix over L . We demonstrate that the rank of a marix over L does not exceed the factorisation rank of the matrix and prove that the rank of product of matrices over L does not exceed the ranks of the factors. Keywords: Distributive lattice, Dually Browerian lattice, Lattice matrix, Determinant of a lattice matrix, Rank of a lattice matrix 2010 MSC: 15B99 1 INTRODUCTION The notion of lattice matrices appeared firstly in the work, ‘Lattice matrices’ [4] by G. Give’on in 1964. A matrix is called a lattice matrix if its entries belong to a distributive lattice. All Boolean matrices and fuzzy matrices are lattice matrices. Lattice matrices in various special cases become useful tools in various domains like the theory of switching nets, automata theory and the theory of finite graphs [4]. 1746 Geena Joy and K. V. Thomas The theory of determinant of Boolean matrices appeared firstly in the work of O. B. Sokolov [9]. Since then, a number of researchers have studied the determinant theory for Boolean matrices and lattice matrices (see [2, 6, 7, 8]). In [8], V. B. Poplavskii introduced the notion of minor rank of a Boolean matrix and discussed some of its properties. Further E. E. Marenich [6] discussed the determinant rank for matrices over a Browerian, distributive lattice with 1 and 0 . The eigenproblems and characteristic roots of matrices over a complete and completely distributive lattice with the greatest element 1 and the least element 0 are studied in [3, 10, 14]. In [10], Y. J. Tan discussed the eigenproblems of lattice matrices and provided the least element for the set of all characteristic roots of a lattice matrix. Further, G. Joy and K. V. Thomas [3] discussed the eigenproblems of nilpotent lattice matrices and introduced the concept of non-singular lattice matrices. Also, K. V. Thomas and G. Joy [14] studied the characteristic roots of different types of lattice matrices and introduced the concept of similar lattice matrices. The least element for the set of all characteristic roots of a lattice matrix is taken as the determinant of a lattice matrix. In the present work, we define the determinant of a matrix over a dually Browerian, distributive lattice L with the greatest element 1 and the least element 0 , and prove that determinant of product of matrices over L is less than equal to the product of determinants of the matrices. Also, this paper introduces the concept of rank of a matrix over L . We demonstrate that the rank of a marix over L does not exceed the factorisation rank of the matrix and prove that the rank of product of matrices over L does not exceed the ranks of the factors. 2 PRELIMINARIES We recall some basic definitions and results on lattice theory and lattice matrices which will be used in the sequel. For details see [1, 4, 5, 10, 11, 12, 13, 14, 15]. A partially ordered set (,)L is a lattice if for all a, b L , the least upper bound of {,}a b and the greatest lower bound of {,}a b exist in L . For any a, b L , the least upper bound and the greatest lower bound is denoted by a b and a b (or ab ), respectively. A lattice L is called a complete lattice if for any HL , both the least upper bound {y | y H } and the greatest lower bound {y | y H } of H exist in L . A lattice (,,,)L is a distributive lattice if the operations and are distributive with respect to each other. An element a L is called greatest element of L if x a , x L . An element b L is called least element of L if b x , x L . We use 1 and 0 to denote the greatest element and the least element of L , respectively. A complement of an element a L is an element b L for which a b =1 and a b = 0. The complement of a is denoted by a . If for any a L , a exists, then The Determinant and Rank of a Lattice Matrix 1747 L is said to be a complemented lattice. A complemented distributive lattice is called a Boolean lattice. For any a, b L , the least element x L satisfying the inequality b x a is called the relative lower pseudocomplement of b in a and is denoted by a b. If for any a, b L , a b exists, then L is said to be a dually Brouwerian lattice. If L is a Boolean lattice, then a b= a b . A lattice L is said to be completely distributive, if for any x L and any {yi | i I } L , where I is an index set, (a). x( i I y i ) = i I ( x y i ) and (b). x( i I y i ) = i I ( x y i ) holds. It is known [1] that a complete lattice L is dually Brouwerian if and only if (b) is satisfied in L . Therefore, a complete and completely distributive lattice L is dually Brouwerian. Lemma 2.1 [10, 12, 13] Let L be a distributive and dually Brouwerian lattice with 1 and 0. Then for any a,, b c L , we have (a). a a = 0 (b). a b a (c). a b a b = 0 (d). b( a b ) = a b (e). a() b c ab ac (f). (a b ) ( b c ) = ( a b ) ( b c ) (g). 1i n()()()a i b i 1i n a i 1i n b i . Let L be a distributive lattice with the greatest element 1 and the least element 0 . Let MLm, n () be the set of all m n matrices over L (Lattice Matrices). We shall denote MLn, n () by MLn () . Also, aij denotes the element of L which stands in the th (,)i j entry of AML m, n () . For A= ( aij ), B= ( bij ), C= ( cij ) Mm, n ( L ) , define A B= C cij = a ij b ij ( i =1,2, , m , j =1,2, , n ) T A= C cij = a ji ( i =1,2, , n , j =1,2, , m ) aA =C cij =aa ij fora L ,( i =1,2, , m , j =1,2, , n ) 1748 Geena Joy and K. V. Thomas For Aa= (ij ) MLBbMLCcm, k ( ), = (ij ) k, n ( ), = (ij ) MLm, n ( ) , define AB =C c = a b( i =1,2, , m , j =1,2, , n ) . ij 1l k il lj 1, ifi= j (I )ij = (0n )ij =0 (,i j =1,2, ,) n 0, ifi j 0 k1 k For A= ( aij ) Mn ( L ) , define A= I , A = A A for k 0, k is an integer. Lemma 2.2 [4] For any Aa= (ij ) MLBbm, k ( ), = (ij ) MLCck, r ( ), = (ij ) MLr, n ( ) , (a). A(BC ) = ( AB )C (b). (AATT ) = . Let AML n () . Then A is called nilpotent, if there exists some integer k 1 such k that A = 0n ; if aij = 0, i> j , i, j =1,2, n , then A is called upper triangular; if aij = 0, i< j , i, j =1,2, n , then A is called lower triangular. The matrix A is said to be invertible, if there is a BML n () such that AB= BA = I . Theorem 2.3 [12, 15] Let AML () be nilpotent. Then a a a = 0 , for n ii1i 1 i 2 imi {i , i1 , i 2 , , im } {1,2, ,n }. Let ABML,() n . If there exists an invertible matrix PML n () such that BP= 1AP , then B is said to be similar to A . For AML n () , the permanent |A | of A is defined as | |= , A a1 (1) a 2 (2) an () n Sn where Sn denotes the symmetric group of all permutations of the indices {1,2,n }. Let us use the following notations: SSn = { n | is even} SSn = { n | is odd}. Now the semi-permanants of AML n () are defined as follows: The Determinant and Rank of a Lattice Matrix 1749 pn ( A ) = a1 (1) a 2 (2) an () n Sn qn ( A ) = a1 (1) a 2 (2) an () n . Sn Lemma 2.4 [6] Let AML n () . Then (a). pn ()() A qn A=| A| T T (b). pn ( A ) = pn ( A ) and qn ( A ) = qn ( A ) (c). let B be the matrix obtained from A by interchanging columns A and i Aj . Then pn ( B ) = qn ( A ) and qn ( B ) = pn ( A ) (d). let B be the matrix obtained from A by replacing Aj by Aj . Then pn ( B ) = pn ( A ) and qn ( B ) = qn ( A ) (e). let A= b c , for some j=1,2, , n, where b,() c M1 L . Suppose that j n B is the matrix obtained from A by replacing Aj by b and C is the matrix obtained from A by replacing Aj by c . Then pn ( A ) = pn ( B ) pn ( C ) and qn ( A ) = qn ( B ) qn ( C ) .