there are many equivalent ways of defining the rank of a matrix over a field . the rank of an m by n matrix a could be defined as the largest k for which there exists a k by k submatrix of a with nonzero determinant , or the dimension of the row space of a , or the dimension of the column space of a or the smallest k for which there exists an m by k matrix b and a k by n matrix c with a = bc . for matrices over semirings , all of these definitions are no longer equivalent and each of these generalizes to a distinct rank function for matrices over semirings . there are many different rank functions for matrices over semirings and their properties and the relationships between them have been much studied ( see , e.g. , [ 13 ] ) . in this paper , we use the -determinant of tan [ 4 , 5 ] to define a new family of rank functions for matrices over semirings . we examine the properties of these rank functions as well as their relationship to some of the other rank functions found in the literature . in this section , we review some background material on semirings . in section 2 , we list the definition and some important properties of -determinant of tan first introduced in [ 4 , 5 ] . we then use the -determinant to introduce a new family of rank functions called the (  , i)-rank functions and show that these rank functions satisfy some of the usual inequalities such as the rank - sum inequality and the sylvester inequality . in section 3 , we look at bijective linear preservers of (  , i)-rank for all matrices over commutative antinegative semirings . in section 4 , we introduce the sign pattern semiring and show that our rank function in this case is equal to the dimension of the largest sign - nonsingular submatrix . our new rank functions depend on the ideal structure of the semirings and this leads us to study semirings which have a unique maximal ideal in section 5 . vandiver introduced the concept of semiring in , in connection with the axiomatization of the arithmetic of the natural numbers . a semiring is a set s together with two operations  and  and two distinguished elements 0 , 1 in s with 0  1 , such that ( 1)(s ,  , 0 ) is a commutative monoid,(2)(s ,  , 1 ) is a monoid,(3) is both left and right distributive over ,(4)the additive identity 0 satisfies the property r  0 = 0  r = 0 , for all r  s. 
  a semiring is a set s together with two operations  and  and two distinguished elements 0 , 1 in s with 0  1 , such that ( 1)(s ,  , 0 ) is a commutative monoid,(2)(s ,  , 1 ) is a monoid,(3) is both left and right distributive over ,(4)the additive identity 0 satisfies the property r  0 = 0  r = 0 , for all r  s. ( s ,  , 0 ) is a commutative monoid , ( s ,  , 1 ) is a monoid ,  is both left and right distributive over  , the additive identity 0 satisfies the property r  0 = 0  r = 0 , for all r  s. in other words , semirings are unital rings without the requirement that each element has additive inverse . if ( s ,  , 1 ) is a commutative monoid then s is called a commutative semiring . a semiring is said to be antinegative or zerosumfree if the only element with an additive inverse is the additive identity 0 . an element of a semiring s is called a unit if it has a multiplicative inverse in s. the natural numbers form a semiring under the usual addition and multiplication . a much studied example of a semiring is the max - plus semiring ( max ,  ,  ) , where max =   {  } with a  b = max{a , b } and a  b = a + b. in this case 0 =  and 1 = 0 . a totally ordered set s with greatest element 1 and least element 0 forms a semiring with a  b = max{a , b } and a  b = min{a , b}. this is called chain semiring . the chain semiring with two elements { 0 , 1 } is called the boolean semiring and it is denoted by . let mm , n(s ) denote the set of all m by n matrices over s and mn(s ) denotes the set of all n by n matrices over s. addition and multiplication of these matrices can be defined in the usual way . then the element aij is called the ( i , j)-entry of a. the ( i , j)-entry of a is sometimes denoted by aij . let a = [ aij ] and b = [ bij ] be m by n matrices over a semiring ( s ,  ,  ) and let c = [ cij ] be an n by p matrix over the same semiring . then a + b = [ aij  bij ] and ac = [ k=1aik  ckj ] . the set of n by n matrices over a semiring is itself a semiring . for any k if a = [ aij ] is an n by n matrix over a commutative ring , then the standard determinant expression of a is ( 1)deta=snsgna11a22ann , where sn is the symmetric group of order n and sgn( ) = + 1 if  is even permutation and sgn( ) = 1 if  is odd permutation . here an(n ) is called a term of the determinant . 
  if a = [ aij ] is an n by n matrix over a commutative ring , then the standard determinant expression of a is ( 1)deta=snsgna11a22ann , where sn is the symmetric group of order n and sgn( ) = + 1 if  is even permutation and sgn( ) = 1 if  is odd permutation . here sgn()a1(1)a2(2 )  an(n ) is called a term of the determinant . since we do not have subtraction in a semiring , we can not write the determinant of a matrix over a semiring in this form . we split the determinant into two parts , the positive determinant and the negative determinant . let a be an n by n matrix over a commutative semiring s. we define the positive and the negative determinant as ( 2)det+a=anni=1aii , deta=snani=1naii . here an is the alternating group of order n , that is , the set of all even permutations of order n and snan is the set of all odd permutations of order n. 
  let a be an n by n matrix over a commutative semiring s. we define the positive and the negative determinant as ( 2)det+a=anni=1aii , deta=snani=1naii . here an is the alternating group of order n , that is , the set of all even permutations of order n and snan is the set of all odd permutations of order n. as such we note that the determinant of a matrix a over a commutative ring takes the form ( 3)deta = det+adeta . in the semiring case , one can not subtract the negative determinant from the positive determinant and so the positive determinant and the negative determinant are listed as a pair . the definition of the permanent involves no subtractions ; hence it carries over to the semiring case unchanged . let a = [ aij ] be an n by n matrix over a semiring ; then the permanent of a is ( 4)pera=sna11a22ann . 
  let a = [ aij ] be an n by n matrix over a semiring ; then the permanent of a is ( 4)pera=sna11a22ann . the permanent of a square matrix is the sum of its positive and negative determinants : per(a ) = det+(a )  det(a ) . finally we note that we have a canonical preorder ( a reflexive transitive relation ) called the difference preorder on semirings . we define the difference preorder  on s as follows : if x , y  s then x  y if there exists z  s such that x = y  z. 
  let s be a semiring . we define the difference preorder  on s as follows : if x , y  s then x  y if there exists z  s such that x = y  z. the difference preorder may not be a partial order . however for many semirings such as the nonnegative semiring , max - plus semiring , and any boolean algebra or distributive lattice , the difference semiring corresponds to the natural order on the set . in [ 4 , 5 ] , tan introduced a new type of determinant for semirings . we begin with a concept formulated independently by akian et al . in   and tan in . definition 7 ( see [ 1 , 4 , 5 ] ) . let s be a semiring . a bijection  : s  s is called a symmetry if ((a ) ) = a for all a  s and (a  b ) = a  (b ) = (a )  b for all a , b  s. 
  let s be a semiring . a bijection : s  s is called a symmetry if ((a ) ) = a for all a  s and (a  b ) = a  (b ) = (a )  b for all a , b  s. the term symmetry is from ; this similar concept is called an -function in [ 4 , 5 ] . we note that all of these references also required a symmetry to be additive ( i.e. , (a  b ) = (a )  (b ) ) . in , a symmetry  must further satisfy (0 ) = 0 . we have removed these conditions from the definition as we will show they follow from other properties of a symmetry . a function  : s  s is a symmetry if and only if there exists an   s such that  = 1 and (x ) =   x for all x  s. 
  let s be a commutative semiring . a function  : s  s is a symmetry if and only if there exists an   s such that  = 1 and (x ) =   x for all x  s. 
 proofsuppose  : s  s is a symmetry . let  = (1 ) ; then (x ) = (1  x ) =   x. furthermore  =   (1 ) = (  1 ) = 1 . let  = (1 ) ; then (x ) = (1  x ) =   x. furthermore  =   (1 ) = (  1 ) = 1 . it follows easily from the previous proposition that if  : s  s is a symmetry and x , y  s then (x  y ) = (x )  (y ) , (0 ) = 0 , and (x )  (y ) = x  y. we use this observation to slightly restate the definition of an -determinant given in [ 4 , 5 ] . we will use  to denote the element whose square is the identity rather than a symmetry or -function as in [ 4 , 5 ] ; this allows us to use the same terminology and notation as in [ 4 , 5 ] while taking advantage of the characterization of symmetries given in proposition 8 . let s be a commutative semiring and let   s satisfy  = 1 . then det : mn(s )  s is defined as the following function : det(a ) = det+(a)(  det(a ) ) . let s be a commutative semiring and let   s satisfy  = 1 . then det : mn(s )  s is defined as the following function : det(a ) = det+(a)(  det(a ) ) . we will get a distinct symmetry on s and hence a distinct det from every choice of   s which satisfies  = 1 . one candidate for  that exists in every semiring is the multiplicative identity 1 . if  = 1 , we get det(a ) = per(a ) . tan has shown that the -determinant satisfies two very important identities analogous to the binet - cauchy theorem and the laplace expansion of the ordinary determinant . in order to state them , we introduce the following notation . let  = { 1 < 2 < <k } and  = { 1 < 2 < <k } be two subsets of { 1,2 ,  , let a[ ] denote the k by k submatrix of a whose ( i , j)th entry is ai,j . theorem 10 ( see [ 5 , theorem 3.5 ] ( the generalized binet - cauchy theorem ) ) . let s be a commutative semiring and let   s satisfy  = 1 . let a  mm , n(s ) , b  mn , p(s ) , 1  k  min(m , n , p ) . let  and  be two subsets of { 1,2 , n } of cardinality k. then there exists a   s such that det((ab)[ ,  ] ) = ( {1,2,  ,n};||=kdet(a[ ,  ] )  det(b[ , ]))[  ( 1 ) ] . let s be a commutative semiring and let   s satisfy  = 1 . let a  mm , n(s ) , b  mn , p(s ) , 1  k  min(m , n , p ) . let  and  be two subsets of { 1,2 , n } of cardinality k. then there exists a   s such that det((ab)[ ,  ] ) = ( {1,2,  ,n};||=kdet(a[ ,  ] )  det(b[ , ]))[  ( 1 ) ] . theorem 11 ( see [ 5 , theorem 3.3 ] ( the generalized laplace expansion ) ) . let s be a commutative semiring and let   s satisfy  = 1 . if a  mn(s ) and {1,2 ,  , n } , then det(a ) = {1,2,  ,n};||=||( )  det(a[ ] )  det(a[ ] ) . let s be a commutative semiring and let   s satisfy  = 1 . if a  mn(s ) and {1,2 ,  , n } , then det(a ) = {1,2,  ,n};||=||( )  det(a[ ] )  det(a[ ] ) . in the case where s is a commutative ring and  = 1 , the -determinant reduces to the regular determinant and the two theorems above reduce to the usual binet - cauchy theorem and the laplace expansion . one corollary of the generalized binet - cauchy theorem is the following difference preorder inequality for square matrices . let s be a commutative semiring and let   s satisfy  = 1 . the inequality det(ab )  det(a )  det(b ) holds for all a , b  mn(s ) . let s be a commutative semiring and let   s satisfy  = 1 . the inequality det(ab )  det(a )  det(b ) holds for all a , b  mn(s ) . the special case of this result where s is a boolean algebra ( and by necessity  = 1 which means det is the permanent ) has appeared in . we can use tan 's generalization of the laplace expansion to obtain a relationship between the -determinant of a + b and those of various submatrices of a and b. 
 corollary 13 . let n   , let s be a commutative semiring , and let   s satisfy  = 1 . the identity ( 5)deta+b=,1,2,  ,n;=+det(a[    ])detb[c    c ] holds for all a , b  mn(s ) . let n   , let s be a commutative semiring , and let   s satisfy  = 1 . the identity ( 5)deta+b=,1,2,  ,n;=+det(a[    ])detb[c    c ] holds for all a , b  mn(s ) . prooflet c be the n by n matrix whose ith row is equal to the ith row of a if i   and whose ith row is equal to the ith row of b if i  . then using the multilinearity of the -determinant , det(a + b ) = {1,2,  ,n}det(c ) . now we use the laplace expansion on det(c ) , for any fixed  , to get ( 6)detc=1,2,  ,n;=+det(a[    ])det(b[c    c ] ) . hence ( 7)deta+b=,1,2,  ,n;=+det(a[    ])det(b[c    c ] ) . let c be the n by n matrix whose ith row is equal to the ith row of a if i   and whose ith row is equal to the ith row of b if i  . then using the multilinearity of the -determinant , det(a + b ) = {1,2,  ,n}det(c ) . now we use the laplace expansion on det(c ) , for any fixed  , to get ( 6)detc=1,2,  ,n;=+det(a[    ])det(b[c    c ] ) . hence ( 7)deta+b=,1,2,  ,n;=+det(a[    ])det(b[c    c ] ) . let s be a commutative semiring and let a be an m by n matrix over s. if 1  k  min(m , n ) , let ik(a ) be the ideal in s generated by the set of all the k by k  -minors of a. one defines i0(a ) = s and ik(a ) = { 0 } when k > min(m , n ) . 
  let s be a commutative semiring and let a be an m by n matrix over s. if 1  k  min(m , n ) , let ik(a ) be the ideal in s generated by the set of all the k by k  -minors of a. one defines i0(a ) = s and ik(a ) = { 0 } when k > min(m , n ) . it follows immediately from the laplace expansion that ik(a)ik1(a ) for all k  . 
 definition 15 . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be any proper ideal of s which contains 1  . the (  , i)-rank of an m by n matrix a ( denoted by rankdet(a ) ) is the largest nonnegative integer k such that ik(a ) is not contained in i. 
  let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be any proper ideal of s which contains 1  . the (  , i)-rank of an m by n matrix a ( denoted by rankdet(a ) ) is the largest nonnegative integer k such that ik(a ) is not contained in i. for certain semirings , it may be possible that there is no choice of  for which 1   is not a unit . an example of this is the semiring of nonnegative real numbers  with the usual addition and multiplication . in this case the only    satisfying  = 1 is 1 itself and 2 = 1 + 1 is a unit . the max - plus and max - min semirings are other examples of this . for semirings such as these we will examine how to handle cases like this in the last section of the paper . the principal ideal generated by 1   is a natural choice for our ideal i. 
 definition 16 . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i1 be the principal ideal generated by 1  . the -rank of an m by n matrix a ( denoted by rankdet(a ) ) is the largest nonnegative integer k such that ik(a ) is not contained in i1. 
  let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i1 be the principal ideal generated by 1  . the -rank of an m by n matrix a ( denoted by rankdet(a ) ) is the largest nonnegative integer k such that ik(a ) is not contained in i1. it is clear that any i containing 1   contains i1 and therefore rankdet(a )  rankdet(a ) . recall that the standard definition of the rank of a matrix over a ring is the size of the largest k for which the ideal generated by all k by k subdeterminants of the matrix is nonzero [ 12 , page 82 ] . if s is a ring , then the (  , i)-rank of a matrix a over s is equal to the standard ring - theoretic rank of the matrix (a ) over the quotient ring s / i where  is the natural entrywise quotient map . we now examine some inequalities satisfied by these ranks that are implied by the binet - cauchy theorem , the laplace expansion , and our determinant sum identity . the factor rank ( or schein rank ) of a is the smallest integer r for which there exists an m by r matrix b and an r by n matrix c such that a = bc . the factor rank ( or schein rank ) of a is the smallest integer r for which there exists an m by r matrix b and an r by n matrix c such that a = bc . the inequality rankdet(a )  f(a ) holds whenever s is a commutative semiring and a  mm , n(s ) . the inequality rankdet(a )  f(a ) holds whenever s is a commutative semiring and a  mm , n(s ) . prooflet r = f(a ) , then there exist matrices b  mm , r(s ) and c  mr , n(s ) such that a = bc . let b^mm , r+1(s ) be the matrix obtained by adding a zero column to the right of b and let c^mr+1,n(s ) be the matrix obtained by adding a zero row to the bottom of c. clearly a = b^c^. now we compute the -minors of a(=b^c^ ) of order r + 1 , using the binet - cauchy theorem ; that is , det((b^c^)[,])=det(b^[,{1,2, ,r , r+1}])det(c^[{1,2,  ,r , r+1 } ,  ] )    [ 1   ] . since the first summand is 0 , so ir+1 is contained in the ideal generated by 1  . hence rankdet(a ) r. 
  let r = f(a ) , then there exist matrices b  mm , r(s ) and c  mr , n(s ) such that a = bc . let b^mm , r+1(s ) be the matrix obtained by adding a zero column to the right of b and let c^mr+1,n(s ) be the matrix obtained by adding a zero row to the bottom of c. clearly a = b^c^. now we compute the -minors of a(=b^c^ ) of order r + 1 , using the binet - cauchy theorem ; that is , det((b^c^)[,])=det(b^[,{1,2,  ,r , r+1}])det(c^[{1,2,  ,r , r+1 } ,  ] )    [ 1   ] . since the first summand is 0 , so ir+1 is contained in the ideal generated by 1  . hence rankdet(a )  r. we can also prove a version of sylvester 's inequality for the (  , i)-rank . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be an proper ideal of s which contains 1  . the inequality rankdet(ab )  min(rankdet(a ) , rankdet(b ) ) holds for all a  mm , n(s ) and b  mn , p(s ) . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be an proper ideal of s which contains 1  . the inequality rankdet(ab )  min(rankdet(a ) , rankdet(b ) ) holds for all a  mm , n(s ) and b  mn , p(s ) . if r  min(m , n , p ) we are done so suppose r < min(m , n , p ) and let  and  be both arbitrary subsets of { 1,2 ,  , then either ir+1(a ) or ir+1(b ) is contained in i. it follows from the binet - cauchy theorem that det(ab[ ,  ] )  i. 
  let r = min(rankdet(a ) , if r  min(m , n , p ) we are done so suppose r < min(m , n , p ) and let  and  be both arbitrary subsets of { 1,2 ,  , then either ir+1(a ) or ir+1(b ) is contained in i. it follows from the binet - cauchy theorem that det(ab[ ,  ] )  i. it should be noted that the condition 1    i is required for our version of sylvester 's inequality to hold ; this is largely our motivation for insisting on this condition . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be an proper ideal of s which contains 1  . the inequality rankdet(a + b )  rankdet(a ) + rankdet(b ) holds for all a , b  mm , n(s ) . let s be a commutative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be an proper ideal of s which contains 1  . the inequality rankdet(a + b )  rankdet(a ) + rankdet(b ) holds for all a , b  mm , n(s ) . m = n and rankdet(a ) + rankdet(b ) = n  1 . hence a , b  mn(s ) . we can use corollary 13 to show that det(a + b )  i. note that every term in the expansion of det(a + b ) is of a power of  times det(a[ ] )  det(b[ ] ) where  and  are subsets of { 1,2 ,  , n } satisfying | | = || . if k  r , then n  k  n  r > rankdet(b ) and since | | = | | = n  k we must have det(b[ ] )  i. similarly , if k > r , then det(a[ ] )  i. therefore every term in the expansion of det(a + b ) is in i and hence rankdet(a + b )  n  1 = rankdet(a ) + rankdet(b).now we prove the general case . if r  min(m , n ) then we are done so suppose r < min(m , n ) . , n } , respectively , both of cardinality r + 1 . then rankdet((a + b)[ ] )  rankdet(a[ ] ) + rankdet(b[])rankdet(a ) + rankdet(b ) = r. hence det((a + b)[ ] )  i and since ( a + b)[ ] is an arbitrary r + 1 by r + 1 submatrix of a + b , we have rankdet(a + b )  r = rankdet(a ) + rankdet(b ) . we begin by proving the inequality in the special case where m = n and rankdet(a ) + rankdet(b ) = n  1 . hence a , b  mn(s ) . we can use corollary 13 to show that det(a + b )  i. note that every term in the expansion of det(a + b ) is of a power of  times det(a[ ] )  det(b[ ] ) where  and  are subsets of { 1,2 ,  , n } satisfying | | = || . let k = | | = || . if k  r , then n  k  n  r > rankdet(b ) and since | | = | | = n  k we must have det(b[ ] )  i. similarly , if k > r , then det(a[ ] )  i. therefore every term in the expansion of det(a + b ) is in i and hence rankdet(a + b )  n  1 = rankdet(a ) + rankdet(b ) . if r  min(m , n ) then we are done so suppose r < min(m , n ) . now let  and  be subsets of { 1,2 , , m } and { 1,2 ,  , n } , respectively , both of cardinality r + 1 . then rankdet((a + b)[ ] )  rankdet(a[ ] ) + rankdet(b[])rankdet(a ) + rankdet(b ) = r. hence det((a + b)[ ] )  i and since ( a + b)[ ] is an arbitrary r + 1 by r + 1 submatrix of a + b , we have rankdet(a + b )  r = rankdet(a ) + rankdet(b ) . in this section , we look at bijective linear operators which preserve (  , i)-rank of matrices over antinegative commutative semiring . 
 definition 21 ( see ) . let s be a commutative semiring and a be an m by n matrix over s. the term rank of a is the minimum number of lines ( rows and columns ) needed to include all nonzero entries of a. the term rank of a matrix a is denoted by t(a ) . let s be a commutative semiring and a be an m by n matrix over s. the term rank of a is the minimum number of lines ( rows and columns ) needed to include all nonzero entries of a. the term rank of a matrix a is denoted by t(a ) . for any commutative semiring , one has f(a )  t(a ) whenever a is an m by n matrix over s. 
  for any commutative semiring , one has f(a )  t(a ) whenever a is an m by n matrix over s. let s be a semiring and a , b  mm , n(s ) . we write a  b if there exists c  mm , n(s ) such that a  c = b. we note that the relation (  ) is a reflexive and transitive relation but not antisymmetric in general . it is easy to check that any linear operator t : mm , n(s )  mm , n(s ) preserves this preorder . further , if s is an antinegative semiring then the term rank is a monotone function ; that is , if a  b then t(a )  t(b ) . let s be a commutative semiring and a , b  mm , n(s ) . the schur product of a and b , denoted as ab , is an m by n matrix whose ( i , j)th entry is aij  bij . let s be a commutative semiring and a , b  mm , n(s ) . the schur product of a and b , denoted as ab , is an m by n matrix whose ( i , j)th entry is aij  bij . is called a submonomial matrix if every line ( row or column ) of a contains at most one nonzero entry . a matrix a  mn(s ) is called a monomial matrix if every line ( row or column ) of a contains exactly one nonzero entry . is called a submonomial matrix if every line ( row or column ) of a contains at most one nonzero entry . a matrix a  mn(s ) is called a monomial matrix if every line ( row or column ) of a contains exactly one nonzero entry . the concept of ( p , q , b ) operator is a fundamental concept in the theory of linear preservers over semirings . one says that t is a strong ( p , q , b ) operator if there exist p  mm(s ) , q  mn(s ) , and b  mm , n(s ) such that p and q are permutation matrices , and all of the entries of b are units and either t(x ) = p(xb)q or m = n and t(x ) = p(xb)q . one says that t is a strong ( p , q , b ) operator if there exist p mm(s ) , q  mn(s ) , and b  mm , n(s ) such that p and q are permutation matrices , and all of the entries of b are units and either t(x ) = p(xb)q or m = n and t(x ) = p(xb)q . we note though there is an error earlier in   for the definition of the term rank , the following theorem is correct as stated as it only uses correct properties of the term rank . if rank:mm , n(s )   is a function which satisfies 0  rank(a )  t(a ) for all a  mm , n(s ) with equality whenever a is a submonomial matrix , then any bijective linear operator which preserves this rank function must be a strong ( p , q , b ) operator . if rank:mm , n(s )   is a function which satisfies 0  rank(a )  t(a ) for all a  mm , n(s ) with equality whenever a is a submonomial matrix , then any bijective linear operator which preserves this rank function must be a strong ( p , q , b ) operator . since the (  , i)-rank satisfies the hypotheses of the above theorem , we now have the following corollary which classifies all bijective linear operators which preserve the (  , i)-rank . let s be a commutative antinegative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be any proper ideal of s which contains 1  . any bijective (  , i)-rank preserver on mm , n(s ) must be a strong ( p , q , b ) operator . let s be a commutative antinegative semiring and let  be an element of s such that  = 1 and that 1   is not a unit . let i be any proper ideal of s which contains 1  . any bijective (  , i)-rank preserver on mm , n(s ) must be a strong ( p , q , b ) operator . in this section , we explore connections between the sign pattern matrices and -rank . a matrix whose entries are from the set { + 1 , 1,0 } is called a sign pattern matrix . if a = [ aij ] is a real matrix , then the sign pattern of a is obtained from a , by replacing each entry by its signs [ 14 , 15 ] . the sign pattern of a is denoted by sg(a ) = [ sg(aij ) ] , where ( 8)sg(aij)=0if  aij=0 + 1if  aij>01if  aij<0 . thus in a sign pattern matrix all we know is the sign of each entry . sometimes we may not know the signs of certain entries , so a new symbol , # , has been introduced to denote such entries . 
 definition 28 ( see ) . the generalized sign pattern matrices are the matrices over the set { + 1 , 1,0 , # } , where # corresponds to entries where the sign is unknown . the generalized sign pattern matrices are the matrices over the set { + 1 , 1,0 , # } , where # corresponds to entries where the sign is unknown . the set { + 1 , 1,0 , # } can be viewed as a semiring . if s = { + 1 , 1,0 , # } , then ( s ,  ,  ) is a commutative semiring with identity , where the operations of addition and multiplication are defined as follows : ( 9)0 + 11#00 + 11#+1 + 1 + 1##11#1######0 + 11#00000 + 10 + 11#101 + 1##0 # # # clearly all the properties of a semiring are satisfied where 0 is the additive identity and + 1 is multiplicative identity . here + 1 and 1 are the units of s. more about the sign pattern semiring can be found in . let a be a real matrix . the qualitative class of a is q(a ) , the set of all real matrices with the same sign pattern as a. 
  let a be a real matrix . the qualitative class of a is q(a ) , the set of all real matrices with the same sign pattern as a. 
 definition 30 ( see ) . a sign pattern matrix a is called sign - nonsingular ( sns ) if every matrix in its qualitative class is nonsingular . a sign pattern matrix a is called sign - nonsingular ( sns ) if every matrix in its qualitative class is nonsingular . for matrices over the sign pattern semiring the sign pattern semiring has only two elements whose square is the identity , namely , 1 and 1 . the ideal generated by 1 = 1 + 1 is the entire semiring but # = 1 + 1 generates the unique proper ideal { # , 0}. therefore 1 is the only available choice for  and we have a unique -rank . hence det(a ) = det+(a)(1  det(a ) ) . it is easy to show that an n by n sign pattern matrix has -rank n if and only if it is an sns matrix . hence the -rank of a sign pattern matrix a is the largest integer k for which there exists a k by k sns submatrix of a. it was remarked earlier that rankdet(a )  rankdet(a ) . in other words amongst the family of (  , i)-ranks , choosing i to be the ideal generated by 1   gives us the largest possible rank function from this family . the minimal rank functions from this family arise from the choice of i to be a maximal ideal which contains 1  . in general , there may be many maximal ideals . in this section , we will look at semirings which have a unique maximal ideal . we will use the term sublocal semiring to denote a semiring which has a unique maximal ideal . sublocality in semirings is essentially the straightforward generalization of the very useful concept of locality in rings . we use the term sublocal semiring because the term local semiring has been used to define a slightly different concept . an ideal i of a commutative semiring s is called a k - ideal if for any y  s with x , x + y  i one has y  i. 
  an ideal i of a commutative semiring s is called a k - ideal if for any y  s with x , x + y  i one has y  i. we note that if we consider a commutative ring r to be semiring , the semiring ideals of r are exactly the semiring k - ideals of r which are also exactly the ring ideals of r. 
 definition 32 ( see ) . a proper ideal i of a commutative semiring s is called a maximal ( resp . , k - maximal ) ideal if there exists no other proper ideal ( resp . , k - ideal )  j such that i  j. 
  a proper ideal i of a commutative semiring s is called a maximal ( resp . , k - maximal ) ideal if there exists no other proper ideal ( resp . , k - ideal )  j such that i  j. 
 definition 33 ( see ) . one says that s is a local semiring if s has only one k - maximal ideal . one says that s is a local semiring if s has only one k - maximal ideal . now we will define sublocal semirings using maximal ideals instead of k - maximal ideals . for example , the sign pattern semiring has only one proper ideal { 0 , # } and this is not a k - ideal . one says that s is a sublocal semiring if s has only one maximal ideal . one says that s is a sublocal semiring if s has only one maximal ideal . we note that both local and sublocal semirings are direct but different semiring generalizations of the concept of a local ring . local semirings have been useful in semiring theory ; see   for examples of this . there are many examples of sublocal semirings ; we list some of the notable ones . the sign pattern semiring is a sublocal semiring having only one maximal ideal i = { 0 , # } . we note that this maximal ideal is contained in the proper subsemiring p = { 0 , + 1 , # } ; p however fails to be an ideal in the sign pattern semiring . the set of all natural numbers ,  = { 0,1 , 2 ,  } , forms a sublocal semiring whose only one maximal ideal i = { /{1}}. all chain semirings are sublocal semirings with s/{1 } as a unique maximal ideal . a semifield is a commutative semiring in which all elements except 0 have a multiplicative inverse . we begin with the following elementary lemma whose proof is identical to the corresponding result for rings . an element a of a commutative semiring s is a unit of s if and only if a lies outside all maximal ideals of s. 
  an element a of a commutative semiring s is a unit of s if and only if a lies outside all maximal ideals of s. 
 prooflet a be a unit of s ; then the ideal generated by a must be s itself and hence a lies outside all maximal ideals of s. if a is not a unit of s , then 1  as and hence there exits a maximal ideal m of s such that a  asm . let a be a unit of s ; then the ideal generated by a must be s itself and hence a lies outside all maximal ideals of s. if a is not a unit of s , then 1  as and hence there exits a maximal ideal m of s such that a  asm . one of the key results of   is that a semiring is local if the set of all of its nonunits forms a k - ideal . a commutative semiring s is a sublocal semiring if and only if the set of all nonunits of s forms an ideal . a commutative semiring s is a sublocal semiring if and only if the set of all nonunits of s forms an ideal . since every k - ideal is an ideal , it follows that every local semiring is a sublocal semiring . the unique maximal ideal is {1 } ; the maximal k - ideals are of the form p for any prime p. since the set of nonunits in any sublocal semiring is an ideal , the nonunits are closed under addition . let a1 , a2  s. if a1  a2 is a unit of s , then either a1 or a2 is a unit of s. 
  let s be a sublocal semiring . let a1 , a2  s. if a1  a2 is a unit of s , then either a1 or a2 is a unit of s. the ranks introduced in the previous sections all require an element  satisfying the condition that  = 1 and 1   is not a unit . such an element may not exist in a given semiring ; the max - min and max - plus semirings are examples of semiring which lack an . fortunately , there is a known construction which allows us to append such an element . this construction is from , in which it was applied to the max - plus semiring . in this paper we explore applications of this construction both to general semirings and to the specific examples such as the boolean semiring and the sign pattern semiring . if ( s ,  ,  ) is a commutative semiring then s = { ( a , b)a , b  s } is also a commutative semiring with addition and multiplication defined as follows : for all a , b , c and d  s , ( 10)a , bc , d = ac , bd , a , bc , d = acbd , bcad . we can see that all the properties of a semiring are satisfied with ( 0 , 0 ) being the additive identity of s. essentially this construction allows us to append an element  = ( 0 , 1 ) with the property  = 1 to the semiring s in a natural way giving us a way to apply the -determinant theory to semirings which do not have nontrivial self inversive elements . the ideal in s generated by ( 1 , 1 ) = ( 1 , 0 ) +  is  = { ( x , x ) : x  s } which we will call the diagonal ideal . the -determinant in this case is the standard bideterminant and the -rank is the standard determinantal rank defined as follows . let a be an m by n matrix over a commutative semiring s. the determinantal rank of a is the largest k for which there exists b , a k by k submatrix of a with det+(b )  det(b ) . let a be an m by n matrix over a commutative semiring s. the determinantal rank of a is the largest k for which there exists b , a k by k submatrix of a with det+(b )  det(b ) . our results in section two generalize the known results on the determinantal rank of max - plus matrices to the (  , i)-rank of matrices over general semirings . remark 39 . recall that  denotes boolean semiring which has two elements { 0,1}. then  = { ( 0,0 ) , ( 1,0 ) , ( 0,1 ) , ( 1,1 ) } is also a semiring with the addition and the multiplication defined above for s. moreover it is isomorphic to the sign pattern semiring as ( 0,0 ) = 0 , ( 1,0 ) = + 1 , ( 0,1 ) = 1 , and ( 1,1 ) = # . recall that  denotes boolean semiring which has two elements { 0,1}. then  = { ( 0,0 ) , ( 1,0 ) , ( 0,1 ) , ( 1,1 ) } is also a semiring with the addition and the multiplication defined above for s. moreover it is isomorphic to the sign pattern semiring as ( 0,0 ) = 0 , ( 1,0 ) = + 1 , ( 0,1 ) = 1 , and ( 1,1 ) = # . we complete our paper by showing that the symmetrized semiring s inherits some important properties from s. 
 theorem 40 . if s is antinegative and has no zero divisors then s is also antinegative and has no zero divisors . if s is antinegative and has no zero divisors then s is also antinegative and has no zero divisors . let us suppose that ( a1 , b1 )  s has an additive inverse , so there exists ( a2 , b2 )  s , such that ( a1 , b1)(a2 , b2 ) = ( 0 , 0 ) . consequently a1  a2 = 0 and b1  b2 = 0 . since s is antinegative so a1 = a2 = b1 = b2 = 0 . hence ( a1 , b1 ) thus only the additive identity has an additive inverse in s which means s is an antinegative semiring . now suppose that ( a1 , b1 )  s is a zero divisor , so there exists a nonzero ( a2 , b2 )  s , such that ( a1 , b1)(a2 , b2 ) = ( 0 , 0 ) . it follows that ( ( a1  a2)(b1  b2 ) ) = 0 and ( ( a1  b2)(a2  b1 ) ) = 0 . since s is antinegative , a1  a2 = 0 and b1  b2 = 0 . also s has no zero divisors so either a1 = 0 or a2 = 0 and either b1 = 0 or b2 = 0 ; combining this with ( ( a1  b2)(a2  b1 ) ) = 0 , we get either ( a1 , b1 ) = ( 0 , 0 ) or ( a2 , b2 ) = ( 0 , 0 ) . hence s has no zero divisors . let us suppose that ( a1 , b1 )  s has an additive inverse , so there exists ( a2 , b2 )  s , such that ( a1 , b1)(a2 , b2 ) = ( 0 , 0 ) . consequently a1  a2 = 0 and b1  b2 = 0 . since s is antinegative so a1 = a2 = b1 = b2 = 0 . hence ( a1 , b1 ) thus only the additive identity has an additive inverse in s which means s is an antinegative semiring . now suppose that ( a1 , b1 )  s is a zero divisor , so there exists a nonzero ( a2 , b2 )  s , such that ( a1 , b1)(a2 , b2 ) = ( 0 , 0 ) . it follows that ( ( a1  a2)(b1  b2 ) ) = 0 and ( ( a1  b2)(a2  b1 ) ) = 0 . since s is antinegative , a1  a2 = 0 and b1  b2 = 0 . also s has no zero divisors so either a1 = 0 or a2 = 0 and either b1 = 0 or b2 = 0 ; combining this with ( ( a1  b2)(a2  b1 ) ) = 0 , we get either ( a1 , b1 ) = ( 0 , 0 ) or ( a2 , b2 ) = ( 0 , 0 ) . hence s has no zero divisors . if s is a semiring , we let u(s ) denote the set of units of s. there is a very close relation between the units of s and the units of s. 
 lemma 41 . if s is a commutative antinegative semiring with no zero divisors then u(s ) = { ( x , 0 ) : x  u(s)}{(0 , x ) : x  u(s)}. 
  if s is a commutative antinegative semiring with no zero divisors then u(s ) = { ( x , 0 ) : x  u(s)}{(0 , x ) : x  u(s)}. 
 prooflet x be a unit in s. then there exists a nonzero element a in s such that x  a = 1 . consider ( x , 0)(a , 0 ) = ( x  a , 0 ) = ( 1 , 0 ) and ( 0 , x)(0 , a ) = ( x  a , 0 ) = ( 1 , 0 ) . consequently ( x , 0 ) and ( 0 , x ) are units of s. now we have to prove that these are the only units for s. suppose ( x , y )  s is a unit in s. then there exists a nonzero element ( a , b )  s such that ( x , y)(a , b ) = ( 1 , 0 ) . thus ( ( a  x)(b  y ) , ( b  x)(a  y ) ) = ( 1 , 0 ) . consequently ( a  x)(b  y ) = 1 and ( b  x)(a  y ) = 0 . since ( b  x)(a  y ) = 0 and s is an antinegative semiring , so b  x = 0 and a  y = 0 . also given that s has no zero divisors it follows that either b = 0 or x = 0 ( note that both b and x can not be zero because ( a  x)(b  y ) = 1 ) and either a = 0 or y = 0 ( here also both a and y can not be zero because ( a  x)(b  y ) = 1 ) . since ( x , y ) and ( a , b ) are nonzero elements of s so the units of s , ( x , y ) , have only two choices which are ( x , 0 ) and ( 0 , y ) . putting ( x , y ) = ( x , 0 ) in ( a  x)(b  y ) = 1 , we get a  x = 1 , and this means that x is a unit of s. putting ( x , y ) = ( 0 , y ) in ( a  x)(b  y ) = 1 , we get b  y = 1 , and this means that y is a unit of s. thus all the units in s are of the type ( x , 0 ) and ( 0 , x ) where x is a unit in s. 
  let x be a unit in s. then there exists a nonzero element a in s such that x  a = 1 . consider ( x , 0)(a , 0 ) = ( x  a , 0 ) = ( 1 , 0 ) and ( 0 , x)(0 , a ) = ( x  a , 0 ) = ( 1 , 0 ) . consequently ( x , 0 ) and ( 0 , x ) are units of s. now we have to prove that these are the only units for s. suppose ( x , y )  s is a unit in s. then there exists a nonzero element ( a , b )  s such that ( x , y)(a , b ) = ( 1 , 0 ) . thus ( ( a  x)(b  y ) , ( b  x)(a  y ) ) = ( 1 , 0 ) . consequently ( a  x)(b  y ) = 1 and ( b  x)(a  y ) = 0 . since ( b  x)(a  y ) = 0 and s is an antinegative semiring , so b  x = 0 and a  y = 0 . also given that s has no zero divisors it follows that either b = 0 or x = 0 ( note that both b and x can not be zero because ( a  x)(b  y ) = 1 ) and either a = 0 or y = 0 ( here also both a and y can not be zero because ( a  x)(b  y ) = 1 ) . since ( x , y ) and ( a , b ) are nonzero elements of s so the units of s , ( x , y ) , have only two choices which are ( x , 0 ) and ( 0 , y ) . putting ( x , y ) = ( x , 0 ) in ( a  x)(b  y ) = 1 , we get a  x = 1 , and this means that x is a unit of s. putting ( x , y ) = ( 0 , y ) in ( a  x)(b  y ) = 1 , we get b  y = 1 , and this means that y is a unit of s. thus all the units in s are of the type ( x , 0 ) and ( 0 , x ) where x is a unit in s. we can now prove that if s is a sublocal antinegative semiring with no zero divisors then so is s. 
 theorem 42 . if s is a sublocal antinegative semiring with no zero divisors then s is also a sublocal antinegative semiring with no zero divisors . if s is a sublocal antinegative semiring with no zero divisors then s is also a sublocal antinegative semiring with no zero divisors . proofsuppose s is a sublocal antinegative semiring with no zero divisors ; then by theorem 40 , s is an antinegative semiring with no zero divisors . hence we only need to prove that s is sublocal ; we show this by proving that the set of all nonunits in s forms an ideal of s. let m = { ( a , b ) : where  ( a , b )  is  not  a  unit  of  s } be the set of all nonunits of s. let ( a1 , b1 ) and ( a2 , b2 )  m such that ( a1 , b1)(a2 , b2 ) = ( x , 0 ) , where x is a unit in s. hence a1  a2 = x and b1  b2 = 0 . since s is antinegative so b1 = 0 and b2 = 0 , and also a1  a2 = x , where x is a unit in s so ( using corollary 37 ) either a1 is a unit or a2 is a unit . thus either ( a1 , b1 ) = ( a1 , 0 ) , where a1 is a unit in s , or ( a2 , b2 ) = ( a2 , 0 ) , where a2 is a unit in s. it follows that either ( a1 , b1 ) is a unit or ( a2 , b2 ) is a unit in s , which is a contradiction to the fact that both ( a1 , b1 ) and ( a2 , b2 )  m. thus the sum of nonunits in s is a nonunit . a similar argument works if ( a1 , b1)(a2 , b2 ) = ( 0 , x ) , where x is a unit in s. now suppose that for ( a , b )  m and ( s1 , s2 )  s we have ( a , b)(s1 , s2 ) = ( x , 0 ) , where x is a unit in s. then ( a  s1)(b  s2 ) = x and ( a  s2)(b  s1 ) = 0 . since s is antinegative so a  s2 = 0 and b  s1 = 0 , and also s has no zero divisors so either a = 0 or s2 = 0 and either b = 0 or s1 = 0 . clearly ( a , b ) or ( s1 , s2 ) can not be ( 0 , 0 ) since ( a  s1)(b  s2 ) = x , so we have ( a , b ) = ( 0 , b ) or ( a , 0 ) . further x is a unit in s and ( a  s1)(b  s2 ) = x , so either a  s1 is a unit in s or b  s2 is a unit in s. hence either a is a unit in s or b is a unit in s. we get ( a , b ) = ( 0 , b ) or ( a , 0 ) is a unit in s , which is a contradiction to the fact that ( a , b )  m. thus ( a , b)(s1 , s2 )  m for all ( a , b )  m and ( s1 , s2 )  s. hence the set of all nonunits in s forms an ideal of s. consequently s is a sublocal semiring . a similar argument works if ( a , b)(s1 , s2 ) = ( 0 , x ) , where x is a unit in s. 
  suppose s is a sublocal antinegative semiring with no zero divisors ; then by theorem 40 , s is an antinegative semiring with no zero divisors . hence we only need to prove that s is sublocal ; we show this by proving that the set of all nonunits in s forms an ideal of s. let m = { ( a , b ) : where  ( a , b )  is  not  a  unit  of  s } be the set of all nonunits of s. let ( a1 , b1 ) and ( a2 , b2 )  m such that ( a1 , b1)(a2 , b2 ) = ( x , 0 ) , where x is a unit in s. hence a1  a2 = x and b1  b2 = 0 . since s is antinegative so b1 = 0 and b2 = 0 , and also a1  a2 = x , where x is a unit in s so ( using corollary 37 ) either a1 is a unit or a2 is a unit . thus either ( a1 , b1 ) = ( a1 , 0 ) , where a1 is a unit in s , or ( a2 , b2 ) = ( a2 , 0 ) , where a2 is a unit in s. it follows that either ( a1 , b1 ) is a unit or ( a2 , b2 ) is a unit in s , which is a contradiction to the fact that both ( a1 , b1 ) and ( a2 , b2 )  m. thus the sum of nonunits in s is a nonunit . a similar argument works if ( a1 , b1)(a2 , b2 ) = ( 0 , x ) , where x is a unit in s. now suppose that for ( a , b )  m and ( s1 , s2 )  s we have ( a , b)(s1 , s2 ) = ( x , 0 ) , where x is a unit in s. then ( a  s1)(b  s2 ) = x and ( a  s2)(b  s1 ) = 0 . since s is antinegative so a  s2 = 0 and b s1 = 0 , and also s has no zero divisors so either a = 0 or s2 = 0 and either b = 0 or s1 = 0 . clearly ( a , b ) or ( s1 , s2 ) can not be ( 0 , 0 ) since ( a  s1)(b  s2 ) = x , so we have ( a , b ) = ( 0 , b ) or ( a , 0 ) . further x is a unit in s and ( a  s1)(b  s2 ) = x , so either a  s1 is a unit in s or b  s2 is a unit in s. hence either a is a unit in s or b is a unit in s. we get ( a , b ) = ( 0 , b ) or ( a , 0 ) is a unit in s , which is a contradiction to the fact that ( a , b )  m. thus ( a , b)(s1 , s2 )  m for all ( a , b )  m and ( s1 , s2 )  s. hence the set of all nonunits in s forms an ideal of s. consequently s is a sublocal semiring . a similar argument works if ( a , b)(s1 , s2 ) = ( 0 , x ) , where x is a unit in s.
we use the -determinant introduced by ya - jia tan to define a family of ranks of matrices over certain semirings . 
 we show that these ranks generalize some known rank functions over semirings such as the determinantal rank . 
 we also show that this family of ranks satisfies the rank - sum and sylvester inequalities . 
 we classify all bijective linear maps which preserve these ranks .