| #### D.15.6.11 grtranspose Procedure from library `gradedModules.lib` (see gradedModules_lib). Usage: grtranspose(M), graded object M Return: graded object Purpose: graded transpose of M Note: no reordering is performend by this procedure Example:
    LIB "gradedModules.lib";
    ring r=32003,(x,y,z),dp;
    module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);
    ==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\
       x.
    module N = grsyz( grtranspose( M ) ); grview(N);
    ==> Graded homomorphism: r(2) + r + r(-4)^2 <- r(2) + r + r(-4)^2, given by a\
        diagonal matrix, with degrees:
    ==>      ..1 ..2 ..3 ..4 ....
    ==>      --- --- --- --- +...
    ==>  -2 :  0   -   -   - |..1
    ==>   0 :  -   0   -   - |..2
    ==>   4 :  -   -   0   - |..3
    ==>   4 :  -   -   -   0 |..4
    ==>      === === === ===
    ==>       -2   0   4   4
    module L = grtranspose(N); grview( L );
    ==> Graded homomorphism: r(-2) + r + r(4)^2 <- r(-2) + r + r(4)^2, given by a\
        diagonal matrix, with degrees:
    ==>      ..1 ..2 ..3 ..4 ....
    ==>      --- --- --- --- +...
    ==>   2 :  0   -   -   - |..1
    ==>   0 :  -   0   -   - |..2
    ==>  -4 :  -   -   0   - |..3
    ==>  -4 :  -   -   -   0 |..4
    ==>      === === === ===
    ==>        2   0  -4  -4
    module K = grsyz( L ); grview(K);
    ==> Graded homomorphism: r(-2) + r + r(4)^2 <- 0, given by zero (4 x 0) matri\
       x.
    // Corner cases: 0 <- 0!
    module Z = grzero(); grview(Z);
    ==> Graded homomorphism: 0 <- 0, given by zero (0^2) matrix.
    grview( grtranspose( Z ) );
    ==> Graded homomorphism: 0 <- 0, given by zero (0^2) matrix.
    // Corner cases: * <- 0
    matrix M1[3][0];
    module Z1 = grobj( M1, intvec(-1, 0, 1) ); grview(Z1);
    ==> Graded homomorphism: r(1) + r + r(-1) <- 0, given by zero (3 x 0) matrix.
    grview( grtranspose( Z1 ) );
    ==> Graded homomorphism: 0 <- r(-1) + r + r(1), given by zero (0 x 3) matrix.
    // Corner cases: 0 <- *
    matrix M2[0][3];
    module Z2 = grobj( M2, 0:0, intvec(-1, 0, 1) ); grview(Z2);
    ==> Graded homomorphism: 0 <- r(1) + r + r(-1), given by zero (0 x 3) matrix.
    grview( grtranspose( Z2 ) );
    ==> Graded homomorphism: r(-1) + r + r(1) <- 0, given by zero (3 x 0) matrix.

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