Singular Manual: Examples of use of Letterplace

| ### 7.7.1 Examples of use of LETTERPLACE First, define a commutative ring in , equipped with a monomial well-ordering and call it, say, `r`. Then, decide what should be the degree (length) bound , that is how long may the words (monomials in the free algebra) become and run the procedure `freeAlgebra(r, d)`. In the case you wish to work with subbimodules of the free bimodule of rank , use `freeAlgebra(r, d, k)` instead of the previous. The `freeAlgebra`. procedure creates free algebra resp. the free bimodule of rank over subject to a monomial (module) ordering, corresponding to the one in the original commutative ring , see Monomial orderings on free algebras. Polynomial (vector) arithmetics in this -algebra is the usual one: `+,-,*,^` while of course, `x*y` and `y*x` are different monomials while `x*7=7*x`. Let us define an ideal `I` as a list of polynomials in the free algebra and run, for example, `twostd` (see twostd (letterplace)). The answer is a two-sided Groebner basis `J` of the two-sided ideal up to the length bound `d`. Then, we want to compute the following: 1. The two-sided normal form of `xyzy` with respect to `J` using the function `reduce` (see reduce (letterplace)). 2. By introducing a factor algebra of type `qring`, and demonstrate the functions `reduce` and `rightstd` (for right Groebner bases) over the factor algebra. 3. By creating the free `R`-bimodule of rank 8, we demonstrate how embeddins works with `imap` and also, how to express a subbimodule (or a single element) in terms of bimodule generators with `lift`. In other words, we compute and compare presentations of a polynomials with respect to the original generating set of ideal and with respect to a Groebner basis. 4. In the same free `R`-bimodule we will compute the module of bisyzygies of `J` and do some syzygy tests. 5. We demonstrate the bimodule membership problem: a boolean answer via `NF` and the certified version (with a Groebner presentation) via `lift`. 6. We show how elimination of module components works for bimodules. We illustrate the approach with the following example:
    //******* Part 1 *******//
    LIB "freegb.lib";
    ring r = 0,(x,y,z),dp; // the ordering on the free algebra will be degree right lex
    ring R = freeAlgebra(r, 5);  // 5 the is degree (length) bound;
    ideal I = x*y + y*z, x*x + x*y - z; // define an ideal via the set of polynomials
    ideal J = twostd(I);
    J; // as we see, with respect to the current ordering this Groebner basis
    ==> J[1]=x*y+y*z
    ==> J[2]=x*x-y*z-z
    ==> J[3]=y*z*y-y*z*z+z*y
    ==> J[4]=y*z*x+y*z*z+z*x-x*z
    ==> J[5]=y*z*z*y-y*z*z*z-x*z*y
    ==> J[6]=y*z*z*x+y*z*z*z-x*z*x+y*z*z+z*z
    ==> J[7]=y*z*z*z*y-y*z*z*z*z+y*z*z*z+x*z*y+z*z*y
    ==> J[8]=y*z*z*z*x+y*z*z*z*z+x*z*x+z*z*x-x*z*z-y*z*z-z*z
    // tends to be infinite. Increasing the bound and recomputing helps to check it.
    poly p = reduce(x*y*z*y,J);
    p; // since p!=0, x*y*z*y is not contained in J up to length 5
    ==> -y*z*z*z-x*z*y
    // however this does not imply a definite answer on whether p is in J
    poly q = x*(y+1)*z*y-x*y*z^2;
    reduce(q, J); // 0, thus q is in J
    ==> 0
    //******* Part 2 *******//
    qring Q = J; // J is a Groebner basis, computed above
    poly p = reduce(x*x, twostd(0)); // the canonical representative of x*x in Q
    p;
    ==> y*z+z
    rightstd(ideal(p)); // right Groebner basis of the right ideal, generated by p in Q
    ==> _[1]=z*z
    ==> _[2]=y*z+z
    ==> _[3]=x*z
    //******* Part 3 *******//
    setring r;
    ring R8 = freeAlgebra(r, 5, 8);  // 5 the is length bound; 8 is the rank of the free bimodule
    ideal J = imap(R, J); // we map J identically from R (of rank 1)
    J = twostd(J);
    poly q = imap(R, q);
    NF(q, J); // NF is an alias to reduce, we have rechecked that q is in J
    ==> 0
    matrix L = lift(J, q); // creates the presentation for q in terms of J
    // since J is a Groebner basis, this is a Groebner presentation of q
    print(transpose(matrix(L))); // J has 8 generators and these are the needed coefficients
    ==> ncgen(1)*z*y-ncgen(1)*z*z,0,0,0,-ncgen(5),0,0,0
    // here, the generators of the free bimodule are ncgen(1)*gen(1), ... , ncgen(8)*gen(8)
    // the output means, that substituting ncgen(i) by the i-th generator of J, we get q
    J[1]*z*y - J[1]*z*z  - J[5] - q; // 0, so this is the seeked expression of q
    ==> 0
    testLift(J,L); // recovers q from the lift matrix
    ==> _[1]=x*y*z*y-x*y*z*z+x*z*y
    // Let us compare now this nice Groebner presentation with the one
    // obtained from the original set of generators
    ideal I = imap(R,I); // note: I is not a Groebner basis of itself
    matrix M = lift(I, q); // creates the presentation for q in terms of I
    M; // presentation is longer and more complicated than the one in L
    ==> M[1,1]=-ncgen(1)*x*y+x*ncgen(1)*y-ncgen(1)*y*z-x*ncgen(1)*z+y*z*ncgen(1)+\
       z*ncgen(1)
    ==> M[2,1]=ncgen(2)*x*y-x*ncgen(2)*y+ncgen(2)*y*z
    testLift(I,M); // a routine test to ensure that indeed we recover q
    ==> _[1]=x*y*z*y-x*y*z*z+x*z*y
    //******* Part 4 *******//
    // Let us compute the module of bisyzygies of J and analyze it
    module S = syz(J); size(S); // 18
    ==> 18
    S[6]; // consider, for example, this element
    ==> ncgen(1)*z*y*gen(1)-ncgen(1)*z*z*gen(1)+y*z*ncgen(1)*gen(1)-ncgen(4)*y*ge\
       n(4)-ncgen(3)*z*gen(3)+z*ncgen(1)*gen(1)-x*ncgen(3)*gen(3)
    // plugging the i-th generator of J instead of ncgen(i), we obtain a bisyzygy:
    J[1]*z*y - J[1]*z*z - x*J[3] - J[5]; //0
    ==> 0
    module S2 = S[6..8]; // pick just three generators
    testSyz(J,S2); // tests the bisyzygy property for the generators
    ==> _[1]=0
    ==> _[2]=0
    ==> _[3]=0
    //******* Part 5 *******//
    option(redSB); option(redTail); // to compute minimal and tail-reduced bases
    module GS = twostd(S); size(GS); // 30
    ==> 30
    // let us construct a vector, belonging to GS:
    vector v = GS[11]*y - x*GS[7] + z*GS[3]*z;
    print(v);
    ==> [-x*ncgen(1)*x*y-x*ncgen(1)*y*z+x*x*y*ncgen(1)-x*z*ncgen(1),y*z*z*ncgen(2\
       )*y+z*y*z*ncgen(2)*z-x*z*ncgen(2)*y+z*z*ncgen(2)*z,z*ncgen(3)*z*z+x*ncgen\
       (3)*z,-z*ncgen(4)*x*z+x*ncgen(4)*y,ncgen(5)*z*y-x*ncgen(5),-ncgen(6)*x*y+\
       z*ncgen(6)*z+ncgen(6)*y,0,ncgen(8)*y]
    NF(v, GS); // 0, by the construction
    ==> 0
    // now we wish to compute the expression of v via GS
    ring r3 = 0,(x,y,z),(c,dp);
    ring R30 = freeAlgebra(r3,5,30);
    module GS = imap(R8,GS);
    vector v = imap(R8,v);
    matrix L = lift(GS,v); // via printing we see only three components involved:
    L[3,1]; // =z*ncgen(3)*z, as well as
    ==> z*ncgen(3)*z
    L[7,1]; // =-x*ncgen(7) and
    ==> -x*ncgen(7)
    L[11,1]; // =ncgen(11)*y
    ==> ncgen(11)*y
    //******* Part 6 *******//
    // Notice, that the module ordering is (c,dp): it is a position-over-term ordering
    // which eliminates module components in an descending way.
    GS = GS[1..5]; // consider just first five syzygies,  for a smaller example
    GS = twostd(GS); // a nice finite Groebner basis
    print(matrix(GS)); // shows the structure
    ==> 0,       0,       0,     0,     0,     _[1,6],   _[1,7],   _[1,8],
    ==> 0,       _[2,2],  _[2,3],_[2,4],_[2,5],_[2,6],   _[2,7],   0,
    ==> _[3,1],  _[3,2],  _[3,3],_[3,4],_[3,5],-ncgen(3),0,        _[3,8],
    ==> 0,       _[4,2],  _[4,3],_[4,4],_[4,5],-ncgen(4),-ncgen(4),_[4,8],
    ==> ncgen(5),0,       0,     _[5,4],_[5,5],0,        0,        ncgen(5),
    ==> 0,       ncgen(6),0,     _[6,4],_[6,5],0,        0,        0,
    ==> ncgen(7),0,       0,     0,     _[7,5],0,        0,        0
    // As we can see, intersections of the subbimodule GS with the free bimodules
    // generated by all but first resp. all but first two bimodule generators
    // are not empty and given by vectors having zero in the first resp.
    // in the first two components.
See Functions (letterplace) for the list of all available kernel functions. There are various conversion routines in the library `freegb_lib` (see freegb_lib). Many algebras are predefined in the library `fpalgebras_lib` (see fpalgebras_lib). Important ring-theoretic properties can be established with the help of the library `fpaprops_lib` (see fpaprops_lib), while K-dimension and monomial bases and Hilbert data - with the help of the library `fpadim_lib` (see fpadim_lib). We work further on implementing more algorithms for non-commutative ideals and modules over free associative algebra.

User manual for Singular version 4.4.1, 2025, generated by texi2html.