Singular

7.7.3 Functionality and release notes of LETTERPLACE

With the present functionality it is possible to compute two-sided Groebner basis of an arbitrary two-sided ideal in a free associative algebra up to a given degree.

The weights of variables are nonnegative and are determined by the current monomial ordering.

Restrictions/conventions of the LETTERPLACE subsystem:

• Since free algebra is not Noetherian, one has to work with explicitly fixed degree (length) bound, up to which a partial Groebner basis will be computed. The initialization routine freeAlgebra (letterplace) constructs the ring with this bound. For increasing the length bound one needs to define another ring and to use imap for mapping the objects over.
• All the computations happen up to the explicitly fixed length bound.
• The options redSB, redTail are effective for computations involving Groebner bases,
• The options prot, mem are effective for the whole LETTERPLACE subsystem.
• For monomial orderings, which are not compatible with the length, the following error message might appear: degree bound of Letterplace ring is 11, but at least 12 is needed for this multiplication In such a situation, activating option(redSB), option(redTail) and increaing the length (degree) bound might help. Though there are situations, where nothing leads to a finite computation.

Operations for polynomials in Letterplace rings are the usual ones: + (addition), - (subtraction), * (multiplication) and ^ (power).

The functions bracket, maxideal and std (an alias for twostd (letterplace)) also work within letterplace rings:

LIB "freegb.lib";
ring r = 0,(x,y,z),dp; // the ordering will be degree right lex
ring R = freeAlgebra(r, 5);  // degree (length) bound is 5
// maxideal in a letterplace ring:
print(matrix(maxideal(2))); // all monomials of length 2
==> x*x,y*x,z*x,x*y,y*y,z*y,x*z,y*z,z*z
// bracket in a letterplace ring:
bracket(x,y);
==> -y*x+x*y
poly f = x*x + x*y - z;
bracket(f,x);
==> x*y*x-x*x*y-z*x+x*z
bracket(f,x,2);
==> -x*y*x*x*x+x*x*y*x*x+x*x*x*y*x+x*y*x*y*x-x*x*x*x*y-2*x*y*x*x*y+x*x*y*x*y+\
   z*x*x*x-x*z*x*x-z*x*y*x-x*x*z*x-x*y*z*x+2*z*x*x*y-x*z*x*y+x*x*x*z+2*x*y*x\
   *z-x*x*y*z+z*z*x-2*z*x*z+x*z*z

Further functionality is provided in the libraries for the LETTERPACE subsystem: see LETTERPLACE libraries for details.

In the freegb_lib one finds e.g. Letterplace initialization together with legacy, conversion and convenience tools.

The fpadim_lib contains procedures for computations with vector space basis of a factor algebra including finiteness check and dimension computation.

The fpaprops_lib contains procedures for determining important ring-theoretic properties including Gelfand-Kirillov dimension.

The fpalgebras_lib contains procedures for the generation of various algebras, including group algebras of finitely presented groups in the Letterplace ring.

The ncfactor_lib contains the procedure ncfactor for factorizing polynomials in the Letterplace ring.

See bracket; maxideal; reduce (letterplace); rightstd (letterplace); std (letterplace); twostd (letterplace).