Code bachelor's thesis, computer science

This GAP code is intended to work with the NumericalSgps GAP package. There are three main parts to this code.

Ideals

We give two methods for ideal decomposition.

• IdealDecomposition(I) gives the decomposition the relative Ideal, I, into Z-irreducible ideals.

• ProperIdealDecomposition(I) gives the decomposition the proper Ideal, I, into proper irreducible ideals.

Characterization

The first part is focused in alternative characterizations of numerical semigroups, based on nu, tau and oplus.

• Nu(S,i) computes the i-th value of the Nu-sequence, and NuSequence(S) computes the Nu-Sequence of the semigroup S, up to Nu_2*c-g (from that point onwards, it increases 1 by 1)

• SemigroupFromNu(Nu) computes the numerical semigroup with nu sequence nu, assuming Nu contains enough elements to determine the semigroup.

• TauSemigroup(S,i) computes the i-th value of the Tau-sequence, and TauSequence(S) computes the Nu-Sequence of the semigroup S, up to Nu_2*c-g (from that point onwards, it increases by 1 every two elements)

• TauSequence(S) Computes the TauSequence of the semigrup S, up to 2c-g+1. For numbers greater than that, it increases 1 by 1, every 2, Tau_{2c-g+1}=Tau_{2c-g+2}.

• SemigroupFromTau(Tau) computes the numerical semigroup with tau sequence Tau, assuming Tau contains enough elements to determine the semigroup.

• Oplus(i,j,S) computes the operation oplus for indexes i and j in the semigroup S. Returns k=i oplus j.

• SemigroupFromOplus(O) computes the semigroup, using the Oplus operation, given as a list of triplets. For testing, there is a createO function in the Test folder that can generate this sequence from a list of tuples. In the same way, NuFromOplus computes the nu sequence, given the oplus operation as triplets.

Feng Rao distance

The third part focuses on the computation of Feng-Rao distances, for special cases.

• FengRaoDistance2(s,r,m) Computes the r-th Feng-Rao distance, for element m in semigrup s. It is based on the default function, FengRaoDistance, implemented in the package NumericalSgps. It incorporates all the functions that are listed below.

• FengRaoDistanceArf(S,m) Computes the 1-st Feng-Rao distance of S, an Arf numerical Semigroup.

• FengRaoDistance2(S,m) Computes the second Feng-Rao distance of m<2c-1, m in S, an Arf numerical Semigroup.

• FengRaoDistanceBruteForce(S,m) Comptes the first Feng-Rao distance of S, in a non recursive way.

• FengRaoDistanceOrdinary2(S,m) Computes the second Feng-Rao distance of c<m<2c-1, in the semigrup S, an ordinary Semigroup.

• FengRaoDistanceSymmetric(S,m) Computes the first Feng-Rao Distance of a Symmetric semigrup, for m in [c+1, 2c-2]

• FengRaoDistanceSymmetricGeneralized(S,r,m) Computes the r-th Feng-Rao distance of m, in a Symmetric semigroup, S, for m in [c+1, 2c-2], if m-2*g+1 in s.

• FengRaoDistanceTwoGenerators(S,m) Computes the first Feng-Rao distance of m<2c-1, an element of S, a semigroup generated by 2 elements.