basics2.gd

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#W  basics2.gd              Manuel Delgado <[email protected]>
#W                          Pedro A. Garcia-Sanchez <[email protected]>
#W                          Jose Morais <[email protected]>
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##
#Y  Copyright 2005 by Manuel Delgado,
#Y  Pedro Garcia-Sanchez and Jose Joao Morais
#Y  We adopt the copyright regulations of GAP as detailed in the
#Y  copyright notice in the GAP manual.
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#O  IsSubsemigroupOfNumericalSemigroup(S,T)
#O  IsSubset (is a synonym)
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##  Test whether the numerical semigroup T is contained in the
##  numerical semigroup S
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DeclareOperation( "IsSubsemigroupOfNumericalSemigroup",[IsNumericalSemigroup,IsNumericalSemigroup]);
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#DeclareOperation( "IsSubset",[IsNumericalSemigroup,IsNumericalSemigroup]);

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#F DifferenceOfNumericalSemigroups(S,T)
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## returns the set difference S\T 
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DeclareGlobalFunction("DifferenceOfNumericalSemigroups");


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#F  IntersectionOfNumericalSemigroups(S,T)
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##  Returns the intersection of the numerical
##  semigroups S and T.
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DeclareGlobalFunction( "IntersectionOfNumericalSemigroups" );




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#F  RepresentsGapsOfNumericalSemigroup(L)
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##  Tests if the given list L represents the gaps of
##  some numerical semigroup.
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DeclareGlobalFunction("RepresentsGapsOfNumericalSemigroup");


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#F  NumericalSemigroupsWithFrobeniusNumber(g)
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##  Computes the set of numerical semigroups with Frobenius number g.
##  The algorithm is based on
##  "Fundamental gaps in numerical semigroup".
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DeclareGlobalFunction("NumericalSemigroupsWithFrobeniusNumber");

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#F NumericalSemigroupsWithGenus
##computes the set of numerical semigroups with genus g,
# that is, numerical semigroups with exactly g gaps
#
#
# numerical semigroups are encoded in lists containing the apery set with
# respect to the multiplicity removing the zero element. The multiplicity
# is thus the lenght of the list plus one. In this way deciding membership
# to a numerical semigroup is straightforward (belongs). The computation of
# the Frobenius number is performed using Selmer's idea (frob). Removing a new
# generator is easy (removegen), as well as computing those minimal generators
# greater than the Frobenius number (minimalgeneratorsf).
# Given a numerical semigroup of genus g, removing minimal generators, one
# obtains numerical semigroups of genus g+1. In order to avoid repetitions,
# we only remove minimal generators greater than the frobenius number of
# the numerical semigroup (this is accomplished with the local function sons).
# References:
# -J. C. Rosales, P. A. García-Sánchez, J. I. García-García and 
#  J. A. Jimenez-Madrid, The oversemigroups of a numerical semigroup.
#  Semigroup Forum 67 (2003), 145--158.
# -M. Bras-Amorós, Fibonacci-like behavior of the number of numerical
#  semigroups of a given genus. Semigroup Forum 76 (2008), 379--384.
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DeclareGlobalFunction("NumericalSemigroupsWithGenus");