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Project: cocalc-sagemath-dev-slelievre
Path: gap4r8 / pkg / NumericalSgps-1.1.5 / doc / Numerical_semigroups_with_maximal_embedding_dimension.xml
Views: 418346<?xml version="1.0" encoding="UTF-8"?>1<Section>2<Heading>3Numerical semigroups with maximal embedding dimension4</Heading>5If <M>S</M> is a numerical semigroup and <M>m</M> is its multiplicity (the least6positive integer belonging to it), then the embedding dimension <M>e</M> of <M>S</M> (the7cardinality of the minimal system of generators of <M>S</M>) is less than or equal8to <M>m</M>. We say that <M>S</M> has maximal embedding dimension (MED for short) when <M>e=m</M>.9The intersection of two numerical semigroups with the same multiplicity and maximal embedding10dimension is again of maximal embedding dimension. Thus we define the MED closure of a non-empty subset11of positive integers <M>M=\{m < m_1 < \cdots < m_n <\cdots\}</M> with <M>\gcd(M)=1</M>12as the intersection of all MED numerical semigroups with multiplicity <M>m</M>.1314<P/>15Given a MED numerical semigroup <M>S</M>, we say that <M>M=\{m_1 < \cdots< m_k\}</M> is a MED system16of generators if the MED closure of <M>M</M> is <M>S</M>. Moreover, <M>M</M> is a minimal MED generating17system for <M>S</M> provided that every proper subset of <M>M</M> is not a MED system of generators18of <M>S</M>. Minimal MED generating systems are unique, and in general are smaller than the19classical minimal generating systems (see <Cite Key="RGGB03"></Cite>).202122<ManSection>23<Prop Arg="S" Name="IsMED"></Prop>24<Prop Arg="S" Name="IsMEDNumericalSemigroup"></Prop>25<Description>26<A>S</A> is a numerical semigroup.27Returns true if <A>S</A> is a MED numerical semigroup and false otherwise.28<Example><![CDATA[29gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5,7));30true31gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,5));32false33]]></Example>34</Description>35</ManSection>363738<ManSection>39<Func Arg="S" Name="MEDNumericalSemigroupClosure"></Func>40<Oper Arg="S" Name="MEDClosure" Label="for numerical semigroups"></Oper>41<Description>42<A>S</A> is a numerical semigroup.43Returns the MED closure of <A>S</A>.44<Example><![CDATA[45gap> MEDNumericalSemigroupClosure(NumericalSemigroup(3,5));46<Numerical semigroup>47gap> MinimalGenerators(last);48[ 3, 5, 7 ]49]]></Example>50</Description>51</ManSection>5253<ManSection>54<Func Arg="S" Name="MinimalMEDGeneratingSystemOfMEDNumericalSemigroup"></Func>55<Description>56<A>S</A> is a MED numerical semigroup.57Returns the minimal MED generating system of <A>S</A>.58<Example><![CDATA[59gap> MinimalMEDGeneratingSystemOfMEDNumericalSemigroup(60> NumericalSemigroup(3,5,7));61[ 3, 5 ]62]]></Example>63</Description>64</ManSection>6566</Section>6768<Section>69<Heading>70Numerical semigroups with the Arf property and Arf closures71</Heading>72Numerical semigroups with the Arf property are a special kind of numerical semigroups with73maximal embedding dimension. A numerical semigroup <M>S</M> is Arf if for every <M>x,y,z</M> in74<M>S</M> with <M>x\geq y\geq z</M>, one has that <M>x+y-z\in S</M>.7576<P/>77The intersection of two Arf numerical semigroups is again Arf, and thus we can consider the78Arf closure of a set of nonnegative integers with greatest common divisor equal to one. Analogously79as with MED numerical semigroups, we define Arf systems of generators and minimal Arf generating80system for an Arf numerical semigroup. These are also unique(see <Cite Key="RGGB04"></Cite>).818283<ManSection>84<Prop Arg="S" Name="IsArf"></Prop>85<Prop Arg="S" Name="IsArfNumericalSemigroup"></Prop>86<Description>87<A>S</A> is a numerical semigroup.88Returns true if <A>S</A> is an Arf numerical semigroup and false otherwise.89<Example><![CDATA[90gap> IsArfNumericalSemigroup(NumericalSemigroup(3,5,7));91true92gap> IsArfNumericalSemigroup(NumericalSemigroup(3,7,11));93false94gap> IsMEDNumericalSemigroup(NumericalSemigroup(3,7,11));95true96]]></Example>97</Description>98</ManSection>99100<ManSection>101<Func Arg="S" Name="ArfNumericalSemigroupClosure"></Func>102<Oper Arg="S" Name="ArfClosure" Label="of numerical semigroup"></Oper>103<Description>104<A>S</A> is a numerical semigroup.105Returns the Arf closure of <A>S</A>.106<Example><![CDATA[107gap> ArfNumericalSemigroupClosure(NumericalSemigroup(3,7,11));108<Numerical semigroup>109gap> MinimalGenerators(last);110[ 3, 7, 8 ]111]]></Example>112</Description>113</ManSection>114115<ManSection>116<Func Arg="S" Name="ArfCharactersOfArfNumericalSemigroup"></Func>117<Func Arg="S" Name="MinimalArfGeneratingSystemOfArfNumericalSemigroup"></Func>118<Description>119<A>S</A> is an Arf numerical semigroup.120121Returns the minimal Arf generating system of <A>S</A>.122123The current version of this algorithm is due to G. Zito.124125<Example><![CDATA[126gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup(127> NumericalSemigroup(3,7,8));128[ 3, 7 ]129]]></Example>130</Description>131</ManSection>132133<ManSection>134<Func Arg="f" Name="ArfNumericalSemigroupsWithFrobeniusNumber"></Func>135<Description>136<A>f</A> is an integer greater than or equal to -1. The output is the set of all Arf137numerical semigroups with Frobenius number <A>f</A>.138The current version of this algorithm is due to G. Zito.139140<Example><![CDATA[141gap> ArfNumericalSemigroupsWithFrobeniusNumber(10);142[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,143<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,144<Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup> ]145gap> Set(last,MinimalGenerators);146[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],147[ 6, 11, 13, 14, 15, 16 ], [ 7, 9, 11, 12, 13, 15, 17 ],148[ 7, 11, 12, 13, 15, 16, 17 ], [ 8, 11, 12, 13, 14, 15, 17, 18 ],149[ 9, 11, 12, 13, 14, 15, 16, 17, 19 ], [ 11 .. 21 ] ]150]]></Example>151</Description>152</ManSection>153154<ManSection>155<Func Arg="f" Name="ArfNumericalSemigroupsWithFrobeniusNumberUpTo"></Func>156<Description>157<A>f</A> is an integer greater than or equal to -1. The output is the set of all Arf158numerical semigroups with Frobenius number less than or equal to <A>f</A>.159The current version of this algorithm is due to G. Zito.160<Example><![CDATA[161gap> Length(ArfNumericalSemigroupsWithFrobeniusNumberUpTo(10));16246163]]></Example>164</Description>165</ManSection>166167<ManSection>168<Func Arg="g" Name="ArfNumericalSemigroupsWithGenus"></Func>169<Description>170<A>g</A> is a nonnegative integer. The output is the set of all Arf171numerical semigroups with equal to <A>g</A>.172The current version of this algorithm is due to G. Zito.173<Example><![CDATA[174gap> Length(ArfNumericalSemigroupsWithGenus(10));17521176]]></Example>177</Description>178</ManSection>179180<ManSection>181<Func Arg="g" Name="ArfNumericalSemigroupsWithGenusUpTo"></Func>182<Description>183<A>g</A> is a nonnegative integer. The output is the set of all Arf184numerical semigroups with genus less than or equal to <A>g</A>.185The current version of this algorithm is due to G. Zito.186<Example><![CDATA[187gap> Length(ArfNumericalSemigroupsWithGenusUpTo(10));18886189]]></Example>190</Description>191</ManSection>192193<ManSection>194<Func Arg="g,f" Name="ArfNumericalSemigroupsWithGenusAndFrobeniusNumber"></Func>195<Description>196<A>f</A> is an integer greater than or equal to -1, and <A>g</A> is a nonnegative integer. The output is the set of all Arf197numerical semigroups with genus <A>g</A> and Frobenius number <A>f</A>. The algorithm is explained in <Cite Key="arf-frob-gen"></Cite>.198<Example><![CDATA[199gap> ArfNumericalSemigroupsWithGenusAndFrobeniusNumber(10,13);200[ <Numerical semigroup>, <Numerical semigroup>, <Numerical semigroup>,201<Numerical semigroup>, <Numerical semigroup> ]202gap> List(last,MinimalGenerators);203[ [ 8, 10, 12, 14, 15, 17, 19, 21 ], [ 6, 10, 14, 15, 17, 19 ],204[ 5, 12, 14, 16, 18 ], [ 6, 9, 14, 16, 17, 19 ], [ 4, 14, 15, 17 ] ]205]]></Example>206</Description>207</ManSection>208209210</Section>211<Section>212<Heading>213Saturated numerical semigroups214</Heading>215Saturated numerical semigroups with the Arf property are a special kind of numerical semigroups with216maximal embedding dimension. A numerical semigroup <M>S</M> is saturated if the following condition holds: <M> s, s_1 , \ldots , s_r</M> in <M>S</M> are such that <M>s_i \leq s</M> for all <M>i</M> in <M>\{1, \ldots , r\}</M> and <M>z_1 , \ldots , z_r</M> in <M>\mathbb Z</M> are such that <M>z_1 s_1 + \cdots + z_r s_r\geq 0</M>, then <M>s + z_1 s_1 + \cdots + z_r s_r</M> in <M>S</M>.217218<P/>219The intersection of two saturated numerical semigroups is again saturated, and thus we can consider the220saturated closure of a set of nonnegative integers with greatest common divisor equal to one (see <Cite Key="RGbook"></Cite>).221222223<ManSection>224<Prop Arg="S" Name="IsSaturated"></Prop>225<Prop Arg="S" Name="IsSaturatedNumericalSemigroup"></Prop>226<Description>227<A>S</A> is a numerical semigroup.228Returns true if <A>S</A> is a saturated numerical semigroup and false otherwise.229<Example><![CDATA[230gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(4,6,9,11));231true232gap> IsSaturatedNumericalSemigroup(NumericalSemigroup(8, 9, 12, 13, 15, 19 ));233false234]]></Example>235</Description>236</ManSection>237238<ManSection>239<Func Arg="S" Name="SaturatedNumericalSemigroupClosure"></Func>240<Oper Arg="S" Name="SaturatedClosure" Label="for numerical semigroups"></Oper>241<Description>242<A>S</A> is a numerical semigroup.243Returns the saturated closure of <A>S</A>.244<Example><![CDATA[245gap> SaturatedNumericalSemigroupClosure(NumericalSemigroup(8, 9, 12, 13, 15));246<Numerical semigroup>247gap> MinimalGenerators(last);248[ 8 .. 15 ]249]]></Example>250</Description>251</ManSection>252253254<ManSection>255<Func Arg="f" Name="SaturatedNumericalSemigroupsWithFrobeniusNumber"></Func>256<Description>257<A>f</A> is an integer greater than or equal to -1. The output is the set of all Saturated258numerical semigroups with Frobenius number <A>f</A>.259<Example><![CDATA[260gap> SaturatedNumericalSemigroupsWithFrobeniusNumber(10);261[ <Numerical semigroup with 3 generators>,262<Numerical semigroup with 4 generators>,263<Numerical semigroup with 6 generators>,264<Numerical semigroup with 6 generators>,265<Numerical semigroup with 7 generators>,266<Numerical semigroup with 8 generators>,267<Numerical semigroup with 9 generators>,268<Numerical semigroup with 11 generators> ]269gap> List(last,MinimalGenerators);270[ [ 3, 11, 13 ], [ 4, 11, 13, 14 ], [ 6, 9, 11, 13, 14, 16 ],271[ 6, 11, 13, 14, 15, 16 ], [ 7, 11, 12, 13, 15, 16, 17 ],272[ 8, 11, 12, 13, 14, 15, 17, 18 ], [ 9, 11, 12, 13, 14, 15, 16, 17, 19 ],273[ 11 .. 21 ] ]274]]></Example>275</Description>276</ManSection>277278</Section>279280281