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Catenary and Tame degrees of numerical semigroups
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<div class="ChapSects"><a href="chap9.html#X7C76B03A84BA7574">9 <span class="Heading">
Catenary and Tame degrees of numerical semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap9.html#X7FDB54217B15148F">9.1 <span class="Heading">
Factorizations in Numerical Semigroups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X78C6D3BF7C7C2760">9.1-1 FactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FDE4F94870951B1">9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X85C2987C7827D18D">9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FE2D6F77BE96716">9.1-4 ElasticityOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X8384DBFE7D82A634">9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7E3ED34D78F3A8CA">9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FAF204E85D9C21B">9.1-7 CatenaryDegreeOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X787334508257C510">9.1-8 CatenaryDegreeOfElementNS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X860BDF5B85975B73">9.1-9 TameDegreeOfNumericalSemigroup</a></span>
</div>
</div>
<h3>9 <span class="Heading">
Catenary and Tame degrees of numerical semigroups
</span></h3>
<p><a id="X7FDB54217B15148F" name="X7FDB54217B15148F"></a></p>
<h4>9.1 <span class="Heading">
Factorizations in Numerical Semigroups
</span></h4>
<p>Let S be a numerical semigroup minimally generated by {m_1,...,m_n}. A factorization of an element sin S is an n-tuple a=(a_1,...,a_n) of nonnegative integers such that n=a_1 n_1+cdots+a_n m_n. The lenght of a is |a|=a_1+cdots+a_n. Given two factorizations a and b of n, the distance between a and b is d(a,b)=max { |a-gcd(a,b)|,|b-gcd(a,b)|}, where gcd((a_1,...,a_n),(b_1,...,b_n))=(min(a_1,b_1),...,min(a_n,b_n)).</p>
<p>If l_1>cdots > l_k are the lenghts of all the factorizations of s in S, the Delta set associated to s is Delta(s)={l_1-l_2,...,l_k-l_k-1}.</p>
<p>The catenary degree of an element in S is the least positive integer c such that for any two of its factorizations a and b, there exists a chain of factorizations starting in a and ending in b and so that the distance between two consecutive links is at most c. The catenary degree of S is the supremum of the catenary degrees of the elements in S.</p>
<p>The tame degree of S is the least positive integer t for any factorization a of an element s in S, and any i such that s-m_iin S, there exists another factorization b of s so that the distance to a is at most t and b_inot = 0.</p>
<p>The basic properties of these constants can be found in <a href="chapBib.html#biBGHKb">[GH06]</a>. The algorithm used to compute the catenary and tame degree is an adaptation of the algorithms appearing in <a href="chapBib.html#biBCGLPR">[PR06]</a> for numerical semigroup (see <a href="chapBib.html#biBCGL">[CL07]</a>). The computation of the elascitiy of a numerical semigroup reduces to m/n with m the multiplicity of the semigroup and n its largest minimal generator (see <a href="chapBib.html#biBCHM06">[CM06]</a> or <a href="chapBib.html#biBGHKb">[GH06]</a>).</p>
<p><a id="X78C6D3BF7C7C2760" name="X78C6D3BF7C7C2760"></a></p>
<h5>9.1-1 FactorizationsElementWRTNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> FactorizationsElementWRTNumericalSemigroup</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a nonnegative integer. The output is the set of factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> FactorizationsElementWRTNumericalSemigroup(1100,s);
[ [ 0, 0, 0, 2, 2, 0 ], [ 0, 2, 3, 0, 0, 1 ], [ 0, 8, 1, 0, 0, 0 ],
[ 5, 1, 1, 0, 0, 1 ] ]
</pre></td></tr></table>
<p><a id="X7FDE4F94870951B1" name="X7FDE4F94870951B1"></a></p>
<h5>9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> LengthsOfFactorizationsElementWRTNumericalSemigroup</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a nonnegative integer. The output is the set of lengths of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(1100,s);
[ 4, 6, 8, 9 ]
</pre></td></tr></table>
<p><a id="X85C2987C7827D18D" name="X85C2987C7827D18D"></a></p>
<h5>9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ElasticityOfFactorizationsElementWRTNumericalSemigroup</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a positive integer. The output is the maximum length divided by the minimum length of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> ElasticityOfFactorizationsElementWRTNumericalSemigroup(1100,s);
9/4
</pre></td></tr></table>
<p><a id="X7FE2D6F77BE96716" name="X7FE2D6F77BE96716"></a></p>
<h5>9.1-4 ElasticityOfNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ElasticityOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. The output is the elasticity of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> ElasticityOfNumericalSemigroup(s);
286/101
</pre></td></tr></table>
<p><a id="X8384DBFE7D82A634" name="X8384DBFE7D82A634"></a></p>
<h5>9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DeltaSetOfFactorizationsElementWRTNumericalSemigroup</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a nonnegative integer. The output is the Delta set of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> DeltaSetOfFactorizationsElementWRTNumericalSemigroup(1100,s);
[ 1, 2 ]
</pre></td></tr></table>
<p><a id="X7E3ED34D78F3A8CA" name="X7E3ED34D78F3A8CA"></a></p>
<h5>9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MaximumDegreeOfElementWRTNumericalSemigroup</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> a nonnegative integer. The output is the maximum length of the factorizations of <var class="Arg">n</var> in terms of the minimal generating set of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> MaximumDegreeOfElementWRTNumericalSemigroup(1100,s);
9
</pre></td></tr></table>
<p><a id="X7FAF204E85D9C21B" name="X7FAF204E85D9C21B"></a></p>
<h5>9.1-7 CatenaryDegreeOfNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CatenaryDegreeOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. The output is the catenary degree of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> CatenaryDegreeOfNumericalSemigroup(s);
8
</pre></td></tr></table>
<p><a id="X787334508257C510" name="X787334508257C510"></a></p>
<h5>9.1-8 CatenaryDegreeOfElementNS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CatenaryDegreeOfElementNS</code>( <var class="Arg">n, S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">n</var> is a nonnegative integer and <var class="Arg">S</var> is a numerical semigroup. The output is the catenary degree of <var class="Arg">n</var> relative to <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> CatenaryDegreeOfElementNS(157,NumericalSemigroup(13,18));
0
gap> CatenaryDegreeOfElementNS(1157,NumericalSemigroup(13,18));
18
</pre></td></tr></table>
<p><a id="X860BDF5B85975B73" name="X860BDF5B85975B73"></a></p>
<h5>9.1-9 TameDegreeOfNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> TameDegreeOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. The output is the tame degree of <var class="Arg">S</var>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(101,113,196,272,278,286);
<Numerical semigroup with 6 generators>
gap> TameDegreeOfNumericalSemigroup(s);
14
</pre></td></tr></table>
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