<p>An irreducible numerical semigroup is a semigroup that cannot be expressed as the intersection of numerical semigroups properly containing it.</p>
<p>It is not difficult to prove that a semigroup is irreducible if and only if it is maximal (with respect to set inclusion) in the set of all numerical semigroups having its same Frobenius number (see <a href="chapBib.html#biBRB03">[RB03]</a>). Hence, according to <a href="chapBib.html#biBFGH87">[FGR87]</a> (respectively <a href="chapBib.html#biBBDF97">[BDF97]</a>), symmetric (respectively pseudo-symmetric) numerical semigroups are those irreducible numerical semigroups with odd (respectively even) Frobenius number.</p>
<p>In <a href="chapBib.html#biBRGGJ03">[RGGJ03]</a> it is shown that a nontrivial numerical semigroup is irreducible if and only if it has only one special gap. We use this characterization.</p>
<p>In old versions of the package, we first constructed an irreducible numerical semigroup with the given Frobenius number (as explained in <a href="chapBib.html#biBRGS04">[RG04]</a>), and then we constructed the rest from it. The present version uses a faster procedure presented in <a href="chapBib.html#biBBR13">[BR13]</a>.</p>
<p>Every numerical semigroup can be expressed as an intersection of irreducible numerical semigroups. If <span class="SimpleMath">S</span> can be expressed as <span class="SimpleMath">S=S_1∩ ⋯∩ S_n</span>, with <span class="SimpleMath">S_i</span> irreducible numerical semigroups, and no factor can be removed, then we say that this decomposition is minimal. Minimal decompositions can be computed by using Algorithm 26 in <a href="chapBib.html#biBRGGJ03">[RGGJ03]</a>.</p>
<p>In this section we provide membership tests to the two families that conform the set of irreducible numerical semigroups. We also give a procedure to compute the set of all irreducible numerical semigroups with fixed Frobenius number (or equivalently genus, since for irreducible numerical semigroups once the Frobenius number is fixed, so is the genus). Also we give a function to compute the decomposition of a numerical semigroup as an intersection of irreducible numerical semigroups.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AnIrreducibleNumericalSemigroupWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. When <span class="SimpleMath">f=0</span> or <span class="SimpleMath">f≤ -2</span>, the output is <code class="code">fail</code>. Otherwise, the output is an irreducible numerical semigroup with Frobenius number <var class="Arg">f</var>. From the way the procedure is implemented, the resulting semigroup has at most four generators (see <a href="chapBib.html#biBRGS04">[RG04]</a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleNumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of all irreducible numerical semigroups with Frobenius number <var class="Arg">f</var>. The algorithm is inspired in <a href="chapBib.html#biBBR13">[BR13]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity</code>( <var class="Arg">f</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> and <var class="Arg">m</var> are integers. The output is the set of all irreducible numerical semigroups with Frobenius number <var class="Arg">f</var> and multiplicity <var class="Arg">m</var>. The implementation appears in <a href="chapBib.html#biBBOR19">[BOR21]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecomposeIntoIrreducibles</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 a set of irreducible numerical semigroups containing it. These elements appear in a minimal decomposition of <var class="Arg">s</var> as intersection into irreducibles.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DecomposeIntoIrreducibles(NumericalSemigroup(5,6,8));</span>
[ <Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators> ]
</pre></div>
<p>The cardinality of a minimal presentation of a numerical semigroup is always greater than or equal to its embedding dimension minus one. Complete intersection numerical semigroups are numerical semigroups reaching this bound, and they are irreducible. It can be shown that every complete intersection (other that <span class="SimpleMath">N</span>) is a complete intersection if and only if it is the gluing of two complete intersections. When in this gluing, one of the copies is isomorphic to <span class="SimpleMath">N</span>, then we obtain a free semigroup in the sense of <a href="chapBib.html#biBBC77">[BC77]</a>. Two special kinds of free semigroups are telescopic semigroups (<a href="chapBib.html#biBKP95">[KP95]</a>) and those associated to an irreducible planar curve (<a href="chapBib.html#biBZ86">[Zar86]</a>). We use the algorithms presented in <a href="chapBib.html#biBAGS13">[AG13]</a> to find the set of all complete intersections (also free, telescopic and associated to irreducible planar curves) numerical semigroups with given Frobenius number.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGluingOfNumericalSemigroups</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. Returns all partitions <span class="SimpleMath">{A_1,A_2}</span> of the minimal generating set of <var class="Arg">s</var> such that <var class="Arg">s</var> is a gluing of <span class="SimpleMath">⟨ A_1⟩</span> and <span class="SimpleMath">⟨ A_2⟩</span> by <span class="SimpleMath">gcd(A_1)gcd(A_2)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCompleteIntersection</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsACompleteIntersectionNumericalSemigroup</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is true if the numerical semigroup is a complete intersection, that is, the cardinality of a (any) minimal presentation equals its embedding dimension minus one.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of all complete intersection numerical semigroups with Frobenius number <var class="Arg">f</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFree</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFreeNumericalSemigroup</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is true if the numerical semigroup is free in the sense of <a href="chapBib.html#biBBC77">[BC77]</a>: it is either <span class="SimpleMath">N</span> or the gluing of a copy of <span class="SimpleMath">N</span> with a free numerical semigroup.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeNumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of all free numerical semigroups with Frobenius number <var class="Arg">f</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTelescopic</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTelescopicNumericalSemigroup</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is true if the numerical semigroup is telescopic in the sense of <a href="chapBib.html#biBKP95">[KP95]</a>: it is either <span class="SimpleMath">N</span> or the gluing of <span class="SimpleMath">⟨ n_e⟩</span> and <span class="SimpleMath">s'=⟨ n_1/d,..., n_e-1/d⟩, and s'</span> is again a telescopic numerical semigroup, where <span class="SimpleMath">n_1 < ⋯ < n_e</span> are the minimal generators of <var class="Arg">s</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TelescopicNumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of all telescopic numerical semigroups with Frobenius number <var class="Arg">f</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUniversallyFree</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUniversallyFreeNumericalSemigroup</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is true if the numerical semigroup is free for all the arrangements of its minimal generators.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is true if the numerical semigroup is associated to an irreducible planar curve singularity (<a href="chapBib.html#biBZ86">[Zar86]</a>). These semigroups are telescopic.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumericalSemigroupsPlanarSingularityWithFrobeniusNumber</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer. The output is the set of all numerical semigroups associated to irreducible planar curves singularities with Frobenius number <var class="Arg">f</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAperySetGammaRectangular</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.</p>
<p>Test for the <span class="SimpleMath">γ</span>-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in <a href="chapBib.html#biBDAMSClasses">[DMS14]</a>. Numerical Semigroups with this property are free and thus complete intersections.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAperySetBetaRectangular</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.</p>
<p>Test for the <span class="SimpleMath">β</span>-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in <a href="chapBib.html#biBDAMSClasses">[DMS14]</a>; <span class="SimpleMath">β</span>-rectangularity implies <span class="SimpleMath">γ</span>-rectangularity.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAperySetAlphaRectangular</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.</p>
<p>Test for the <span class="SimpleMath">α</span>-rectangularity of the Apéry Set of a numerical semigroup. This test is the implementation of the algorithm given in <a href="chapBib.html#biBDAMSClasses">[DMS14]</a>; <span class="SimpleMath">α</span>-rectangularity implies <span class="SimpleMath">β</span>-rectangularity.</p>
<p>A numerical semigroup is almost-symmetric (<a href="chapBib.html#biBBF97">[BF97]</a>) if its genus is the arithmetic mean of its Frobenius number and type. We use a procedure presented in <a href="chapBib.html#biBMR3169635">[RG14]</a> to determine the set of all almost-symmetric numerical semigroups with given Frobenius number. In order to do this, we first calculate the set of all almost-symmetric numerical semigroups that can be constructed from an irreducible numerical semigroup.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlmostSymmetricNumericalSemigroupsFromIrreducible</code>( <var class="Arg">s</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">s</var> is an irreducible numerical semigroup. The output is the set of almost-symmetric numerical semigroups that can be constructed from <var class="Arg">s</var> by removing some of its generators (as explained in <a href="chapBib.html#biBMR3169635">[RG14]</a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType</code>( <var class="Arg">s</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">s</var> is an irreducible numerical semigroup and <var class="Arg">t</var> is a positive integer. The output is the set of almost-symmetric numerical semigroups with type <var class="Arg">t</var> that can be constructed from <var class="Arg">s</var> by removing some of its generators (as explained in <a href="chapBib.html#biBBOR18">[BOR18]</a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlmostSymmetric</code>( <var class="Arg">s</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlmostSymmetricNumericalSemigroup</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 <code class="code">true</code> if the numerical semigroup is almost symmetric.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber</code>( <var class="Arg">f</var>[, <varclass="Arg">ts</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer, and so is <var class="Arg">ts</var>. The output is the set of all almost symmetric numerical semigroups with Frobenius number <var class="Arg">f</var>, and type greater than or equal to <var class="Arg">ts</var>. If <var class="Arg">ts</var> is not specified, then it is considered to be equal to one, and so the output is the set of all almost symmetric numerical semigroups with Frobenius number <var class="Arg">f</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType</code>( <var class="Arg">f</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">f</var> is an integer and so is <var class="Arg">t</var>. The output is the set of all almost symmetric numerical semigroups with Frobenius number <var class="Arg">f</var> and type <var class="Arg">t</var>.</p>
<h4>6.4 <span class="Heading">
Several approaches generalizing the concept of symmetry
</span></h4>
<p>Let <span class="SimpleMath">S</span> be a numerical semigroup and let <span class="SimpleMath">R</span> be its semigroup ring <span class="SimpleMath">K[[S]]</span>. We say that <span class="SimpleMath">S</span> has the generalized Gorenstein property if its semigroup ring has this property. For the definition and characterization of generalized Gorenstein rings please see <a href="chapBib.html#biBG-I-K-T">[tttt17]</a>.</p>
<p>A numerical semigroup is said to be nearly Gorenstein if its maximal ideal is contained in its trace ideal <a href="chapBib.html#biBtrace-canonical">[HHS19]</a>. Every almost symmetric numerical semigroup is nearly Gorenstein.</p>
<p>A numerical semigroup <span class="SimpleMath">S</span> with canonical ideal <span class="SimpleMath">K</span> is a generalized almost symmetric numerical semigroup if either <span class="SimpleMath">2K=K</span> (symmetric) or <span class="SimpleMath">2K∖ K={F(S)-x_1,dots, F(S)-x_r,F(S)}</span> for some <span class="SimpleMath">x_1,dots,x_r ∈ M∖ 2M</span> (minimal generators) and <span class="SimpleMath">x_i-x_jnot\in (S-M)∖ S</span> (not pseudo-Frobenius numbers), see <a href="chapBib.html#biBgas-semigroups">[DS21]</a>. As expected, every almost symmetric numerical semigroup is a generalized almost symmetric numerical semigroup.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedGorenstein</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is <code class="code">true</code> if the semigroup ring <span class="SimpleMath">K[[S]]</span> is generalized Gorenstein using the characterization by Goto-Kumashiro <a href="chapBib.html#biBG-K">[MK17]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNearlyGorenstein</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is <code class="code">true</code> if the semigroup is nearly Gorenstein, and <code class="code">false</code> otherwise.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NearlyGorensteinVectors</code>( <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. The output is a lists of lists (making the cartesian product of them yields all possible NG-vectors). If <span class="SimpleMath">n_i</span> is the ith generator of <var class="Arg">s</var>, in the ith position of the list it returns all pseudo-Frobenius numbers <span class="SimpleMath">f</span> of <var class="Arg">s</var> such that <span class="SimpleMath">n_i+f-f' is in s for all f a pseudo-Frobenius number of s, [MS21].
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedAlmostSymmetric</code>( <var class="Arg">s</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><var class="Arg">s</var> is a numerical semigroup. Determines whether or not <var class="Arg">s</var> is a generalized almost symmetric numerical semigroup.</p>
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