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SUMMARY:Pedro A. Garcia-Sanchez (Universidad de Granada)
DTSTART:20250520T133000Z
DTEND:20250520T135500Z
DTSTAMP:20260423T010356Z
UID:CANT2025/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/12/
 ">Some problems related to the ideal class monoid of a numerical semigroup
 </a>\nby Pedro A. Garcia-Sanchez (Universidad de Granada) as part of Combi
 natorial and additive number theory (CANT 2025)\n\nLecture held in CUNY Gr
 aduate Center - Science Center (4th floor).\n\nAbstract\nLet $S$ be a nume
 rical semigroup (a submonoid of the set of non-negative integers under add
 ition such that $\\max(\\mathbb{Z}\\setminus S)$ exists). A non-empty set 
 of integers $I$ is said to be an ideal of $S$ if $I+S\\subseteq I$ and $I$
  has a minimum. If $I$ and $J$ are ideals of $S$\, we write $I\\sim J$ if 
 there exists an integer $z$ such that $I=z+J$. The ideal class monoid of $
 S$ is defined as the set of ideals of $S$ modulo this relation\, where add
 ition of two classes $[I]$ and $[J]$ is defined as $[I]+[J]=[I+J]$\, with 
 $I+J=\\{i+j\\mid i\\in I\, j\\in J\\}$. \n\nAn ideal $I$ is said to be nor
 malized if $\\min(I)=0$. The set of normalized ideals of $S$\, denoted by 
 $\\mathfrak{I}_0(S)$\, is a monoid isomorphic to the ideal class monoid of
  $S$ [1]. \n\nIt is known that if $S$ and $T$ are numericals semigroups fo
 r which $\\mathfrak{I}_0(S)$ is isomorphic to $\\mathfrak{I}_0(T)$\, then 
 $S$ and $T$ must be the same numerical semigroup [2].\n\nOn $\\mathfrak{I}
 _0(S)$ we can define a partial order $\\preceq$ as $I\\preceq J$ if there 
 exists $K\\in \\mathfrak{I}_0(S)$ such that $I+K=J$. We know that if $S$ a
 nd $T$ are numerical semigroups with multiplicity three such that the pose
 t $(\\mathfrak{I}_0(S)\,\\preceq)$ is isomorphic to the poset $(\\mathfrak
 {I}_0(T)\,\\preceq)$\, then $S$ and $T$ are the same numerical semigroup [
 3]. However\, if we remove the multiplicity three condition\, this poset i
 somorphsm problem is still open. \n\nIn [4]\, we study the case when the p
 oset $(\\mathfrak{I}_0(S)\,\\preceq)$ is a lattice. We show that this is t
 he case if and only if the multiplicity of $S$ is at most four. \n\nRefere
 nces:\n\n1.  L. Casabella\, M. D'Anna\, P. A. García-Sánchez\, Apéry se
 ts and the ideal class monoid of a numerical semigroup\, Mediterr. J. Math
 . 21\, 7 (2024). \n\n2. P. A. García-Sánchez\, The isomorphism problem f
 or ideal class monoids of numerical semigroups\, Semigroup Forum 108 (2024
 )\, 365--376. \n\n3. S. Bonzio\, P. A. García-Sánchez\, The poset of nor
 malized ideals of numerical semigroups with multiplicity three\, to appear
  in Comm. Algebra. \n\n4. S. Bonzio\, P. A. García-Sánchez\, When the po
 set of the ideal class monoid of a numerical \nsemigroup is a lattice\, ar
 Xiv:2412.07281.\n
LOCATION:https://researchseminars.org/talk/CANT2025/12/
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