BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Scott Chapman (Sam Houston State University) DTSTART:20250522T173000Z DTEND:20250522T175500Z DTSTAMP:20260423T041812Z UID:CANT2025/44 DESCRIPTION:Title: Betti elements and non-unique factorizations\nby Scott Chapman (Sam Houston State University) as part of Combinatorial and additive number the ory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Science Center ( 4th floor).\n\nAbstract\nLet $M$ be a commutative cancellative reduced ato mic monoid with set of atoms (or irreducibles) $\\mathcal{A}(M)$. Given a nonunit $x$ in $M$\, let\n$Z(x)$ represent the set of factorizations of $ x$ into atoms. Define a graph $\\nabla_x$ whose vertex set is $Z(x)$ wher e two vertices are joined\nby an edge if these factorizations share an ato m. Call $x$ a \\textit{Betti element} of $M$ if the graph $\\nabla_x$ is disconnected.\nBetti elements have proven to be a powerful tool in the stu dy of nonunique factorizations of elements in monoids. In particular\, \n over the past several years many papers have used Betti elements to study factorizaton properties in \\textit{affine monoids} (i.e.\, finitely gene rated additive submonoids of $\\mathbb{N}_0^k$ for some positive integer $ k$). Several strong results have been obtained when $M$ is a numerical mo noid (i.e.\, $k=1$ above). In this talk\, we will\nreview the basic prope rties of Betti elements and some of the results regarding affine monoids m entioned above. \nWe will then extend this study to\nmore general rings a nd monoids which are commutative and cancellative. We focus on two cases : (I) when the monoid $M$ has a single Betti element\, (II) when each atom of $M$ divides every Betti element. We call those monoids satisfying con dition (II) as having \\textit{full atomic support}. We show using elemen tary arguments that a monoid of type (I) is actually of full atomic suppor t. We close by showing for a monoid of full atomic support that the cate nary degree\, the tame degree\, and the omega primality constant (three w ell studied invariants in the nonunique factorization literature) can be e asily computed from the monoid's set of Betti elements.\n LOCATION:https://researchseminars.org/talk/CANT2025/44/ END:VEVENT END:VCALENDAR