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:20260714T150000Z DTEND:20260714T152500Z DTSTAMP:20260710T111709Z UID:CANT2026/24 DESCRIPTION:Title: A surprising characterization of unique factorization domains\nby Sc ott Chapman (Sam Houston State University) as part of Combinatorial and ad ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nA surprising charac terization of unique factorization domains \\\\\nAbstract: & We address so me recent work on the generalization of the UFD propery which has pointed back to an open problem first mentioned in a paper by myself\, Dan Anderso n\, Muhammad Zafrullah\, and Franz Halter-Koch (Criteria for unique factor ization in integral domains\, J. Pure Appl. Algebra 127(1998)\, 205--218)\ , which we abbreviate as ACHKZ. Fix a positive integer $n>1$. Call an atom ic integral domain $D$ quasi-$n$-factorial if\, for any irreducible elemen ts \n$x_1\, \\ldots \, x_n\, y_1\, \\ldots \, y_n$\, the equality\n$x_1\\c dots x_n=y_1\\cdots y_n$ implies that $x_i=u_iy_{\\sigma(i)}$ for some uni t $u_i$ and permutation $\\sigma$ of $\\{1\,\\ldots \,n\\}$. Further\, $D$ is length-factorial if it is quasi-$n$-factorial for all $n>1$. Jim Coyke ndall and William W. Smith showed in 2011 the surprising result that an at omic monoid is a UFD if and only if it is length-factorial. This allows on e to alter the classic definition of a UFD. in a surprising manner. The au thors in ACHKZ offer examples of monoids which are quasi-$n$-factorial for specific $n$\, but are not factorial. They offer no such example of an in tegral domain. Hence\, the Coykendall-Smith result makes the following pro blem explored in ACHKZ all the more relevant. Open Problem: Does there exi st an atomic integral domain $D$ which is quasi-$n$-factorial for some $n> 1$\, but not factorial?\n LOCATION:https://researchseminars.org/talk/CANT2026/24/ END:VEVENT END:VCALENDAR