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SUMMARY:Ana Agore (Free University of Brussels\, Belgium)
DTSTART:20241007T150000Z
DTEND:20241007T160000Z
DTSTAMP:20260513T125807Z
UID:ENAAS/89
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ENAAS/89/">S
 olutions of the set-theoretic Yang-Baxter equation of Frobenius-Separabili
 ty (FS) type</a>\nby Ana Agore (Free University of Brussels\, Belgium) as 
 part of European Non-Associative Algebra Seminar\n\n\nAbstract\nWe investi
 gate a special class of solutions of the set-theoretic Yang-Baxter equatio
 n\, called Frobenius-Separability (FS) type solutions. In particular\, we 
 show that the category of solutions of the set-theoretic Yang-Baxter equat
 ion of Frobenius-Separability (FS) type is equivalent to the category of p
 ointed Kimura semigroups. As applications\, all involutive\, idempotent\, 
 nondegenerate\, surjective\, finite order\, unitary or indecomposable solu
 tions of FS type are classified. For instance\, if $|X| = n$\, then the nu
 mber of isomorphism classes of all such solutions on $X$ that are (a) left
  non-degenerate\, (b) bijective\, (c) unitary or (d) indecomposable and le
 ft-nondegenerate is: (a) the Davis number $d(n)$\, (b) $\\sum_{m|n} \\\, p
 (m)$\, where $p(m)$ is the Euler partition number\, (c) $\\tau(n) + \\sum_
 {d|n}\\left\\lfloor \\frac d2\\right\\rfloor$\, where $\\tau(n)$ is the nu
 mber of divisors of $n$\, or (d) the Harary number. The automorphism group
 s of such solutions can also be recovered as automorphism groups $\\mathrm
 {Aut}(f)$ of sets $X$ equipped with a single endo-function $f\\colon X\\to
  X$. We describe all groups of the form $\\mathrm{Aut}(f)$ as iterations o
 f direct and (possibly infinite) wreath products of cyclic or full symmetr
 ic groups\, characterize the abelian ones as products of cyclic groups\, a
 nd produce examples of symmetry groups of FS solutions not of the form $\\
 mathrm{Aut}(f)$. Based on joint work with A. Chirvasitu and G. Militaru.\n
LOCATION:https://researchseminars.org/talk/ENAAS/89/
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