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SUMMARY:Paolo Lipparini (Universita' di Tor Vergata\, Rome\, Italy)
DTSTART:20220329T190000Z
DTEND:20220329T200000Z
DTSTAMP:20260423T004656Z
UID:PALS/28
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PALS/28/">Re
 lative lengths of Maltsev conditions</a>\nby Paolo Lipparini (Universita' 
 di Tor Vergata\, Rome\, Italy) as part of PALS Panglobal Algebra and Logic
  Seminar\n\n\nAbstract\nThe study of Maltsev conditions is a significant p
 art of universal algebra\, with classical characterizations of families of
  varieties (congruence permutable\, distributive\, modular...) and recent 
 advanced results by Hobby\, McKenzie\, Kearnes\, Kiss\, among others. In p
 articular\, the interplay between distinct Maltsev conditions for congruen
 ce modular varieties has led to a refined theory for such varieties.\n\nRe
 call that a Maltsev condition is\, roughly\, a statement of the form "ther
 e are some n and terms t1\,...\,tn such that a certain finite set of equat
 ions hold". As we mentioned\, many deep and sophisticated results are know
 n about Maltsev conditions. On the other hand\, when two conditions are co
 mpared\, really little is known about the exact value of the smallest n as
  above. For example\, a simple observation by A. Day asserts that if some 
 variety V has k Jónsson terms witnessing congruence distributivity\, then
  V has 2k-1 Day terms witnessing congruence modularity. About fifty years 
 ago Day asked whether this result is best possible\, but\, to the best of 
 our knowledge\, an exact solution is not yet known.\n\nA deeper problem (a
 sked by Lakser\, Taylor\, Tschantz in 1985) concerns the relative lengths 
 of sequences of Day and Gumm terms characterizing congruence modularity. M
 ore recently\, Kazda\, Kozik\, McKenzie\, Moore provided still another cha
 racterization of congruence distributive and modular varieties by means of
  "directed" terms. Again\, the exact relationships between the lengths of 
 the sequences of terms is not known. A solution of the above problems is s
 upposed to provide either interesting exotic examples of congruence modula
 r and distributive varieties\, or more refined structure theorems.\n\nWe s
 hall present recent results about the above Day\, LTT and KKMM problems\, 
 with an unexpected application to congruence distributive varieties.\n
LOCATION:https://researchseminars.org/talk/PALS/28/
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