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UID:/NewsandEvents/Archives/2006/newsitem/1591/14-
 November-2006-Intervals-in-the-Medvedev-lattice-Ba
 s-Terwijn
DTSTAMP:20061109T000000
SUMMARY:Intervals in the Medvedev lattice, Bas Ter
 wijn
ATTENDEE;ROLE=Speaker:Bas Terwijn (RUU and Technic
 al University of Vienna)
DTSTART;TZID=Europe/Amsterdam:20061114T160000
DTEND;TZID=Europe/Amsterdam:20061114T170000
LOCATION:Room 3.27, Plantage Muidergracht 24, 1018
  TV, Amsterdam
DESCRIPTION:The Medvedev lattice is a structure fr
 om computability theory with ties to constructive 
 logic. We will briefly describe this connection an
 d the relation to structures such as the Turing de
 grees. We will then discuss structural properties 
 of the Medvedev lattice, in particular, the size o
 f its intervals. We prove that every interval in t
 he lattice is either finite, in which case it is i
 somorphic to a finite Boolean algebra, or contains
  an antichain of size 22^\\aleph_0, the size of th
 e lattice itself. We also prove that it is consist
 ent that the lattice has chains of this size, and 
 in fact that these big chains occur in every inter
 val that has a big antichain. We also study embedd
 ings of lattices and algebras. We show that large 
 Boolean algebras can be embedded into the Medvedev
  lattice as upper semilattices, but that a Boolean
  algebra can be embedded as a lattice only if it i
 s countable. Finally we discuss which of these res
 ults hold for the closely related Muchnik lattice.
  The talk was given previously in the Mathematical
  Logic Seminar but many people missed it.    For m
 ore information, please contact marjanv at science
 .uva.nl
X-ALT-DESC;FMTTYPE=text/html:\n      <p>\n        
 The Medvedev lattice is a structure from computabi
 lity theory with ties to constructive logic. We wi
 ll briefly describe this connection and the relati
 on to structures such as the Turing degrees. We wi
 ll then discuss structural properties of the Medve
 dev lattice, in particular, the size of its interv
 als. We prove that every interval in the lattice i
 s either finite, in which case it is isomorphic to
  a finite Boolean algebra, or contains an antichai
 n of size 22^\\aleph_0, the size of the lattice it
 self.\n        We also prove that it is consistent
  that the lattice has chains of this size, and in 
 fact that these big chains occur in every interval
  that has a big antichain. We also study embedding
 s of lattices and algebras. We show that large Boo
 lean algebras can be embedded into the Medvedev la
 ttice as upper semilattices, but that a Boolean al
 gebra can be embedded as a lattice only if it is c
 ountable. Finally we discuss which of these result
 s hold for the closely related Muchnik lattice. \n
         The talk was given previously in the Mathe
 matical Logic Seminar but many people missed it.\n
       </p>\n    \n      <p>\n        For more info
 rmation, please contact <a class="email">marjanv <
 span class="at">at</span> science.uva.nl</a>\n    
   </p>\n    
URL:/NewsandEvents/Archives/2006/newsitem/1591/14-
 November-2006-Intervals-in-the-Medvedev-lattice-Ba
 s-Terwijn
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