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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 \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

\n \n \n For more info
rmation, please contact marjanv <
span class="at">at science.uva.nl\n

\n
URL:/NewsandEvents/Archives/2006/newsitem/1591/14-
November-2006-Intervals-in-the-Medvedev-lattice-Ba
s-Terwijn
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