Jonny Stephenson, Pure Mathematics, University of Waterloo
"Embedding Lattices in the Computably Enumerable Degrees (continued)"
This talk is a continuation of one given August 6th.
The
question
of
which
finite
lattices
can
be
embedded
into
the
c.e.
degrees
first
arose
with
the
construction
of
a
minimal
pair
by
Yates,
and
independently
by
Lachlan,
showing
the
4
element
Boolean
algebra
can
be
embedded.
This
result
was
rapidly
generalised
to
show
any
finite
distributive
lattice
can
also
be
embedded.
For
non-distributive
lattices,
the
situation
is
more
complicated.
There
are
two
minimal
nondistributive
lattices
M_5
and
N_5
with
the
property
that
a
lattice
is
nondistributive
if
and
only
if
it
contains
one
as
a
sublattice.
Both
of
these
lattices
are
embeddable
(but
not
all
nondistributive
lattices
are).
We will continue with our construction of an embedding of the nondistributive lattice M_5 into the computably enumerable Turing degrees.