Jonny Stephenson, Pure Mathematics, University of Waterloo
"Embedding Lattices into the Computably Enumerable Degrees"
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).
In
this
talk
we
will
discuss
the
use
of
the
pinball
machine
method
to
give
an
embedding
of
M_5.