Robert Raphael: The countable lifting property. The case of finite fields

Place: Fields Institute (Room 210)

Date: March 18, 2016 (13:30-15:00)

Speaker: Robert Raphael

Title: The countable lifting property. The case of finite fields

Abstract:

This talk continues the topic discussed in the seminar in the spring of
2015. One is interested in vector lattices (vector spaces with a lattice
structure). Elements u and v of a vector lattice are called disjoint, or
orthogonal if u inf v = 0. Morphisms are linear transformations that
preserve the lattice structure. Suppose that f:V —> W is such a
surjective morphism. One says that one has the countable lifting property
(for f) if every countable set of (pairwise) disjoint elements in W has a
preimage in V which is also pairwise disjoint.

Topping originally established the countable lifting property for vector
lattices over the real field. This was done via a flawed induction and a
counterexample was later provided by Conrad. In joint work with Hager the
countable lifting property was established for maps between rings of the
form C(X) (lifting sets of positive functions).

At the seminar in 2015 the question arose as to whether the countable
lifting property holds for vector lattices over a finite field. The talk
will present details of a counterexample obtained in work with W.D.
Burgess of the University of Ottawa. One will restrict the discussion to
the case of Z_2.