Veblen triple systems

The Veblen Axiom (or Veblen-Youngs Axiom) is one of the axioms used to define a projective space. In the case of finite geometries where lines have size three it gives rise to the Projective Steiner triple systems. In turn Steiner triple systems are equivalent to quasigroups that satisfy certain algebraic conditions. Using these equivalences the Veblen axiom can be interpreted as an algebraic condition. In this talk we will consider a generalisation of this algebraic condition, namely we ‘throw away’ commutativity. In many cases commutativity comes back as a consequence of other conditions, but not in all cases…

This is ongoing work with Terry Griggs.