Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleAlgebras of incidence structures: representations of regular double p-algebras,
PresenterChristopher Taylor (La Trobe University)
Author(s)Mr Christopher Taylor

Reyes and Zolfaghari proved that the lattice of subgraphs of a graph naturally
forms a double-Heyting algebra. We generalise this result and show that the
lattice of point-preserving substructures of an incidence structure naturally
forms a regular double p-algebra.

An incidence structure is a standard geometric object consisting of a set of
points, a set of lines and an incidence relation specifying which points lie on
which lines. This concept generalises, for example, graphs, hypergraphs and
projective planes. 

The result given in this talk is a characterisation of the regular double
p-algebras which are isomorphic to a lattice of point-preserving substructures.
In addition to the corollary that every finite regular double p-algebra is
isomorphic to such a lattice, a special case of the result is a standard theorem
for boolean algebras: a boolean algebra is isomorphic to a powerset lattice if
and only if it is complete and atomic.