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

Register for: VAC33

Details of talk

TitleQuasi-primal Cornish algebras
PresenterBrian Davey (La Trobe University)
Author(s)Prof Brian A. Davey and Asha Gair
Sessionn/a
Time
Abstract


Varieties generated by quasi-primal algebras are a natural generalisation of
Boolean algebras and play an important role in the interface between logic and
universal algebra. A Cornish algebra is a bounded distributive lattice equipped
with a family of unary operations each of which is either an endomorphism, and
so models a strong form of modal operator, or a dual endomorphism, and so models
a De Morgan negation. We characterise quasi-primal Cornish algebras. The results
yield as a special case a recent result by Davey, Nguyen and Pitkethly
describing quasi-primal Ockham algebras. Our characterisation is in terms of the
Priestley dual of the algebra. 

(Cornish algebras are named for the South Australian mathematician Bill Cornish
who introduced them
as a natural generalisation of Ockham algebras, in an invited lecture entitled
``Monoids acting on distributive lattices" at the annual meeting of the
Australian Mathematical Society at La Trobe University in May 1977. The notes
from that lecture were never published but were distributed privately. They
first appeared in print as part of Cornish's far-reaching monograph
``Antimorphic Action: Categories of Algebraic Structures with Involutions or
Anti-endomorphisms" published nine years later.)