1 research outputs found
Restricted Priestley dualities and discriminator varieties
Anyone who has ever worked with a variety~ of
algebras with a reduct in the variety of bounded distributive lattices will
know a restricted Priestley duality when they meet one---but until now there
has been no abstract definition. Here we provide one. After deriving some basic
properties of a restricted Priestley dual category
of such a variety, we give a characterisation, in terms of
, of finitely generated discriminator subvarieties
of~.
As a first application of our characterisation, we give a new proof of
Sankappanavar's characterisation of finitely generated discriminator varieties
of distributive double p-algebras.
A substantial portion of the paper is devoted to the application of our
results to Cornish algebras. A Cornish algebra is a bounded distributive
lattice equipped with a family of unary operations each of which is either an
endomorphism or a dual endomorphism of the bounded lattice. They are a natural
generalisation of Ockham algebras, which have been extensively studied. We give
an external necessary-and-sufficient condition and an easily applied,
completely internal, sufficient condition for a finite set of finite Cornish
algebras to share a common ternary discriminator term and so generate a
discriminator variety. Our results give a characterisation of discriminator
varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen
and Pitkethly's characterisation of quasi-primal Ockham algebras