Abstract
Anyone who has ever worked with a variety $$\varvec{\mathscr {A}}$$ 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 $$\varvec{\mathscr {X}}$$ of such a variety, we give a characterisation, in terms of $$\varvec{\mathscr {X}}$$ , of finitely generated discriminator subvarieties of $$\varvec{\mathscr {A}}$$ . As an 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.