Canonical extensions and completions of posets and lattices

A b s t r a c t. The purpose of this note is to expose a new way of viewing the canonical extension of posets and bounded lattices. Specifically, we seek to expose categorical features of this completion and to reveal its relationship to other completion processes. The theory of canonical extensions is introduced by Jónsson and Tarski [15, 16] for Boolean algebras with operators. Their approach was based on a complete-lattice theoretic characterisation of the dual space of a Boolean algebra and provides access to the benefits of Stone duality in a uniform way for a class of varieties of algebras having a Boolean reduct. The most The support of EPSRC under grant no. EP/E029329/1 is gratefully acknowledged. The first author would also like to thank Sarah Hunt for her invaluable help with childcare during the writing of this paper.