A View of Canonical Extension

This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given. In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras. Acknowledgements: Both authors would like to acknowledge the influence of discussions and work with H.A. Priestley on the content of this paper. In particular, the fact that something like Theorem 5.7 might hold was first discussed with H. A. Priestley in the process of the first author’s work on a book-in-preparation on Lattices in Logic. The expository material in this paper is also based to a large extent on work on the book.