Lattices with additional operations

Mai Gehrke

In this workshop/tutorial we present the theory of canonical extensions for lattice expansions (LEs), that is, lattices with additional operations. This theory provides an algebraic alternative to the topological dualities for LEs and we will illustrate how this theory may be useful in applications.

First we define canonical extensions for arbitrary (bounded) lattices, and show their existence and essential uniqueness. We develop some of the basic properties of these extensions and identify the connection to the various topological dualities for general lattices as well as for distributive lattices and Boolean algebras.

After this point, we concentrate on the distributive case even though much of what will be said generalizes without much trouble to the non-distributive setting. We then introduce several topologies on canonical extensions of distributive lattices (DLs) as these will be useful in understanding the extension of operations on DLs. Using these topologies, we define extensions for additional operations and develop the basic properties of these, culminating in the functoriality of taking canonical extensions for large classes of DLEs. We also discuss the relationship of the extensions of operations obtained here to the relations obtained as duals in the topological setting.

Finally we establish several general results concerning the preservation of identities in canonical extensions and consider particular applications and limitations of these results.

In the final lecture, Hideo Nagahashi will give a talk on Generalized Modal Logic