### 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