Jonathan Smith

Algebraic Methods in Applied Logic

The course offers algebraic techniques for applying logic in various fields. In the context of the school, the topics are chosen to remain complementary to the presentations of the parallel courses.

1. Categories

   A novel presentation treats categories as (directed) graphs with algebra structure. The basic notions are diagrams as graph maps from a graph to a category, and natural transformations as maps between diagrams.

2. Limits and universal algebra

   Limits are presented as natural transformations. Created in word algebras over alphabets, universal algebra relies on two adjunctions: free algebras, and replications.

3. Duality

   Duality is interpreted as dual equivalence between spaces and coordinate algebras. It is specified on finitary objects, and then extended by (co)limits. Diagrams are used to broaden the reach of any given duality.

4. Logic action on sets

   Logical structures in commutative rings act on sets within modules. Over fields from the rationals to the reals, barycentric algebras synthesize convex sets and semilattices into hierarchical structures with multiple levels.

5. Anharmonic action

   Fields are matched to universal algebra: Logical complementation and geometric inversion decompose the projective line over a field into three disjoint unions of a group with a Boolean algebra.

6. The logic of cells

   Barycentric algebras map the gates of Boolean genetic networks to the continuous gates that are actually observed in biological experiments. Geometric inversion from Lecture 5 switches activation and inhibition.