Theory Graphs

A theory graph (slides p.357, "Modular Theories") is a diagram showing how mathematical structures inherit from one another. Nodes are structures; edges are "is-a" or "has-property" relations.

In the SMAI theory graph (notes p.196), arrows go from the MORE GENERAL structure DOWN to the MORE SPECIFIC one. Each arrow adds an axiom.

The canonical SMAI example:
Nat → NatPlus → NatMult → Monoid → Group → AbelGroup → Ring → IntArith
where each step adds an axiom (successor, addition, multiplication, identity, inverse, commutativity, distributivity, etc.). Reading top-down tells you which axioms are inherited; reading bottom-up tells you which structures you can recover as fragments of richer ones.

Two key operations on a theory graph:
1. Interpretation $\varphi$ — a map from one structure to another that preserves the operations. For example, the interpretation $\varphi = \{ G \mapsto \mathbb{N},\ \circ \mapsto +,\ e \mapsto 0 \}$ sends the abstract monoid down to $\langle \mathbb{N}, +, 0 \rangle$.
2. Copying — any axiom, object, or theorem can be copied along any edge to a more specific structure. So once you prove something about Monoid, you immediately get it about Group, AbelGroup, Ring, etc.

This is "object-oriented math" — it prevents the exponential blowup of reproving the same fact for every specialization.