Revise Notes 5: Term Models (pdf), (term-models.tex) to include a running example based on a set, E, of equational axioms (other than just the Ring Axioms for arithmetic equations). Your axioms should have a simple, but not completely trivial term-model.

Some possible examples include

1. The "automorphisms of the square" sketched in class: the signature is a symbol, ∘, of arity two, corresponding to composition, and constants R, F, D corresponding to 90⁰ clockwise rotation, reflection about a vertical axis, reflection about an upper-left/lower-right diagonal, respectively. The axioms would include (x ∘ y) ∘ z = x ∘ (y ∘ z) asserting the associativity of composition, R⁵ = R corresponding to the fact that 5 clockwise 90⁰ rotations are the same a one such rotation, F³ = F
   D³ = D corresponding to the fact that the result of 3 reflections is the same as one, F ∘ R = D corresponding to the fact that a 90⁰ clockwise rotation followed by a reflection about a vertical axis yields the same result as a diagonal reflection, ... and a few more equations like these.
2. Automorphisms of some other simple shape, e.g., a tetrahedron.
3. The Ring Axioms along with x³ = x.

4. The Ring Axioms along with some axiom of the form e = 0 where e is an arithmetic expression whose only variable is x.
5. The Axioms for Boolean operations AND, OR, NOT
6. ... something else.

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