Some homework for Math 736, Model-Theory, given September 28, 2001. Let L be a signature, and let M be a structure in 𝔐𝔬𝔡(L).

Problem 1. We didn’t actually define terms of L; we simply asserted that terms t exist whose interpretations t^{M} have certain properties. Prove this assertion.

Problem 2. Prove the lemma that, if t is an n-ary term of L, and u_{0}, …, u_{n−1} are m-ary terms of L, then there is an m-ary term of L whose interpretation in M is the map

a ↦ t^{M}(u_{0}^{M}(a), …, u_{n−1}^{M}(a)) : M^{m} → M.

Problem 3. Suppose that a structure in the signature {∧, ∨, ¬, 0, 1} can be expanded to a signature containing + in such a way that the identities x∨y = x + y +(x∧y) and ¬x = x+1 are satisfied; suppose further that this expansion, reduced to the signature {+, ∧, 0, 1}, is a Boolean ring. Then the original structure is, by definition, a Boolean algebra.

1. Prove that the Boolean algebras are precisely those structures (B, ∧, ∨, ¬, 0, 1) whose reducts (B, ∧, 1) and (B, ∨, 0) are monoids, and that satisfy the equations ¬¬x = x, and ¬(x ∧ y) = ¬x ∧ ¬y, and also

x ∧ y = y ∧ x,  x ∨ y = y ∨ x,
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z),
x ∧ ¬x = 0  x ∨ ¬x = 1
x ∧ 0 = 0  x ∨ 1 = 1.

2. Find a structure (C, +, ∧, ⋎, ¬, 0, 1) that satisfies

x + y = (x ∧ ¬y) ⋎ (y ∧ ¬x),

and whose reduct (C, +, ∧, 0, 1) is a Boolean ring, but whose reduct (C, ∧, ⋎, ¬, 0, 1) is not a Boolean algebra.

Problem 4. Prove that the map x ↦ [x] from a Boolean algebra to the power-set of its Stone-space is an embedding of Boolean algebras.