Tools and Techniques in Modal Logic Marcus Kracht II ...

4.2. Varieties, Logics and Equationally Definable Classes 169
with all functions. Hence, the calculus without substitution defines the smallest congruence
induced by Γ. Now a substitution is nothing but an endomorphism of the
term algebra so (V5) enshrines the requirement that the congruence derivable from
Γ is fully invariant.
Theorem 4.2.7. Let Γ be a set of equations over L of signature Ω. The smallest
fully invariant congruence on TmΩ(X) containing Γ is the set of all s ≈ t such that
Γ ⊢V s ≈ t.
Let us call a set of the form Eq(K) an equational theory. Then we have
Corollary 4.2.8 (Birkhoff). A set of equations is an equational theory iff it is
closed under the rules of ⊢V.
Proof. First of all, the rules (V1) – (V5) are correct. That is, given an algebra
A and given a rule, if A satisfies every premiss of a rule then A also satisfies
the conclusion. (V1). For all v : X → A we have v(s) = v(s) for any term s.
(V2). Assume that for all v : X → A, v(s) = v(t). Then for all v : X → A
also v(t) = v(s), showing A � t ≈ s. (V3). Assume that A � s ≈ t; t ≈ u.
Take a map v : X → A. Then v(s) = v(t) as well as v(t) = v(u), from which
v(s) = v(u). Thus A � s ≈ u. (V4). Assume that A � si ≈ ti for all i < n. Take v :
X → A. Then v( f (s0, . . . , sn−1)) = f A (v(s0), . . . , v(sn−1)) = f A (v(t0), . . . , v(tn−1)) =
v( f (t0, . . . , tn−1)). Hence A � f (s0, . . . , sn−1) ≈ f (t0, . . . , fn−1). (V5). Assume
A � s ≈ t. Define a substitution σ by σ : xi ↦→ ui, i < n, σ : xi ↦→ xi for
i ≥ n. Then v ◦ σ : X → A, and the homomorphism extending the map is just
v ◦ σ, since it coincides on X with v ◦ σ. Now let v : X → A be given. Then
v(s(u0, . . . , un−1)) = v(σ(s)) = v ◦ σ(s) = v ◦ σ(t) = v(t(u0, . . . , un−1)). Thus
A � s(u0, . . . , un−1) ≈ t(u0, . . . , un−1).
Now let Γ be any set of equations and Θ its closure under (V1) to (V5). Let
A := TmΩ(X)/Γ. Then if s ≈ t � Γ we have A � s ≈ t. For just take the canonical
homomorphism hΘ : TmΩ(X) → A with kernel Θ. Since Θ is a congruence, this
is well–defined and we have hΘ(s) � hΘ(t), as required. Next we have to show
that A � Γ. To see that take any equation s ≈ t ∈ Γ and v : X → A. Since A
is generated by terms over X modulo Θ, we can define a substitution σ such that
v = hΘ ◦ σ. Namely, put σ(x) := t(�y), where t(�y) ∈ h−1 Θ (h(x)) is freely chosen. Then
hΘ(σ(x)) = v(x), as required. Now σ(s) Θ σ(t), by closure under substitution, so that
v(s) = hΘ(σ(s)) = hΘ(σ(t)) = v(t). Thus A � Γ. �
Theorem 4.2.9. There is a one–to–one correspondence between varieties of Ω–
algebras and fully invariant congruences on the freely countably generated algebra.
Moreover, V1 ⊆ V2 iff for the corresponding congruences Θ1 ⊇ Θ2.
One half of this theorem is actually Theorem 2.2.9. The converse direction was
actually much harder to prove but makes the result all the more useful.
Consider now what this means for modal logic. (We will henceforth write again
p and q for variables instead of x and y, as well as ϕ and ψ for formulae instead of