Proposition 24. Given non-associative algebras A and B, an oplax homomorphism h : A −→ B preserves algebraic closure. If h is injective, it also reflects algebraic closure. Proof. Since an oplax homomorphism is a homomorphism between Boolean algebras, it preserves the order. So for any three relations R1, R2, R3 in the algebraically closed CSP over A, with R3 ≤ R1; R2 the preservation of the order implies:
h(R3) ≤ h(R1; R2).

Applying the oplaxness property yields:
h(R3) ≤ h(R1); h(R2).

and hence the image of the CSP under h is also algebraically closed. If h is injective, it reflects equations and inequations, and the converse implication follows.

Definition 25. Following [26], a *constraint network* over a non-associative algebra A can be seen as a function ν : A → P(N × N), where N is the set of nodes (or variables), and ν maps each abstract relation R to the set of pairs
(n1, n2) that are decorated with R. (Note that ν is a weak representation only if the constraint network is algebraically closed.)
Constraint networks can be translated along homomorphisms of nonassociative algebras as follows: Given h : A → B and ν : A → P(N × N),
h(ν) : B → P(N × N) is the network that decorates (n1, n2) with h(R) whenever ν decorates it with R
A *solution* for ν in a weak representation ϕ : A → P(*U × U*) is a function j : N → U such that for all R ∈ A, P(j × j)(ν(R)) ⊆ ϕ(R), or P(j × j) ◦ ν ⊆ ϕ for short.

Proposition 26. *Oplax homomorphisms of weak representations preserve solutions for constraint networks.*
Proof. Let weak representations ϕ : A → P(*U × U*) and ψ : B → P(*V × V*) and an oplax homomorphism of weak representations (*h, i*) : ϕ → ψ be given.

A given solution j : N → U for ν in ϕ is defined by P(j × j) ◦ ν ⊆ ϕ.

From this and the oplax commutation property P(i × i) ◦ ϕ ⊆ ψ ◦ h we infer P(i ◦ j × i ◦ j) ◦ ν ⊆ ψ ◦ h, which implies that i ◦ j is a solution for h(ν).

An important question for a calculus (= weak representation) is whether algebraic closure decides consistency. We will now prove that this property is preserved under certain homomorphisms.