Using properness of the quotient, it is then easily shown that the relation algebra part of the lax homomorphism property carries over from ϕ to ϕ/∼: Concerning composition, by surjectivity of qA, we know that any given relations R1, R2 ∈ A/∼A are of the form R1 = qA(S1) and R2 = qA(S2). Hence, ϕ/∼ (R1; R2) = ϕ/∼ (qA(S1); qA(S2)) = ϕ/∼ (qA(S1; S2)) = P(q × q)(ϕ(S1; S2)) ≥
P(q ×q)(ϕ(S1); ϕ(S2)) = P(q ×q)(ϕ(S1));P(q ×q)(ϕ(S2)) = ϕ/∼ (qA(S1)); ϕ/∼
(qA(S2)) = ϕ/∼ (R1); ϕ/∼ (R2). The inequality of the identity is shown similarly.

Proposition 18. (qA, q) : ϕ → ϕ/∼ is an oplax quotient homomorphism of weak representations. If ∼ is regular w.r.t. ϕ, then the quotient homomorphism is proper, and satisfies the following universal property: if (qB, i) : ϕ → ψ is another oplax homomorphism of weak representations with ψ *injective and*
∼⊆ ker (i)*, then there is a unique oplax homomorphism of weak representations*
(*h, k*) : ϕ/∼→ ψ with (qB, i) = (h, k) ◦ (qA, q). Proof. The oplax homomorphism property for (qA, q) is P(q × q) ◦ ϕ ⊆ ϕ/∼
◦qA, which by definition of ϕ/∼ amounts to P(q × q) ◦ ϕ ⊆ P(q × q) ◦ ϕ ◦ q
−1 A ◦ qA,
which follows from surjectivity of q. Regularity of ∼ is w.r.t. ϕ means that ∼A
is the kernel of P(q × q) ◦ ϕ, which turns the above inequation into an equality.

Concerning the universal property, let (qB, i) : ϕ → ψ with the mentioned properties be given. Since ∼⊆ ker (i), there is a unique function k : U/*∼→ V* with i = k◦q. The homomorphism h we are looking for is determined uniquely by h(qA(b)) = qB(b); this also ensures the proper homomorphism property. All that remains to be shown is well-definedness. Suppose that b1 ∼A b2. By regularity, P(q ×q)(ϕ(b1)) = P(q ×q)(ϕ(b2)). Hence also P(i×i)(ϕ(b1)) = P(i×i)(ϕ(b2))
and ψ(qB(b1)) = ψ(qB(b2)). By injectivity of ψ, we get qB(b1) = qB(b2).

Example 19. Given dipoles d1, d2 ∈ D, let d1 ∼ d2 denote that d1 and d2 have the same start point and point in the same direction. (This is regular w.r.t.

ϕf.) Then D/∼ is the domain OP of oriented points in R
2. Let ϕop : DRAop →
P(OP × OP) and ϕopp : DRAopp → P(OP × OP) be the weak representations obtained as quotients of ϕf and ϕfp, respectively, see Fig. 11. At the level of non-associative algebras, the quotient is given by the tables in Figs. 4 and 10.

This way of constructing DRAop and DRAopp by a quotient gives us their converse and composition tables for no extra effort; we can obtain them by applying the respective congruences to the tables for DRAf and DRAfp, respectively. Moreover, the next result shows that we also can use the quotient to transfer an important property of calculi. Proposition 20. Quotient homomorphism of weak representations preserve strength of composition.

Proof. Let (*h, i*) : ϕ → ψ with ϕ : A → P(*U × U*) and ψ : B → P(*V × V*) be a quotient homomorphism of weak representations. According to Prop. 9, the