Proof. The identity relation = is clearly mapped to the identity relation sese.

For the composition and converse properties, we just inspect the composition and converse tables for the two calculi.14 The mapping of the base-relation is then lifted directly to a mapping of all relations, where the map is applied component-wise on the relations. Using the laws of non-associative algebras, the homomorphism property of these relations follows from that of the baserelations.

In cases stemming from the embedding of Allen's Interval Algebra, the dipoles lie on the same straight lines and have the same direction. DRAf and DRAfp also contain 13 additional relations which correspond to the case with dipoles lying on a line but facing opposite directions.

As we shall see, it is very useful to extend the notion of homomorphisms to weak representations:
Definition 12. Given weak representations ϕ : A → P(*U × U*) and ψ : B →
P(*V × V*), a lax (oplax, proper) homomorphism of weak representations (*h, i*) :
ϕ → ψ is given by
- a proper homomorphism of non-associative algebras h : A → B, and - a map i : *U → V*, such that the diagram

![0_image_0.png](0_image_0.png)

commutes laxly (respectively oplaxly, properly). Here, lax commutation means that for all R ∈ A, ψ(h(R)) ⊆ P(i × i)(ϕ(R)), oplax commutation means the same with ⊇, and proper commutation with =. Note that P(i × i) is the obvious extension of i to a function between relation algebras; note that (unless i is bijective) this is not even a homomorphism of Boolean algebras (it may fail to preserve top, intersections and complements), although it satisfies the oplaxness property (and the laxness property if i is surjective).15 Note that Ligozat [47] defines a more special notion of morphism between weak representations; it corresponds to our oplax homomorphism of weak representations where the component h is the identity.