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By H. Araki, R. V. Kadison

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Hence a Frobenius algebra in Mat 2 (R+ ) has a reduct to one in Rel via this homomorphism. Just as before, we can apply Theorem 30 to this reduct. H*-ALGEBRAS AND NONUNITAL FROBENIUS ALGEBRAS 21 We have the following analogue to Proposition 34, where M : (A×A)×A → R+ is the matrix realizing the Frobenius algebra structure. Proposition 36. The function M is constant on each disjoint summand of A. Proof. We can use the same reasoning as in Proposition 34(a) to show that, if e is an identity element in one of the disjoint summands, then for all a, b in that disjoint summand, M (a, e, a) = M (b, e, b).

E. if R : X ✤ / Y and S : Y ✤ / Z, then S ◦ R(x, z) = R(x, y) · S(y, z). e. relational converse. Thus Rel(Q) is a symmetric monoidal dagger category, and the notion of Frobenius algebra makes sense in it. Note that Rel(2) is just Rel. A homomorphism of quantales h : Q → R induces a (strong) monoidal dagger functor h∗ : Rel(Q) → Rel(R), which transports Frobenius algebras in Rel(Q) to Frobenius algebras in Rel(R). In particular, by Proposition 33, a Frobenius algebra Δ : A ✤ / A × A in Rel(Q) has a reduct h∗ Δ : A ✤ / A × A in Rel.

1. Frobenius algebras in Rel and lbfRel. We assume given a set A, and a Frobenius algebra structure on it given by a relation Δ ⊆ A × (A × A). We shall write ∇ for Δ† . Definition 25. Define x ∼ y if and only if (x, y)∇z for some z. By (M), the relation ∇ is single-valued and surjective. Therefore, we may also use multiplicative notation xy (suppressing the ∇), and write x ∼ y to mean that xy is defined. Lemma 26. The relation ∼ is reflexive. Proof. Let a ∈ A. By (M), we have a = a1 a2 for some a1 , a2 ∈ A.

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