By Adam Brandenburger, H. Jerome Keisler (auth.), Bob Coecke, Luke Ong, Prakash Panangaden (eds.)
This Festschrift quantity, released in honor of Samson Abramsky, includes contributions written by way of a few of his colleagues, former scholars, and pals. In occasion of the sixtieth birthday of Samson Abramsky, a convention was once held in Oxford, united kingdom, in the course of may perhaps 28-30, 2010. The papers during this quantity symbolize his manifold contributions to semantics, common sense, video games, and quantum mechanics.
Read Online or Download Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky: Essays Dedicated to Samson Abramsky on the Occasion of His 60th Birthday PDF
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Extra info for Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky: Essays Dedicated to Samson Abramsky on the Occasion of His 60th Birthday
First, note the following stardard equation for Frobenius algebras: = = = Then, the result follows from associativity: = = = = . Lemma 9. For projections we have: (i) (p · q)∗ = q ∗ · p∗ ; (ii) If p · q is a projection, then p · q = q · p. Proof. (i) We have ⎞∗ ⎛ ⎠= (p·q) = ⎝ ∗ p p p q = q q = p = q ∗ ·p∗ , q where the middle equation follows from Lemma 8. (ii) If p · q = r then, by selfconjugateness of projections and (i), q · p = q ∗ · p∗ = (p · q)∗ = r∗ = r = p · q. Theorem 10. In a dagger symmetric monoidal category, projections on a symmetric dagger Frobenius algebra with a zero projection are partially ordered and come with an orthogonality relation.
A projection of a symmetric dagger Frobenius algebra (A, in a dagger symmetric monoidal category C is a morphism p : I → A satisfying equations (5). The next section studies the structure of these generalised projections of abstract C*-algebras. Before that, we compare abstract projections to copyable points. These played a key role for commutative abstract C*-algebras, because they correspond to the elements of an orthonormal basis that determines the algebra . However, as we will now see, in the noncommutative case, there simply do not exist enough copyable points (whereas the projections do have interesting structure, as the next section shows).
Let G be a groupoid, and (G, ) the corresponding sym- metric dagger Frobenius algebra in Rel. e. subcategories of G that are groupoids themselves. Proof. This follows directly from [25, Theorem 16]. It immediately follows that in Rel, like in FHilb, the abstract projection lattice is a complete lattice, even though we are not dealing with ﬁnite sets. Corollary 19. If (G, Proj(G, ) is a symmetric dagger Frobenius algebra in Rel, then ) forms a complete lattice. Proof. The collection of subgroupoids is closed under arbitrary intersections.