By Erich Hecke

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**Additional info for Vorlesungen über die Theorie der Algebraischen Zahlen**

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To appear in Proceedings of the 3rd International Workshop on Quantum Programming Languages, 2005. Appendix The following equational proofs involve some long typed formulas. To aid in readability, we have annotated each equational step (reading down the page) by underlining each redex, and overlining the corresponding contractum. Proof of Lemma 3 Proof. V TrU A,B (f ) ⊗ TrC,D (g) = {Superposing} TrVA⊗C,B⊗D (TrU A,B (f ) ⊗ g) = {Naturality of τ } TrVA⊗C,B⊗D (τD⊗V,B ◦ (g ⊗ TrU A,B (f )) ◦ τA,C⊗V ) = {Superposing} TrVA⊗C,B⊗D (τD⊗V,B ◦ TrU C⊗V ⊗A,D⊗V ⊗B (g ⊗ f ) ◦ τA,C⊗V ) = {Input/Output Naturality} TrVA⊗C,B⊗D (TrU A⊗C⊗V,B⊗D⊗V ((τD⊗V,B ⊗ 1U ) ◦ (g ⊗ f ) ◦ (τA,C⊗V ⊗ 1U ))) = {SM Coherence} TrVA⊗C,B⊗D (TrU A⊗C⊗V,B⊗D⊗V ((1A ⊗ τU,D⊗V ) ◦ (f ⊗ g) ◦ (1A ⊗ τC⊗V,U ))) = {Vanishing II} ⊗U TrVA⊗C,B⊗D ((1A ⊗ τU,D⊗V ) ◦ (f ⊗ g) ◦ (1A ⊗ τC⊗V,U )).

3 Labels As we mentioned in the Introduction, IPOs have been used by Leifer and Milner to derive labelled transition systems for calculi equipped with a reduction semantics derived from a set of ground rules. Here we give a brief overview of the technique. Leifer and Milner’s framework of choice is their notion of ‘reactive system,’ which consists of a category of contexts with a chosen object 0, a subcategory of evaluation contexts which satisfies certain additional axioms and a set of reduction rules R.

Additionally, one can quotient terms by the commutativity equation P|Q=Q|P; the resulting PROP will be called PACP (Prefix and Associative, Commutative Parallel Composition). 3 Labelled Transitions for Ground Reductions This section introduces the background material we need in later sections. First, we briefly recall Leifer and Milner’s notion of idem-relative-pushout (IPO) as well as its dual, the idem-relative-pullback (IPB). Following a brief informal and discussion on how IPOs have been used in order to generate labelled transition systems (LTS) for calculi with ground reduction rules, we shall demonstrate that IPOs and IPBs can be conveniently studied in a category of factorisations, where they are easily seen to be coproducts and products, respectively.

### Vorlesungen über die Theorie der Algebraischen Zahlen by Erich Hecke

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