Algebraic Combinatorics I: Association Schemes by Eiichi Bannai PDF

By Eiichi Bannai

ISBN-10: 0805304908

ISBN-13: 9780805304909

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The functor ® : Rel x Rel -* Rel is defined as follows. On objects, ® = x, while on maps, A ® B C ®D is given by: (a, b)R ®S(c, d) if aRc & bSd. The tensor unit I = {*}, any one element set. 4. Rel+. This is again the category Rel, except ® = + (disjoint union), I = 0, and where A ® B R®S C ® D is given by: (a, 0)R ® S(c, 0) if and only if aRc (b, 1)R ® S(d, 1) if and only if bSd where disjoint union in Set is given by: X +Y = X x {0} UY x {1} 5. Two important monoidal subcategories of Rel+ are: (i) Pfn: Sets and partial functions.

E. objects of g) using IT, A,=> }. e. arrows of g) together with the axioms (identity), (terminal), (projections), (evaluation) using the rules: (pairing), and (currying). We impose the equations of ccc's between proofs. The operation F(-) is functorial. Indeed, the forgetful functor U has a left adjoint F, CCC F Graph, with F(Cg) the free ccc U as described above. Labels on proofs may be encoded by typed lambda terms, in the familiar manner. This is detailed in [51]. For example, in the currying rule above, f * = A,:: A f ((z, x)) where z : C.

Vecfd and Vec: (finite dimensional) vector spaces over k, where k is a field. Here V ® W is taken to be the usual tensor product, and I = k. R. Blute and Ph. Scott 28 Next, it is natural to ask that the tensor product have an appropriate adjoint, and this leads us to our next definition. e. there is an isomorphism, natural in B, C, satisfying C(C ®A, B) -' C(C, A -o B) . This is the monoidal analog of cartesian closed category; A -o B is the "linear exponential" or "linear function space". In particular there are evaluation and coevaluation maps (A -o B) ® A -- B and C -+ (A -o (C 0 A)), satisfying the adjoint equations.

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Algebraic Combinatorics I: Association Schemes by Eiichi Bannai

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