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2) In other words, we have been doing first-order set theory right from the beginning. All our spaces of initial action vectors may be regarded as being first-order extensions (|a)) of zero -order quantum predicates (a). The map 1 (= | )) together with its interpretation given above now opens the door to a higher-order quantum set theory (for a given finite dimen­ sional W)y in which we may form sets of sets, and so on, to any depth of nest­ ing. The idea is to define a structure in which it is possible to iterate 1.

Two of these acts determine a third, namely the act of injecting the symmetrical arrangement of the union of their constituents, and the familiar argument yields an associative unital algebra structure for Q. This time, however, the product may be taken to be commuta­ tive, since interchanging two given elements in $Q has the effect merely of sub­ jecting the arrangement injected by the act represented by the product element to a permutation, to which it is impervious. Therefore, we may use the same vector to represent the product act regardless of the order of the original fac­ tors.

It follows from general algebra-coalgebra duality theory that (EndVF, ifj) is a simple coalgebra: that is, one having no proper subcoalgebras. 3 Bialgebras and Hopf algebras A bialgebra is a vector space which carries both an algebra structure and a coalgebra structure in such a way that either structure respects the other: it then transpires that this respect is mutual. In detail, a bialgebra is a vector space such that: (i) (Ay 7T, / ) is a (unital) algebra, (ii) (A,if/,c) is a (counital) coalgebra, (iii) c : A—>fc is a map of algebras, (iv) i//: A—>A®A is a map of algebras.

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