Orthocomplemented subspaces of a Hilbert space are in bijection with partial projections, yielding a constructive quantum logic with classical negation.
Hidden constructions in abstract algebra, Krull Dimension, Going Up, Going Down
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abstract
We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content where these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.
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Orthocomplemented subspaces and partial projections on a Hilbert space
Orthocomplemented subspaces of a Hilbert space are in bijection with partial projections, yielding a constructive quantum logic with classical negation.