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Tensor products of orthoalgebras. (English) Zbl 0798.06015
The following definition of a tensor product of orthoalgebras is proposed: Let \(P\) and \(Q\) be orthoalgebras. We say that a pair \((T,\tau)\) consisting of an orthoalgebra \(T\) and a bimorphism \(\tau: P\times Q\to T\) is a tensor product of \(P\) and \(Q\) iff the following conditions are satisfied: (i) If \(L\) is an orthoalgebra and \(B: P\times Q\to L\) is a bimorphism, there exists a morphism \(\varphi: T\to L\) such that \(B= \varphi\circ\tau\). (ii) Every element of \(T\) is a finite orthogonal sum of elements of the form \(\tau(p,q)\) with \(p\in P\) and \(q\in Q\).
It is shown that the tensor product need not exist. A sufficient condition for its existence are sufficiently large spaces of probability measures: for instance, two unital orthoalgebras have always a tensor product.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
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