Perspectivity and congruence in partial abelian semigroups. (English) Zbl 0938.03094

In the paper, the notion of a partial abelian semigroup is introduced as a generalization of various structures considered in the literature on quantum logics: orthomodular posets, orthoalgebras, D-posets (or effect algebras), and their respective non-unital variants. A partial abelian semigroup (PAS) is defined as a structure \((L,\perp , \oplus)\), where \(\perp \) is a binary relation on \(L\) and \(\oplus \) is a partially defined binary operation with domain \(\perp \), which is commutative and associative in the restricted sense: if one side of the relevant equality exists, so does the other and equality holds. To any subset \(M\) of a partial abelian semigroup \(L\), a relation of perspectivity \(\sim _M\) is introduced, where \(a\sim _M b\) if there exists some element \(c\in L\) such that \(a\oplus c\) and \(b\oplus c\) both exist and belong to \(M\). If \(\sim _M\) is a faithful congruence, that is, \(\sim _M\) is an equivalence relation such that for all \(a,b,c\in L\), \(a\sim _M b\) and \(c\perp a\) imply \(c\perp b\), \(M\) is called an algebraic set. The theory of pairs \((L,M)\), where \(L\) is a PAS and \(M\) is a fixed algebraic subset provides a natural generalization of the theory of manuals.


03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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