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Residuated lattices and lattice effect algebras. (English) Zbl 1122.81016

The authors study two partial operations in effect algebras, namely \(a \odot b = (a' \oplus b')'\) if \(a' \oplus b'\) is defined and \(a \to_p b = a' \oplus b\) if \(a' \oplus b\) is defined. They show, e.g., that for an effect algebra \((E, \oplus, 0, 1)\) we obtain a (dual) effect algebra \((E, \odot,0,1)\). Using these operations they construct an adjoint pair in an effect algebra with the Riesz decomposition property to obtain an involutive residuated lattice. On the other hand, they prove that an involutive residuated lattice \((L, \leq \otimes, \to, 0, 1)\) corresponds to an effect algebra with the Riesz decomposition property if and only if \(a \land b = a \otimes (a \to b)\) for every \(a,b \in L\).

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
06F05 Ordered semigroups and monoids
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References:

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