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Lattice effect algebras with total operations. (English) Zbl 1044.03049

Summary: Effect algebras are partial algebras that have been introduced by Foulis and Bennett as an algebraic structure providing an instrument for studying quantum effects that may be unsharp. Lattice effect algebras give a common generalization of orthomodular lattices (including Boolean algebras) and MV-effect algebras. In every lattice effect algebra the partial binary operation can be extended to a total binary operation. We bring some properties of that total operation for mutually compatible elements and consequently for elements of blocks or for elements of MV-effect algebras. Moreover we bring some counterexamples to show that some of those properties are violated for noncompatible elements. We show some equivalent conditions for a lattice effect algebra under which it becomes an MV-effect algebra or an orthomodular lattice.

MSC:

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