Many-valued quantum algebras. (English) Zbl 1219.06013

Summary: We deal with algebras \(\mathbf A = (A,\oplus,\neg,0)\) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) \(\mathbf A\) bears a natural lattice structure, (ii) the elements \(a\) for which \(\neg a\) is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.


06D35 MV-algebras
03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
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