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Weakly linear quantum MV-algebras. (English) Zbl 1093.06011

Quantum MV-algebras (QMV-algebras) were introduced by the author in “Quantum MV algebras” [Stud. Log. 56, No. 3, 393–417 (1996; Zbl 0854.03057)] as a non-lattice-theoretical generalization of both MV-algebras and orthomodular lattices. An important example is the system \(E({\mathcal H})\) of all Hermitian operators of a Hilbert space \({\mathcal H}\) between the zero operators and the identity. In general, every po-group \(G\) with a positive element \(u\) gives another example \(G[0,u] =\{g \in G:\;0\leq g \leq u\}\), where \(a\oplus b = a+b\) if \(a+b \leq u\) otherwise \(a\oplus b = u\), of a GMV-algebra. The accent in the paper is on weakly linear QMV-algebras for which the author studies a finite basis for the variety generated by the class of weakly linear QMV-algebras. Finally, three open problems are formulated.

MSC:

06D35 MV-algebras
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Citations:

Zbl 0854.03057
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