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On pseudo MV-algebras. (English) Zbl 0998.06010
The author investigates a noncommutative generalization of the notion of (Chang) MV-algebra, introduced by Georgescu and Iorgulescu, which is thought of as the unit interval of a lattice-ordered Abelian group with strong unit. Various kinds of conditions are given ensuring commutativity, thus recovering the categorical equivalence between MV-algebras and Abelian lattice-ordered groups with strong unit [see the present reviewer’s paper “Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus”, J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)]. Several notions are imported from the theory of lattice-ordered groups, e.g., representability, states, normal ideals. Appropriate pseudo MV-algebraic versions of certain lattice-group theoretical results are proved.

MSC:
06D35 MV-algebras
03G12 Quantum logic
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F35 BCK-algebras, BCI-algebras (aspects of ordered structures)
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