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States on pseudo MV-algebras. (English) Zbl 0999.06011
MV-algebras are an algebraic counterpart of Łukasiewicz infinite-valued propositional logic. By D. Mundici, they are in a one-to-one correspondence with unital abelian lattice-ordered groups ($$\ell$$-groups). Pseudo MV-algebras are a non-commutative generalization of MV-algebras, and A. Dvurečenskij [“Pseudo MV-algebras are intervals in $$l$$-groups”, J. Aust. Math. Soc. 72, 427-445 (2002)] proved that they correspond to unital (not necessarily abelian) $$\ell$$-groups. The author defines states (i.e. finitely additive probability measures) on pseudo MV-algebras and, among others, shows that extremal states correspond to maximal ideals which are normal. Further, he gives an example of a pseudo MV-algebra that (in contrast to MV-algebras) has no states. Therefore he studies classes of pseudo MV-algebras which admit at least one state and proves that representable and normal-valued ones have this property. Moreover, it is proved there that both classes mentioned form varieties of pseudo MV-algebras.

##### MSC:
 06D35 MV-algebras 03G12 Quantum logic 03B50 Many-valued logic
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