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States and idempotents of pseudo MV-algebras. (English) Zbl 0997.03050

Pseudo MV-algebras are non-commutative generalizations of MV-algebras. The study of states defined on a pseudo MV-algebra was started by the author in “States on pseudo MV-algebras” [Stud. Log. 68, 301-327 (2001; Zbl 0999.06011)].
This paper is concerned with the states on a pseudo MV-algebra \(A\) satisfying general comparability, a condition on the Boolean algebra BCA of idempotents of \(A\). A first result shows that every pseudo MV-algebra satisfying general comparability has at least one state. A second result asserts that in such a pseudo MV-algebra \(A\), the space \({\operatorname {Ext}}\bigl (S(A)\bigr)\) of extremal states on \(A\) is homeomorphic to the space \({\operatorname {Ext}}\bigl (S(\operatorname {BCA})\bigr)\) of extremal states on BCA.

MSC:

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

Citations:

Zbl 0999.06011
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