A Spectral Theorem for Sigma Mv-algebras

MV-algebras were introduced by Chang, 1958 as algebraic bases for multi-valued logic. MV stands for "multi-valued" and MV algebras have already occupied an important place in the realm of nonstandard (mathematical) logic applied in several fields including cybernetics. In the present paper, using the Loomis-Sikorski theorem for cr-MV-algebras, we prove that, with every element a in a cr-MV algebra M, a spectral measure (i. e. an observable) Aa : B([0,1]) —> B(M) can be associated, where B(M) denotes the Boolean cr-algebra of idempotent elements in M. This result is similar to the spectral theorem for self-adjoint operators on a Hilbert space. We also prove that MV-algebra operations are reflected by the functional calculus of observables.