Kybernetika 41 no. 3, 361-374, 2005

A spectral theorem for σ MV-algebras

Sylvia Pulmannová


\small 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 $\sigma$-MV-algebras, we prove that, with every element $a$ in a $\sigma$-MV algebra $M$, a spectral measure (i. e. an observable) $\Lambda_a: {\mathcal B}([0,1])\to {\mathcal B}(M)$ can be associated, where ${\mathcal B}(M)$ denotes the Boolean $\sigma$-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.