A spectral theorem for σ MV-algebras

Sylvia Pulmannová

Kybernetika (2005)

  • Volume: 41, Issue: 3, page [361]-374
  • ISSN: 0023-5954

Abstract

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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 σ -MV-algebras, we prove that, with every element a in a σ -MV algebra M , a spectral measure (i. e. an observable) Λ a : ℬ ( [ 0 , 1 ] ) → ℬ ( M ) can be associated, where ℬ ( M ) denotes the Boolean σ -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.

How to cite

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Pulmannová, Sylvia. "A spectral theorem for $\sigma $ MV-algebras." Kybernetika 41.3 (2005): [361]-374. <http://eudml.org/doc/33759>.

@article{Pulmannová2005,
abstract = {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) $\Lambda _a: \{\mathcal \{B\}\}([0,1])\rightarrow \{\mathcal \{B\}\}(M)$ 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.},
author = {Pulmannová, Sylvia},
journal = {Kybernetika},
keywords = {MV-algebras; Loomis–Sikorski theorem; tribe; spectral decomposition; lattice effect algebras; compatibility; block; MV-algebra; Loomis-Sikorski theorem; tribe; spectral decomposition; lattice effect algebra; compatibility},
language = {eng},
number = {3},
pages = {[361]-374},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A spectral theorem for $\sigma $ MV-algebras},
url = {http://eudml.org/doc/33759},
volume = {41},
year = {2005},
}

TY - JOUR
AU - Pulmannová, Sylvia
TI - A spectral theorem for $\sigma $ MV-algebras
JO - Kybernetika
PY - 2005
PB - Institute of Information Theory and Automation AS CR
VL - 41
IS - 3
SP - [361]
EP - 374
AB - 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) $\Lambda _a: {\mathcal {B}}([0,1])\rightarrow {\mathcal {B}}(M)$ 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.
LA - eng
KW - MV-algebras; Loomis–Sikorski theorem; tribe; spectral decomposition; lattice effect algebras; compatibility; block; MV-algebra; Loomis-Sikorski theorem; tribe; spectral decomposition; lattice effect algebra; compatibility
UR - http://eudml.org/doc/33759
ER -

References

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