Σ-complete MV-algebras Research Articles

Spectral Resolutions and Quantum Observables

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of $\mathbb R^{n}$ with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from $\mathbb R^{n}$ into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to $-\infty $ and it goes to 1 if all variables go to $+\infty $ . The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

Two-Dimensional Observables and Spectral Resolutions

A two-dimensional observable is a special kind of a σ-homomorphism defined on the Borel σ-algebra of the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra which has properties similar to a two-dimensional distribution function in probability theory. We show that there is a one-to-one correspondence between two-dimensional observables and two-dimensional spectral resolutions defined on a σ-complete MV-algebras as well as on the monotone σ-complete effect algebras with the Riesz decomposition property. The result is applied to the existence of a joint two-dimensional observable of two one-dimensional observables.

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.

Loomis-Sikorski theorem for monotone σ-complete effect algebras

AbstractWe show that monotone σ -complete effect algebras under some conditions are σ - homomorphic images of effect-tribes (as monotone σ -complete effect algebras), which are nonempty systems of fuzzy sets closed under complements, sums of fuzzy sets less than 1, and containing all pointwise limits of nondecreasing fuzzy sets. Because effect-tribes are generalizations of Boolean σ -algebras of subsets, we present a generlization of the Loomis-Sikorski theorem for such effect algebras. We show that we can choose an effect-tribe to be a system of affin fuzzy sets. In addition, we present a new version of the Loomis-Sikorski theorem for σ-complete MV-algebras.

Spectral resolutions in Dedekind σ-complete ℓ-groups

Using recent results on a generalized form of the Loomis–Sikorski theorem [A. Dvurečenskij, Loomis–Sikorski theorem for σ-complete MV-algebras and ℓ-groups, J. Austral. Math. Soc. Ser. A 68 (2000) 261–277; D. Mundici, Tensor product and the Loomis–Sikorski theorem for MV-algebras, Adv. Appl. Math. 22 (1999) 227–248], it is shown that a unital Dedekind σ-complete ℓ-group is a compatible Rickart comgroup in the sense of D.J. Foulis [D.J. Foulis, Spectral resolutions in a Rickart comgroup, Rep. Math. Phys. 54 (2004) 229–250]. In particular, elements in unital Dedekind σ-complete ℓ-groups and, consequently, elements in σ-MV-algebras, admit uniquely defined spectral resolutions similar to spectral resolutions of self-adjoint operators. A functional calculus and spectra of elements are considered in relation with the Loomis–Sikorski representation by functions.

The σ-complete MV-algebras which have enough states

We characterize <i>Łukasiewicz tribes</i>, i.e., collections of fuzzy sets that are closed under the standard fuzzy complementation and the Łukasiewicz t-norm with countably many arguments. As a tool, we introduce $\sigma $-<i>McNaughton functions</i> as

Conditional probability on MV-algebras

An appropriate definition of a conditional probability on an MV-algebra was an open problem mentioned in Riečan and Mundici (Handbook of Measure Theory, North-Holland, Amsterdam, 2002). We propose some concept of conditional probability (state) on a σ-complete MV-algebra with product. Its basic properties will be proven and it will also be demonstrated that the conditional probability in classical probability theory is a special case of this definition. Moreover, the paper contains also a discussion of the interpretation of fuzzy sets and conditioning in fuzzy probability theory.

Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups

AbstractWe show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ωand closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.

Weakly Divisible MV-Algebras and Product

We introduce weakly divisible MV-algebras and we show that every weakly divisible σ-complete MV-algebra is isomorphic to the system of all continuous fuzzy functions defined on some compact Hausdorff space which generalizes a result of Di Nola and Sesse ( in“Non Classical Logics and Their Application to Fuzzy Subsets,” Kluwer Academic, Dordrecht, to appear). This enables us to define an associative and commutative product on weakly divisible σ-complete MV-algebras with a multiplicative unity 1. As an application, we show that any system of observables on this MV-algebra admits a joint observable with respect to this product.

Tensor Products and the Loomis–Sikorski Theorem for MV-Algebras

MV-algebras are the models of the time-honored equational theory of magnitudes with unit. Introduced by Chang as a counterpart of the infinite-valued sentential calculus of Łukasiewicz, they are currently investigated for their relations with AFC*-algebras, toric desingularizations, and lattice-ordered abelian groups. Using tensor products, in this paper we shall characterize multiplicatively closed MV-algebras. Generalizing work of Loomis and Sikorski, we shall investigate the relationships between σ-complete multiplicatively closed MV-algebras, and pointwise σ-complete MV-algebras of [0,1]-valued functions.