Sikorski Theorem Research Articles

Lifting, n-dimensional spectral resolutions, and n-dimensional observables

We show that under some natural conditions, we are able to lift an n-dimensional spectral resolution from one monotone $$\sigma $$ -complete unital po-group into another one, when the first one is a $$\sigma $$ -homomorphic image of the second one. We note that an n-dimensional spectral resolution is a mapping from $$\mathbb R^n$$ into a quantum structure which is monotone, left-continuous 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 $$+\infty $$ . Applying this result to some important classes of effect algebras including also MV-algebras, we show that there is a one-to-one correspondence between n-dimensional spectral resolutions and n-dimensional observables on these effect algebras which are a kind of $$\sigma $$ -homomorphisms from the Borel $$\sigma $$ -algebra of $${\mathbb {R}}^n$$ into the quantum structure. An important used tool are two forms of the Loomis–Sikorski theorem which use two kinds of tribes of fuzzy sets. In addition, we show that we can define three different kinds of n-dimensional joint observables of n one-dimensional observables.

THE LOOMIS–SIKORSKI THEOREM FOR -ALGEBRAS

An EMV-algebra resembles an MV-algebra in which a top element is not guaranteed. For$\unicode[STIX]{x1D70E}$-complete$EMV$-algebras, we prove an analogue of the Loomis–Sikorski theorem showing that every$\unicode[STIX]{x1D70E}$-complete$EMV$-algebra is a$\unicode[STIX]{x1D70E}$-homomorphic image of an$EMV$-tribe of fuzzy sets where all algebraic operations are defined by points. To prove it, some topological properties of the state-morphism space and the space of maximal ideals are established.

A Loomis–Sikorski theorem and functional calculus for a generalized Hermitian algebra

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $\xi_a$ on the $\sigma$-orthomodular lattice of projections in $A$ and that $\xi_a$ determines the spectral resolution of $a$. Also, if $f$ is a continuous function defined on the spectrum of $a$, we formulate a definition of $f(a)$, thus obtaining a continuous functional calculus for $A$.

Loomis–Sikorski theorem and Stone duality for effect algebras with internal state

Recently Flaminio and Montagna extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis–Sikorski Theorem for monotone σ -complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices Ω such that ∂ e Ω is an F-space.

Representations of pseudo Vitali spaces and Loomis–Sikorski theorem

We present pseudo Vitali spaces as lattices with a partially defined operation, +, that is not necessarily commutative. We show that they can be embedded into ℓ-groups, and we give a categorical equivalence of the category of pseudo Vitali spaces. Studying states on Vitali spaces with strong unit, we show that every Dedekind σ-complete Vitali space is an epimorphic image of a Vitali space of bounded functions where the operations on functions are defined pointwise.

ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE

AbstractIn Flaminio and Montagna [‘An algebraic approach to states on MV-algebras’, in:Fuzzy Logic 2,Proc. 5th EUSFLAT Conference, Ostrava, 11–14 September 2007 (ed. V. Novák) (Universitas Ostraviensis, Ostrava, 2007), Vol. II, pp. 201–206; ‘MV-algebras with internal states and probabilistic fuzzy logic’,Internat. J. Approx. Reason.50(2009), 138–152], the authors introduced MV-algebras with an internal state, called state MV-algebras. (The letters MV stand for multi-valued.) In Di Nola and Dvurečenskij [‘State-morphism MV-algebras’,Ann. Pure Appl. Logic161(2009), 161–173], a stronger version of state MV-algebras, called state-morphism MV-algebras, was defined. In this paper, we present the Loomis–Sikorski theorem forσ-complete MV-algebras with aσ-complete state-morphism-operator, showing that every such MV-algebra is aσ-homomorphic image of a tribe of functions with an internal state induced by a function where all the MV-operations are defined by points.

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Caratheodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces.

On two versions of the Loomis–Sikorski Theorem for algebraic structures

We present two versions of the Loomis---Sikorski Theorem, one for monotone ?-complete generalized pseudo effect algebras with strong unit satisfying a kind of the Riesz decomposition property. The second one is for Dedekind ?-complete positive pseudo Vitali spaces with strong unit. For any case we can find an appropriate system of nonnegative bounded functions forming an algebra of the given type with the operations defined by points that maps epimorphically onto the algebra.

Topology on the set of R 0 semantics for R 0 algebras

This paper introduces the concepts of R 0 valuation, R 0 semantic, countable R 0 category $$\mathcal{C}R_0$$ , R 0 fuzzy topological category $$\mathcal{R}CG$$ , etc. It is established in a natural way that the fuzzy topology ? and its cut topology on the set Ω M consisting of all R 0 valuations of an R 0 algebra M, and some properties of fuzzy topology ? and its cut topology are investigated carefully. Moreover, the representation theorem for R 0 algebras by means of fuzzy topology is given, that is to say the category $$\mathcal{C}R_0$$ is equivalent to the category $$\mathcal{R}CG^{op}$$ . By studying the relation between valuations and filters, the Loomis---Sikorski theorem for R 0 algebras is obtained. As an application, K-compactness of the R 0 logic $$\mathcal{L}^{*}$$ is discussed.

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.

Measures on MV-algebras

We study sequentially continuous measures on semisimple MV-algebras. Let A be a semisimple MV-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz MV-algebra structure and the natural sequential convergence. Each separating set H of MV-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple MV-algebras carrying an initial sequential convergence induced by a separating set of MV-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous MV-algebra homomorphisms, they form a category SM(I). We describe its epireflective subcategory ASM(I) consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished σ-completion σH(A). The epireflection is the maximal object in SM(I) which contains A as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of σH(A) depending on the choice of H. We show that the coproducts in the category of D-posets [9] of suitable families of I-sequential MV-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: H plays the role of elementary events, the embedding of A into σH(A) generalizes the embedding of a field of events A into the generated σ-field σ(A), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8, 11, 14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the Loomis–Sikorski Theorem, objects in ASM(I) correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, D-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature.

D-homomorphisms and atomic σ-complete boolean D-posets

We study observables as σ-D-homomorphisms defined on Boolean D-posets of subsets into a Boolean D-poset. We show that given an atomic σ-complete Boolean D-poset ? with the countable set of atoms there exist a σ-complete Boolean D-poset of subsets ? and a σ-D-homomorphism h from ? onto ?, more precisely we can choose ? = ℬ(ℝ), which gives an analogy of the Loomis–Sikorski representation theorem for Boolean σ-algebras. We show also that any atomic σ-complete Boolean D-poset with the countable set of atoms is the range of a σ-homomorphism defined on a σ-complete Boolean D-poset of fuzzy sets which gives another type of the Loomis–Sikorski theorem.

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.

On a Representation of Observables in Fuzzy Measurable Spaces

We give a representation of observables in fuzzy measurable spaces by random variables and vice versa, and we prove a Loomis–Sikorski theorem analogue. Moreover, some applications are presented.

Limitations on the Fraenkel-Mostowski method of independence proofs

AbstractThe Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold:1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets.2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.

Lω1ω is enough: a reduction theorem for some infinitary languages

In this paper we deal with infinitary languages which allow arbitrary infinite disjunctions and conjunctions, but finite strings of quantifiers only. Furthermore, we shall assume that the primitive predicate symbols are finitary. (Cf. [K] and §2 for further information and unexplained notations.)Our main result shows that every “reasonable” language of this type is, in a certain sense, reducible to one of the same type which allows countable disjunctions and conjunctions only. More precisely if we let θ be the first measurable cardinal and μ ≤ θ, we have the following result partially announced in [R1]:Theorem. Every theory in a Lμω language is equivalent to some theory in a Lω1ω language in the sense that atomic formulas of one language can be mapped into formulas of the other in such a way that, for every set, these maps establish a bijective correspondence between their models having that set as a universe.The paper is divided into three sections. In the first, we derive the main result from a theorem of Sikorski on ω-homomorphisms of μ-complete boolean algebras. In the second, we give a refined version of Sikorski's theorem from which cardinality bounds can be given for the Lω1ω theories obtained by our reduction. In the last, we use a Skolem ultrapower construction to give examples for which these bounds are actually attained. As a corollary, we obtain the converse of our main result.