NDLI: Measures on MV-algebras

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Measures on MV-algebras

Content Provider	Springer Nature Link
Author	Frič, R.
Copyright Year	2002
Abstract	We study sequentially continuous measures on semisimple M V-algebras. Let A be a semisimple M V-algebra and let I be the interval [0,1] carrying the usual Łukasiewicz M V-algebra structure and the natural sequential convergence. Each separating set H of M V-algebra homomorphisms of A into I induces on A an initial sequential convergence. Semisimple M V-algebras carrying an initial sequential convergence induced by a separating set of M V-algebra homomorphisms into I are called I-sequential and, together with sequentially continuous M V-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 σ H (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 σ H (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 M V-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 σ H (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.
Starting Page	130
Ending Page	137
Page Count	8
File Format	PDF
ISSN	14327643
Journal	Soft Computing
Volume Number	7
Issue Number	2
Language	English
Publisher	Springer-Verlag
Publisher Date	2002-12-01
Publisher Place	Berlin Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Software Geometry and Topology

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