Roman Frič

Łukasiewicz Tribes are Absolutely Sequentially Closed Bold Algebras

We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean MV-algebra) can be uniquely extended to a sequentially continuous measure on the generated Łukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on Łukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields a characterization of the Łukasiewicz tribes as bold algebras absolutely sequentially closed with respect to the extension of probabilities. The result generalizes the relationship between fields of sets and the generated σ-fields discovered by J. Novák. We introduce the category of bold algebras and sequentially continuous homomorphisms and prove that Łukasiewicz tribes form an epireflective subcategory. The restriction to fields of sets yields the epireflective subcategory of σ-fields of sets.

Statistical maps: A categorical approach

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution pf, a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets.

Extension of domains of states

We deal with categorical aspects of the extensions of generalized probability measures. In particular, we study various domains of fuzzy sets, describe the relationships between σ-fields of crisp sets and generated Łukasiewicz tribes of measurable functions, and mention some probabilistic aspects. D-posets and sequential continuity play an important role.

Convergence and Duality

We describe dualities related to the foundations of probability theory in which sequential convergence and sequential continuity play an important role.

On D-posets of fuzzy sets

D-posets of fuzzy sets constitute a natural simple mathematical structure in which relevant notions of generalized probability theory can be formalized. We present a classification of D-posets leading to a hierarchy of distinguished subcategories of D-posets related to probability and study their relationships. This contributes to a better understanding of the transition from classical probability theory to fuzzy probability theory. In particular, we describe the transition from the Boolean cogenerator {0, 1} to the fuzzy cogenerator [0, 1] and prove that the generated Łukasiewicz tribes form an epireflective subcategory of the bold algebras.

Measures on MV-algebras

Abstract 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.

States on Bold Algebras

We study bold algebras and states on bold algebras in the context of transition from classical probability theory to fuzzy probability theory. Our aim is to point out the role of bold algebras and states on bold algebras in a categorical approach to probability theory. In particular, we formulate several fundamental questions related to basic probability notions and constructions and provide possible answers in terms of bold algebras and states on bold algebras. We show that the category ID of D-posets of fuzzy sets and sequentially continuous difference homomorphisms can serve as a base category in which both classical and fuzzy probability theory can be developed and generalized. Classical and fuzzy random events such as fields of sets and measurable real-valued functions into the interval [0,1], considered as bold algebras, become special objects. Observables, considered as morphisms between objects, become dual to generalized random variables. States become morphisms into [0,1], considered as an object of ID. Properties of objects of ID follow from classical theorems of analysis such as the Lebesgue Dominated Convergence Theorem (states are sequentially continuous) and categorical constructions such as the product (the structure of a probability domain is completely determined by the states as the initial structure). We prove that each generated Łukasiewicz tribe is the epireflection of its underlying Butnariu–Klement σ-field of sets. This helps to understand the transition from classical crisp random events to fuzzy random events. Indeed, the corresponding fuzzification is necessary to cover generalized random variables having a quantum character, i.e. fuzzy random variables in the Gudder–Bugajski sense sending a classical elementary event (point measure) to a non-trivial probability measure.

Coproducts of D-Posets and Their Application to Probability

D-posets introduced by F. Chovanec and F. Kôpka ten years ago provide a suitable algebraic structure to model events in probability theory. Generalizing analogous results for fields of sets and bold algebras, we describe a duality between certain coproducts of D-posets and generalized measurable spaces. An important role in the duality is played by sequential convergence. We mention some applications to the foundations of probability.

Upgrading Probability via Fractions of Events

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

Sequential convergences on Boolean algebras defined by systems of maximal filters

We study sequential convergences defined on a Boolean algebra by systems of maximal filters. We describe the order properties of the system of all such convergences. We introduce the category of 2-generated convergence Boolean algebras and generalize the construction of Novak sequential envelope to such algebras.