#### Foreword

This publication is a result of cooperation of three generations of authors: Dr. h. c.mult. prof. RNDr. Beloslav Riečan, DrSc., prof. RNDr. Anna Tirpáková, CSc. and Mgr. Renáta Bartková, PhD. I have been related with all the three of them through cooperation during several years.

From 1985 to 1989 we were colleagues with professor B. Riečan at the Department of Mathematics of Military College in Liptovský Mikuláš. That time I was just a novice lecturer, but professor B. Riečan joined our department as a distinguished and highly regarded Slovak mathematician. In that period his scientific work was focused on theory of measure and integral on ordered structures, and probability theory on non-Boolean structures - orthomodular lattices, posets, so called quantum logics. He also guided some of his new colleagues into these branches, namely B. Harman, J. Haluška, F. Chovanec, M. Jureèková, F. Kôpka, P. Malièký, J. Rybárik, E. Rybáriková-Drobná. It was in those times, in the second half of 1980s, that B.
Riečan suggested introducing the notion of uncertainty based on fuzzy sets
principle into quantum logics. He defined the term of F-quantum space (fuzzy
quantum space) where the main terms were F-state and F-observable. F-quantum
spaces became the subject of intensive study of group known as Slovak school of
fuzzy sets whose standard bearers besides B. Riečan were A. Dvurečenskij and R.
Mesiar. Besides the above mentioned personalities from Liptovský Mikuláš several
others gradually joined the group, namely S. Bodjanová (University of Economics
in Bratislava), J. Bán and M. Kalina (Faculty of Mathematics, Physics and
Informatics of Comenius University in Bratislava), A. Kolesárová and M. Šabo
(Slovak University of Technology in Bratislava), D. Markechová (UKF Nitra), A.
Tirpáková (Institute of Archaeology, Slovak Academy of Sciences in Nitra), B.
Stehlíková (District Housing Enterprise Nové Zámky), P. Vojtáš, R. Friè, M. Papèo
(Mathematical Institute, Slovak Academy of Sciences in Bratislava - separate unit
in Košice), M. Gavalec (Technical University of Košice), V. Janiš (Matej Bel
University in Banská Bystrica), A. Marková-Stupòanová (Slovak University of
Technology in Bratislava). During his stay in Liptovský Mikuláš professor Riečan
chose for another meeting point of people from all around the former
Czechoslovakia interested in fuzzy sets the nearby valley Jánska dolina, and thus
he contributed in great extent to the emergence of Czech-Slovak and later on Czech-
Slovak-Polish conferences Theory and Applications of Fuzzy Sets. After the
change of political system in Czechoslovakia these trilateral conferences grew
bigger and became international biennial conferences FSTA (Fuzzy Sets Theory
and Applications) which have regularly taken place in Liptovský Ján since 1992.
During the first FSTA conference F. Kôpka, at that time a postgraduate student supervised by B. Riečan, proposed fuzzy quantum model in which the primary
operation is the partial operation of difference of fuzzy sets. He named this model
a difference poset of fuzzy sets (D-poset of fuzzy sets). D-posets of fuzzy sets as
well as their generalized abstract structure, difference posets (D-posets), were
highly accepted by international community of mathematicians. D-posets
interrelated algebraic structures which had seemed to be mutually unrelated until
that time - quantum logics and multiple-valued logics (MV-algebras).
Unfortunately, in 2008 a severe illness untimely ended life of F. Kôpka. In order to
honour his memory, B. Riečan named D-poset of fuzzy sets with another special
operation of product of fuzzy sets as Kôpka’s D-poset. In further period professor
Riečan together with his postgraduate students dedicated themselves mostly to
development of probability theory on MV-algebras and IF-sets. In September 1987
I together with my colleague F. Kôpka attended an internship at the Mathematical
Institute of Slovak Academy of Sciences in Bratislava under the supervision of an
expert professor A. Dvurečenskij. There we met A. Tirpáková, a postgraduate
student supervised by professor Dvurečenskij. Together we studied quantum
structures, fuzzy sets theory, and explored F-quantum spaces introduced by B.
Riečan. A. Tirpáková was mainly focused on F-observables, their summability,
various types of their convergences; she studied ergodic theory on F-quantum
spaces and achieved remarkable results in this branch. Since then we regularly met
on scientific conferences, such as PROBASTAT, Winter School on Measure
Theory, Theory and Applications of Fuzzy Sets, FSTA and Nitra Statistical Days.

The third author, R. Bartková, started her postgraduate study under supervision of professor Riečan in 2011, but then the supervision of her study was passed to me. In her dissertation she devoted herself to study of validity of fundamental theorems on extreme values on various algebraic structures: non-additive probability space, fuzzy quantum space, MV-algebras and IF-events. In addition to the above mentioned theoretical contribution, she also achieved practical contribution to statistical processing of concrete data from IF-sets by means of principal component analysis and factor analysis in order to reduce the dimension of data set.The results she achieved are included in several chapters of this publication.

The authors of publication Probability Theory for Fuzzy Quantum Spaces with Statistical Applications introduce the reader into the issues of probability theory, gradually from the traditional Kolmogorov probability space where the domain is set algebra, then Zadeh space focused on fuzzy sets, continuing with probability theory on IF-sets, MV-algebras, and finally on F-quantum spaces. The reader can get a coherent picture about the issues of probability theory on particular algebraic structures, and thus become able to judge mutual relationships and differences.

Thanks to high scientific and methodical erudition of the authors the publication is written in an accessible and comprehensible style. I am confident that it is going to be a sought-after study material for students, postgraduate students, and for all those interested in the given issues.

**doc. RNDr. Ferdinand Chovanec, CSc.**

Department of Natural Sciences

Armed Forces Academy

Liptovký Mikuláš, Slovakia

#### Preface

Probability theory, like other mathematical disciplines, underwent a tumultuous development in the past. In 1933 Kolmogorov published work [39], in which he introduced the axiomatic model of his probability theory. In classical probability theory based on Kolmogorov axiomatic model it is assumed that the events associated with the experiment form the Boolean σ-algebra of subsets 𝒮 of the set Ω. Probability is then the σ-additive nonnegative final function P on 𝒮 with values in interval [0,1], whereby it holds that if {An} is a sequence of mutually exclusive events from S, then

At present, we can say that Kolmogorov axioms deeply infiltrated not only into probability theory and mathematical statistics, but they also encouraged the development of other scientific disciplines, such as physics, biology, economics, social sciences, etc.

Over time, however, it has emerget that for some scientific disciplines, for example quantum mechanics, the concept of σ-algebra is too limiting. It does not describe such situations that arise in connection with Heisenberg uncertainty principle, which claims that if any two observables (such as a position *x* and a momentum *y*) are measured at the same time, the product of the squares of errors of measurement and is connected with the inequality , *i.e.* the accuracy of measurement of one variable happens at the expense of the second one. Birkhoff and von Neumann [5] pointed to the fact that the set of experimentally verifiable statements about quantum-mechanical system has different algebraic structure than the Boolean algebra in [39]. The first attempts at the mathematical formulation of the quantum mechanics come from Heisenberg [35] and Schrödinger [96]. Heisenberg proposed the formalism of matrix mechanics and Schrödinger the one of wave mechanics. Both theories were generalized by von Neumann [63]. He proposed a model of quantum mechanics based on a complex separable Hilbert space.

Nowadays, one of the most used axiomatic models of quantum mechanics is quantum logic, which recorded a rapid development in 1960, when works of Varadarajan [103], Mackey [43], Mac Laren [42], Gunson [29] and others appeared.

The basic mathematical model of the current quantum theory is the von Neumann model, based on the geometry of Hilbert space (Varadarajan, [103]). If we define the system M of all closed subspaces of the given Hilbert space (where the notion "a state of system" means a measure of probability on according to Varadarajan [103]), and this definition of the state is compared with the definition of P-measure on fuzzy sets (according to Piasecki [70]), we can see that both objects have similar algebraic structure. In 1985 Piasecki [70] submitted a model called soft σ-algebra in the fuzzy set theory. This model demonstrated several identical characteristics of this new structure with quantum logics. That analogy, which was for the first time observed by Riečan [79] and later by Pykacz [73], led us to the idea to build a quantum theory based on fuzzy sets.

The theory of fuzzy sets originated in 1960s in connection with the emergence of article by Zadeh [106]. If we return to the analogy between quantum logics and fuzzy sets theory, according to Dvurečenskij [13] a fuzzy set can be viewed as a fuzzy event, respectively as a real-valued function, which is defined on set with values in the interval [0,1], which describes the fuzziness of set (event) a within the meaning of Zadeh [106]. Number a(*x*) indicate the measure of membership of point *x* to set a. If *X* is a non-empty set, called the universum, and is a system of fuzzy subsets of universum *X*, *i.e.* the system of functions on *X* with values in interval [0, 1], then according to Riečan [79] we say that (*X*, ) is an F- quantum space, according to Dvurečenskij and Chovanec [17] also called a fuzzy quantum space, or according to Dvurečenskij [12] a fuzzy measurable space. The set according to Piasecki [70] is also called a soft σ-algebra.

More general structures of fuzzy quantum spaces were studied by Dvurečenskij and Chovanec [16, 17] and Dvurečenskij, Chovanec and Kôpka [18]. The law of exclusions of the third or orthomodular law does not apply on set , which according to Mittelstaedt [57] can play certain role in axiomatic models of quantum mechanics.

In the theory of fuzzy quantum spaces many authors try to prove some known assertions from the classical probability theory. For example, the existence of fuzzy state on fuzzy quantum space was studied by Dvurečenskij [12], Navara [59] and Navara, Pták [60, 61], joint fuzzy observables and joint distributions of fuzzy observables were studied by Dvurečenskij and Riečan [21, 22]. Dvurečenskij, Kôpka and Riečan [19] proved the representation theorem, which also includes thecase in fuzzy quantum space. The theory of an indefinite integral on the fuzzy quantum space was studied by Riečan [82, 83] and the extension of the validity of the Bayes formula for fuzzy sets was investigated by Mesiar [53-55], Piasecki [68, 69] Piasecki and Svitalski [71]. The entropy on fuzzy quantum space was studied by Markechová [46, 48, 49].

An important fact for the study of many assertions in the fuzzy sets theory is the existence of sum of fuzzy observables. Harman and Riečan [34] proved the existence of the sum of compatible fuzzy observables.

Among the important concepts of probability theory belong various types of convergence of random variables, which are important especially for those parts which deal with the validity of various forms of the law of large numbers and the central limit theorem. Thus the problem of generalizations of different types of convergence for fuzzy quantum space (*X*, ) became topical. Several authors studied particular types of convergences on quantum logic. We will mention only those works which were the basic material for the study of various types of convergences of fuzzy observables on fuzzy quantum space (*X*, ), and that is Dvurečenskij and Pulmanová [20], Jajte [37], Ochs [64, 65], Cushen [9], Gudder [26], Révesz [76]. Dvurečenskij [10], Riečan [80, 81], Chovanec and Kôpka [36], Kôpka and Chovanec [40], and others dealt with some types of convergences of fuzzy observables on fuzzy quantum space.

The issue of the ergodic theory on quantum logics was studied by more authors. Here we present only those works which were the basis for the generalization of the ergodic theory for fuzzy quantum space (*X*, ): Pulmanová [72], Dvurečenskij [15] and Mesiar [56]. The first authors to prove the individual ergodic theorem for compatible fuzzy observables on fuzzy quantum space were Harman and Riečan [34]. The proof of the individual ergodic theorem for more general case is provided in this book.

The Hahn-Jordann decomposition on quantum logics exists only in partial cases, for example it exists on the logic of Hilbert space (Šerstnev [99], Dvurečenskij [14], Rüttimann [94]). The Hahn-Jordann decomposition on fuzzy quantum space is studied in this book (Chapter 4).

The following figure displays the position of fuzzy quantum space in various algebraic structures.

#### CONFLICT OF INTEREST

The authors confirm that this eBook content has no conflict of interest.

#### ACKNOWLEDGMENT

Hereby we would like to thank prof. Ferdinand Chovanec for his Foreword, and Tomáš Škraban, PhD. and Zuzana Naštická, M.A. for reviewing the English language correctuess. We also acknowledge the Bentham Science Publishers and referees for their remarks and recommendations.

#### List of Contributors

##### Author(s):

**Renáta Bartková **

Podravka International s.r.o, Zvolen

Slovakia

**Beloslav Riečan **

Department of Mathematics, Faculty of Natural Sciences

Matej Bel University

Tajovského 40, SK-974 01 Banská Bystrica

Slovakia

/

Mathematical Institute Slovak Academy of Sciences

Štefánikova 49, SK-814 73 Bratislava

Slovakia

**Anna Tirpáková **

Department of Mathematics, Faculty of Natural Sciences

Constantine the Philosopher University, T r ieda A. Hlinku 1, SK-949 74 Nitra