ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS

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1 ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS MARTIN PAPČO Abstract. There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. This leads to fuzzy random variables in the Bugajski-Gudder sense. The idea is to admit variables sending the set Ω of elementary events not into the real numbers, but into the set M + 1 (R) of all probability measures on the real Borel sets (each real number r R is considered as the degenerated probability measure δ r concentrated at r). We start with four examples of random experiments (Ω is finite); the last one is more complex, it generalizes the previous three, and it leads to a general model. A fuzzy random variable is a map ϕ of M + 1 (Ω) into M + 1 (Ξ), where M + 1 (Ξ) is the set of all probability measures on another measurable space (Ξ, B(Ξ)), satisfying certain measurability condition. We show that for discrete spaces the measurability condition holds true. We continue in our effort to develop a suitable theory of ID-posets, ID-random variables, and ID-observables. Fuzzy random variables and Markov kernels become special cases. Introduction Information about quantum structures can be found, e.g., in DVUREČENSKIJ and PULMANNOVÁ [4], FOULIS [5], and the references therein. In this section we present some basic information about D-posets and effect algebras related to fuzzy random variables. The purpose is twofold: to make the paper more selfcontained and to point out some ideas from GUDDER [13], BUGAJSKI [1, 2], FRIČ [11], PAPČO [15], leading to generalized fuzzy random variables discussed in the last section of the present paper. D-posets have been introduced in KÔPKA and CHOVANEC [14]. Recall that a D-poset is a partially ordered set A possessing the top element 1, the bottom element 0, and carrying two additional algebric structures: a partial operation called difference (a b is defined iff b a) and a complementation sending a A to a = 1 a. The following axioms are assumed: (D1) a 0 = a for all a A; (D2) c b a implies a b a c and (a c) (a b) = b c. Canonical examples are the unit interval I = [0, 1], the D-posets of fuzzy sets I X carrying the pointwise order, difference, and complementation (g f whenever 1991 Mathematics Subject Classification. Primary 47A15; Secondary 46A32, 47D20. Key words and phrases. Random variable, fuzzy (operational) random variable, D-poset, effect algebra, measurable space, measurable map, probability, state, ID-poset, sober ID-poset, closed ID-poset, random walk, conditional probability, Markov kernel, ID-measurable space, IDmeasurable map, ID-Markov kernel, generalized event, generalized observable, generalized fuzzy random variable. Supported by VEGA Grant 1/9056/24. 1

2 2 MARTIN PAPČO g(x) f(x) for all x X, f g is defined iff g f and then (f g)(x) = f(x) g(x) for all x X, (1 f)(x) = 1 f(x) for all x X), and suitable subsets X I X (containing the constant functions 1 X, 0 X, and closed with respect to the difference and the complementation). Our motivation comes from the so-called evaluation of fields of sets. Let (Ω, A) be a field of subsets of Ω and let P (A) be the set of all probabilities on A. The evaluation is a map ev : A I P (A) sending A A to ev(a) = {p(a); p P (A)} I P (A). Observe that if p P (A) is {0, 1}-valued, then p is a boolean homomorphism into the two-element boolean algebra {0, 1}. Denote P 0 (A) P (A) the {0, 1}-valued probabilities. The restricted evaluation ev 0 : A I P0(A) maps each A A to a {0, 1}-valued function on P 0 (A) and ev 0 (A) is in fact a field of subsets (represented by characteristic functions) of P 0 (A) isomorphic to A. For p P (A) \ P 0 (A) we do not get a boolean homomorphism and ev(a) I P (A) does not yield a field of sets but only a D-poset of fuzzy functions. In PAPČO [15] a theory of ID-posets has been developed. ID-posets are D-posets of fuzzy functions carrying the pointwise convergence of sequences. The corresponding category ID has sequentially continuous D-homomorphisms as morphisms. If X I X is an ID-poset then (X, X ) is called an ID-measurable space. If (X, X ), (Y, Y) are ID-measurable spaces and f is a map of X into Y such that the composition v f of f and each v Y belongs to X, then f is said to be measurable; this generalizes the classical measurability of maps for fields of sets. Each measurable map f defines (via v f) a D-homomorphism f of Y into X and f is sequentially continuous, i.e. a morphism. Morphisms of X into I are called states. As proved in PAPČO [15], the usual categorical constructions concerning classical fields sets can be generalized to ID-posets. Namely, there is a duality between ID-posets and ID-measurable spaces and each ID-poset can be embedded into a σ-complete ID-poset having the same states. Further (cf. FRIČ [10]), the duality and the σ-completions commute. The effect algebras (cf. FOULIS and BENNET [6]) and D-posets are known to be isomorphic. An effect algebra is defined via a dual partial operation from which a partial order and can be derived. The reader is referred to DVUREČENSKIJ and PULMANNOVÁ [4] and FOULIS [5] for more details. As pointed out in FRIČ [10] categorical constructions for ID-posets can easily be translated to the corresponding effect algebras of fuzzy functions. Let (Ω, A) be a (classical) measurable space (i.e. A is a σ-algebra of subsets of Ω). Denote E(Ω) the set of all measurable functions into I. Then E(Ω) carrying the partial addition (the sum f g is defined iff f(x) + g(x) 1 for all x Ω and then f g = f + g) is a canonical example of an effect algebra. Let (Ω, A), (Ξ, B) be measurable spaces. We identify each measurable set with its characteristic function so that A E(Ω) and B E(Ξ). Then (c.f. FRIČ [7]) each classical measurable map f : Ω Ξ induces a sequentially continuous boolean homomorphism of B into A. Suitable effect algebra homomorphisms of B into E(Ω) are called fuzzy observables in GUDDER [12] and play a crucial role in the fuzzy probability theory developed by S. Gudder, S. Bugajski and others (cf. GUDDER [13], BUGAJSKI, HELLWIG and STULPE [3], BUGAJSKI [1, 2]). 1. Examples In this section we present four examples of random experiments in which the notion of a classical random variable does not capture their nature. Recall that in

3 ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS 3 the classical case we start with a probability space (Ω, B(Ω), P ) and a measurable function f of Ω into the real line R. The preimage f maps Borel measurable subset B(R) of R into B(Ω) and the composition of f and P yields the probability distribution D f (P ) = P f on B(R). We can consider each point ω Ω as a degenerated point measure on B(Ω), hence f induces a map D f of the set M 1 + (Ω) of all probability measures on B(Ω) into the set M 1 + (R) of all probability measures on B(R). S. Gudder, S. Bugajski, and others developed a theory of fuzzy random variables the idea is that for some random experiments it is natural to map a point ω Ω to a nondegenerated probability measure on B(R). A fuzzy random variable is a map ϕ of M 1 + (Ω) into M 1 + (Ξ), where M 1 + (Ξ) is the set of all probability measures on another measurable space (Ξ, B(Ξ)) and ϕ satisfies a generalized measurability condition. Identifying ω and δ ω, the measurability condition means that, for each B B(Ξ), ϕ(δ ω (B)) is a measurable map of Ω into [0, 1] and, for each p M 1 + (Ω) and each B B(Ξ), we have (ϕ(p))(b) = (ϕ(δ ω ))(B)dp. All four examples deal with discrete probability spaces; in the next section we show that for discrete spaces each map of M 1 + (Ω) into M 1 + (Ξ) the measurability condition holds true. Example 1.4 is more complex, it generalizes the previous three, and it leads to a general model outlined by R. Frič at the QS 2004 conference in Denver. We start with an example of a natural fuzzy random situation in which random input is, due to some random fluctuations, distorted and hence from the given output only a fuzzy conclusion about input can be drawn. Example 1.1. Consider ten objects O 0,..., O 9, randomly drawn (according to some probability distribution). The output is announced on a one-digit display as a corresponding digit i {0,..., 9}. The display consists of usual seven elements, four vertical: left upper, right upper, left lower, right lower, and three horizontal (positioned between the left and the right horizontal ones): top, middle, bottom. Each of them can be on or off. For example, number 8 is announced if all seven elements are on, number 0 is announced if the middle horizontal element is off and all other elements are on ; other numbers are announced via their usual coding. Each element, after being activated, can remain off (independently on the other ones) with a given probability. This results in a distorted outcome. It can be any of 2 7 combinations; its probability can be calculated as the product of the probabilities that particular segment (when activated) remains off or becomes on. Denote Ω the set of all 2 7 possible distorted outcomes. In the classical model (no distortions) we would have 10 standard outcomes and a classical random variable sending the nondistorted digit to the number i corresponding to the object O i. In the fuzzy model (distorsions) we need a fuzzy random variable mapping each distorted outcome ω Ω to some probability distribution P ω indicating the probability that O i has been drawn (input O i ) provided ω is on the display (output Ω). We can proceed as follows. Knowing the probabilities of (independent) failures of each of the seven segments of the display, for each ω Ω and each O i, i {0,..., 9}, we can calculate the conditional probability P ({ω} {O i }) of the output ω Ω given that O i has been drawn. This enables us to construct an auxiliary probability space Ω {O i ; i = 0, 1,..., 9} and probability P on it. Indeed, P ({ω} {O i }). P ({O i }) = P ({ω} {O i }) is the probability of the simultaneous occurence of ω and O i. The Bayes rule does the rest. This way we can calculate

4 4 MARTIN PAPČO P ({O i } {ω}) for all ω Ω and all O i ; i = 0, 1,..., 9. Finally, this determines a mapping of Ω to the set of all probability measures on {O i ; i = 0, 1,..., 9}. In the next section we shall show that this is really a fuzzy random variable in the Bugajski Gudder sense ([1], [2], [13]). Example 1.2. The previous example can be generalized in several directions. First, instead of the one-digit display represented by seven elements, each admitting two states (i.e. 2 7 outputs), we can consider a more general set of outputs. Second, we can have a different number of objects, say n, drawn at random (i.e. n inputs); naturally, n is at most the number of outputs. Assume that the display now consists of k elements and each element can be exactly in one of m states i {1,..., m}. Hence we have altogether m k outputs of the form of k-tuples (a(1),..., a(k)), a(j) {1,..., m}. We assign each object o i, i {1,..., n}, its code c i represented by a k-tuple of states (c i (1),..., c i (k)), where c i (j) {1,..., m}, meaning that the j-th element, j {1,..., k}, is in the state c i (j). Further, we have a probability distribution {p i ; i {1,..., n}}, where p i is the probability that the object o i has been drawn. If o i has been drawn, then we send to each segment of the display a signal to put it into the corresponding state assigned by the code c i, i.e. the j-th segment should be in state c i (j). Due to some random noise, instead of c i, some distorted outcome can occur. One of the possible ways how the input can be distorted is the following. For each i {1,..., n} and for each c i (j), j {1,..., k}, we have a probability distribution p ijl ; l {1,..., m}, where p ijl is the probability that if o i has been drawn and the j-th coordinate of the code c i is c i (j), then the j-th segment is in the state l (instead of c i (j)); the distribution depends only on i and j. Now, our model can be viewed as a random walk. Indeed, at the starting point s we proceed (random draw) according to the probability distribution p i ; i {1,..., n} to some object o i, from o i we proceed to (o i, a(1)), a(1) {1,..., m}, with the probability p i1a(1), from (o i, a(1)) we proceed to (o i, a(1), a(2)), a(2) {1,..., m} with probability p i2a(2), and so on, until we reach (o i, a(1),..., a(k 1), a(k)), where a(1),..., a(k 1) are already fixed and a(k) {1,..., m} occurs with probability p ika(k) ; since the steps are stochastically independent, the probability of the given path (from s to o i ) and, step by step through a(j), j {1,..., k 1}, to the outcome (a(1),..., a(k)) is p i. p i1a(1)..... p ika(k). This yields an auxiliary (discrete) probability space points (the elementary events) of which are all possible paths of our random walk. Since we know the probability of each path, we can calculate the conditional probability that the object o i has been drawn (input) given that we see the distorted version (a(1),..., a(k)) of the code c i, i {1,..., n} on the display (outcome). So, we have two classical probability spaces: the space of objects and the space of outputs on the display. It is natural to consider a fuzzy random variable, mapping each output not to a particular input (object o i ) but to the conditional probability distribution on objects. Since the space of objects is discrete (events are subsets of {o 1,..., o n }), it is enough to know the probabilities of singletons {o i }, i {1,..., n}. As we shall see, such mapping always satisfies the corresponding measurability condition, hence it gives rise to a fuzzy random variable in the Gudder-Bugajski sense (mapping probabilities to probabilities). Example 1.3. In FRIČ [9, Example 3] a simple situation in which only two objects o 1, o 2 are drawn and the display consists of only two elements s 1, s 2 has been described and the corresponding conditional probabilities (probability of o i given

5 ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS 5 the output) have been calculated. There are four possible outputs: (0, 0), (0, 1), (1, 0), (1, 1) and if the second element s 2 is on, then we know for sure that o 2 has been drawn. This is due to the fact that if o 1 has been drawn, then we try to activate (with positive probability success) only the first element s 1. Hence, if s 2 is on, then o 1 is out of question. If we view the situation as a random walk, then there are two types of path. If o 1 has been drawn, then we stop after the next step and there are only two possible paths. If o 2 has been drawn, then we stop after two additional steps (we procced from o 2 to s 1 and then to s 2 ). Our auxiliary probability space of paths now has six elementary events (points). Our next example describes a random walk with no returns in a finite set {s} M starting in a fixed points s. It can be straightforwardly generalized to a countable set. Example 1.4. Let M be a finite set (containing at least four points) and let s be a point, s / M. Let k be a natural number (i.e. a positive integer) and let M be the union of mutually disjoint nonempty sets M i, i {1,..., k + 1}. For each i {1,..., k + 1}, let M i be the union of disjoint sets T i and F i. We assume that T i for all i k and T k+1 =. The random walk starts at the point s, proceeds to a point of M 1 (step 1) and then, inductively, from a point of T i to a point of M i+1 (step i). Points of T i are transitional, points of F i, i {1,..., k + 1} are final the walk stops whenever we reach a point of F i (after i steps). We consider a transitional probability distribution {P 1 (s, y); y M 1 }, where P 1 (s, y) is the probability that our random walk proceeds from s to y M 1. For each point x T i, we consider a transitional probability distribution {P i (x, y); y M i+1 }, where P i (x, y) is the probability that our random walk proceeds from x T i to a point of M i+1 (independently of the path from s to x). We can visualize {s} M as a subset of the Euclidean plane E 2. We identify s and [0, 0], T i = {[i, j] E 2 ; 0 < j n i }, F i = {[i, j] E 2 ; n i < j n i + l i }, i {1,..., k}, M k+1 = F k+1 = {[k + 1, j] E 2 ; 0 < j l k+1 }, where n i is the cardinality of T i, l i is the cardinality of F i (possibly l i = 0 for some i {1,..., k}); our walk randomly proceeds from s = [0, 0] to the right to a point a = [1, j] T 1 F1 with probability P 1 (s, j). If [1, j] F 1, our random walk stops; otherwise we proceed to [2, m] T 2 F2 with probability P 2 ([1, j], [2, m]). If [2, m] F 2, our random walk stops; otherwise we proceed in a similar way until we stop in some point of F i, i {1,..., k + 1}. Each path (trajectory) starts at the point s = [0, 0], ends at some point [i, j i ] F i and it will be represented by an ordered i-tuple (j 1, j 2,..., j i ), 0 < i k+1, where j i indicates the position after step i. E.g., if the walk ends in the step 1, then its path is represented by (j 1 ), n 1 < j 1 n 1 + l 1, if the walk ends in the step k + 1, then its path is represented by (j 1,..., j k, j k+1 ), where 0 < j i n i for i {1,..., k} and 0 < j k+1 l k+1. Denote Ω the set of all possible paths. It is easy to see that for ω = (j 1,..., j i ) the probability P of {ω} is equal to the product of the corresponding transitional probabilities: P 1 ([0, 0], [1, j 1 ]). P 2 ([1, j 1 ], [2, j 2 ])..... P k+1 ([k, j k ], [k+ 1, j k+1 ]), and (Ω, exp Ω, P ) is the corresponding discrete probability space. Denote H n = {(j 1,..., j i ) Ω; j 1 = n}, n {1,..., n 1,..., n 1 + l 1 }, i.e., H n is the set of all paths such that in the first step we reach [1, n] M 1. Clearly, H n Hm = for n m and the union of all H n is the set Ω, hence {H n ; n {1,..., n 1 + l 1 }} is a partition of Ω. Denote F = k+1 i=1 F i the set of all final points. For f F, denote E f = {(j 1,..., j i ) Ω; f = [i, j i ]}, i.e., E f is the

6 6 MARTIN PAPČO set of all paths stopping at f. Since F i Fj = for i j, {E f, f F } is also a partition of Ω. Observe that, in a sense, the previous examples are special cases of the construction of (Ω, exp Ω, P ). Indeed, Example 1.1 can be recovered as follows. Objects O 0,..., O 9 can be visualized as the points of M 1 (F 1 being empty), the set of outcomes at the one digit display (consisting of seven elements, enumerated by digits 1,..., 7, each of them can be on or off ) can be visualized as the points of F 8, the set M 2 (F 2 being empty) consists of two states on, off of the first segment, the set M 3 consists of four states (4 = 2 2 ) of the first and the second segments,..., the set M 7 consists 2 6 states of the first up to the sixth segments; our random walk starts with drawing (step 1) one of the objects O i, i {0,..., 9}, the second step is the activation of the first segment of the display according to O i (O i sends a signal yes or no to each of the seven segments of the display),..., the eighth step is the activation of the seventh segment of the display according to O i. Other examples (Example 1.1 and Example 1.3) can be reconstructed analogously. In all four examples we have the points of M 1 as the input and the points of F = k+1 i=1 F i as the output. The random walk represents sending signals ( on, off ) to the segments of the display. If we start at a given point of M 1, we can end up at different points of F. For each path, its probability is given by the product of corresponding transitional probabilities, i.e., probabilities of success and failure when activating the corresponding segment according to the input. The fuzzy random variable resulting from our model of the random walk sends each output to the conditional probability that the walk proceeds trough the respective points of the input set M 1 given that we have finished our walk at the given output; it can be calculated via the Bayes rule related to sets H n and E f. 2. Discrete fuzzy random variables Let (X, X ) be an ID-measurable space (we always assume that X is reduced, i.e., if x y, then u(x) u(y) for some u X ). Recall that (X, X ) is called sober if each state on X is fixed (i.e. for each sequentially continuous D-homomorphism s on X into I there exists a unique point x X such that s is the evaluation ev x at x defined by ev x (u) = u(x), u X I X ). Define X to be the set of all states on X and consider X as a subset of X. Each u X I X can be prolonged to u I X by putting u (s) = s(u), s X. Then (cf. PAPČO [15]) (X, X ) is a unique (up to an isomorphism) sober ID-measurable space such that X X, X X = X, and (X, X ) and (X, X ) have the same states; it is called the sobrification of (X, X ). Further, if f is a measurable map of (X, X ) into an IDmeasurable space (Y, Y), then there exists a unique measurable map f of (X, X ) into the sobrification (Y, Y ) of (Y, Y) such that f restricted to X is equal to f. The ID-posets X and X are isomorphic and u u yields an isomorphism. For each sequentially continuous D-homomorphism h of Y into X there exists a unique measurable map f : X Y such that h(u ) = u f. The correspondence between h and f yields a categorical duality between ID-posets and sober IDmeasurable spaces. If X is sequentially closed in I X, then X is said to be closed (or sequentially complete). There exists a unique closed ID-poset σ(x ) I X such that X σ(x ) and X and σ(x ) have the same states. Passing from X to σ(x ) yields an epireflector, also called the sequential completion of ID-posets into

7 ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS 7 the category of closed ID-posets. In particular, each sequentially continuous D- homomorphism h : Y X can be uniquely extended to a sequentially continuous D-homomorphism h σ : σ(y) σ(x ). Example 2.1. Let (Ξ, B(Ξ)) be a classical measurable space and let M 1 + (Ξ) be the set of all probability measures (nonnegative and normed) on B(Ξ). Consider Ξ as a subset of M 1 + (Ξ) consisting of point measures. To each B B(Ξ) there corresponds a unique fuzzy subset u B I M + 1 (Ξ) defined by u B (p) = p(b), p M 1 + (Ξ). It is known (cf. Section 3 in PAPČO [15]) that if X = M 1 + (Ξ) and X = {u b, B B(Ξ)}, then (X, X ) is a sober object of ID and each p M 1 + (Ξ) is a morphism of ID. Let (Ξ, B(Ξ)) and (Ω, B(Ω)) be classical measurable spaces and let f be a measurable map of Ω into Ξ. Denote (X, X ), (Y, Y) the corresponding ID-measurable spaces. Then f can be uniquely extended to a measurable map f of Y = M 1 + (Ω) into X = M 1 + (Ξ) defined by ( f(p))(b) = p(f (B)), p M 1 + (Ω), B B(Ξ)). Here (X, X ) is a sobrification of (Ξ, B(Ξ)), (Y, Y) is a sobrification of (Ω, B(Ω)), f is the unique prolongation of f and the measurability of f means that for each u X the composition u f belongs to Y. Via duality, to f there corresponds a unique sequentially continuous D-homomorphism h of X into Y. Observe that h sends u B X, B B(Ξ), to v f (B) Y. If we consider a genuine fuzzy random variable sending some point measure δ ω, ω Ω, into a nontrivial measure in M 1 + (Ξ), then the dual sequentially continuous D-homomorphism of X into Y (the corresponding observable) will have a distinctive fuzzy nature (cf. BUGAJSKI [1, 2], GUDDER [12], FRIČ [10, 11]). An efficient way how to describe a fuzzy random variable is via a Markov kernel. Let (Ω, B(Ω)) and (Ξ, B(Ξ)) be classical measurable spaces, let E(Ω) and E(Ξ) be the effect algebras of all measurable functions to I = [0, 1]. Recall (cf. GUDDER [13]) that a map K : Ω B(Ξ) I is a Markov kernel if K(, B) is measurable for each B B(Ξ) and K(ω, ) is a probability measure for each ω Ω. Observe that K(, B) yields a sequentially continuous D-homomorphism of B(Ξ) into E(Ω) sending B B(Ξ) to K(, B) E(Ω) and a map of M 1 + (Ω) into M 1 + (Ξ) sending p M 1 + (Ω) into K(p) M 1 + (Ξ) defined by (K(p))(B) = K(ω, B)dp, B B(Ξ). This is how a statistical map, or an operational random variable, is defined (cf. BUGAJSKI [1, 2]). Markov kernels are easy to handle in case (Ω, B(Ω)) and (Ξ, B(Ξ)) are discrete spaces (i.e. Ω and Ξ are at most countable and B(Ω) and B(Ξ) are the fields of all subsets of Ω and Ξ, respectively). Let (Ω, B(Ω)) and (Ξ, B(Ξ)) be discrete. Let K 0 be a map of Ω Ξ into [0, 1] such that ξ Ξ K 0(ω, ξ) = 1 for each ω Ω. Define K : Ω B(Ξ) as follows: K(ω, B) = ξ B K 0 (ω, ξ). Lemma 2.2. K is a Markov kernel. Proof. Since (Ω, B(Ω)) is a discrete measurable space, each function of Ω into [0, 1] is measurable. Hence for each B B(Ξ) the function K(ω, B) is measurable. It follows directly from the definition of K that for each ω Ω the function K(ω, ) is a probability measure on B(Ξ). Let (Ω, B(Ω)) and (Ξ, B(Ξ)) be measurable spaces and let ϕ be a map of M + 1 (Ω) into M + 1 (Ξ) such that, for each B B(Ξ), the assignment ω (ϕ(δ ω))(b),

8 8 MARTIN PAPČO ω Ω, δ ω M 1 + (Ω) is the degenerated probability measure concentrated at ω, yields a measurable function of Ω into [0, 1]. If (2.0.1) (ϕ(p))(b) = (ϕ(δ ω ))(B)dp for all p M 1 + (Ω) and all B B(Ξ), then ϕ is said to be a statistical map (cf. BUGAJSKI [1], GUDDER [13]). Let E(Ω) and E(Ξ) be the effect algebras of measurable functions into I = [0, 1]. Put Y = M 1 + (Ω), Y = ev(e(ω)) IY, X = M 1 + (Ξ), X = ev(e(ξ)) IX. Theorem 2.4 in FRIČ [11] states that ϕ : M 1 + (Ω) M 1 + (Ξ) is a statistical map iff it is measurable as a map of (Y, Y) into (X, X ), i.e., for each u X the composition u ϕ belongs to Y. From this point of view, it is natural to call elements of X and Y fuzzy events and ϕ a fuzzy random variable. Now, let K : Ω B(Ξ) I be a Markov kernel. Then K sends B B(Ξ) to a measurable function K(, B) E(Ω). Clearly, its integral K(ω, B)dp with respect to p M 1 + (Ω) sends p into a measure K(p) defined by (K(p))(B) = K(ω, B)dp. Hence K yields a statistical map of M 1 + (Ω) into M 1 + (Ξ). Since (according to Lemma 2.2) for discrete measurable spaces (Ω, B(Ω)), (Ξ, B(Ξ)) each map K 0 : Ω Ξ [0, 1] yields a Markov kernel K : Ω B(Ξ) [0, 1], each map of Ω Ξ into [0, 1] determines (a unique) fuzzy random variable mapping M 1 + (Ω) into M 1 + (Ξ). Consequently, the maps in Example 1.1, Example 1.2, Example 1.3, Example 1.4 determine fuzzy random variables. 3. Generalizations In this section Markov kernels and fuzzy random variables are generalized to ID-posets. Definition 3.1. Let X I X and Y = σ(y) I Y be ID-posets. Let K be a map of Y X into I such that (i) for each u X, K(, u) Y; (ii) for each y Y, K(y, ) is sequentially continuous D-homomorphism of X into I (i. e. a state). Then K is said to be an ID-Markov kernel. Theorem 3.2. Let K : Y X I be an ID-Markov kernel. (i) There exists a unique sequentially continuous ID-homomorphism h : σ(x ) Y such that for each u X we have h(u) = K(u, ). (ii) There exists a unique sequentially continuous ID-homomorphism h : σ(x ) Y such that for each u X we have h (u ) = K(u, ). Proof. (i) Define a map h 0 : X Y as follows: for u X put (h 0 (u))(y) = K(y, u). It is easy to see that h 0 is a sequentially continuous D-homomorphism. Since σ is an epireflector sending X into its sequential completion σ(x ), there is a unique extension of h 0 to a sequentially continuous D-homomorphism h of σ(x ) into Y = σ(y). (ii) The sobrification is an epireflector and hence there is a unique sequentially continuous D-homomorphism h : σ(x ) Y = σ(y) such that h (u ) = (h(u)) = K(, u). It follows from Corollary 3.3 in FRIČ [10] that σ(x ) = σ(x ) and the assertion follows.

9 ON FUZZY RANDOM VARIABLES: EXAMPLES AND GENERALIZATIONS 9 In FRIČ [11] it was proposed to extend basic probability notions (events, probabilities, random variables, observables) to the realm of ID-posets: if (X, X ) is an ID-measurable space, then the elements of X are called generalized events, each ID-morphism of X into I is said to be a state, each ID-morphism of X into Y I Y is a generalized observable, and each measurable map of (Y, Y) into (X, X ) is said to be a generalized fuzzy random variable. Our results about ID-Markov kernels together with the duality between ID-posets and sober ID-measurable maps provide technical tools for further study of probability on ID-structures. References [1] BUGAJSKI, S.: Statistical maps I. Basic properties, Math. Slovaca 51 (2001), [2] BUGAJSKI, S.: Statistical maps II. Operational random variables, Math. Slovaca 51 (2001), [3] BUGAJSKI, S. HELLWIG, K.-E. STULPE, W.: On fuzzy random variables and statistical maps, Rep. Mat. Phys. 51 (1998), [4] DVUREČENSKIJ, A. PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Academic Publ./Ister Science, Dordrecht/Bratislava, [5] FOULIS, D. J.: Algebraic measure theory, Atti Sem. Fis. Univ. Modena 48 (2000), [6] FOULIS, D. J. BENNETT, M. K.: Effect algebras and unsharp quantum logics, Found. Phys 24 (1994), [7] FRIČ, R.: A Stone type duality and its applications to probability, Topology Proc. 22 (1999), [8] FRIČ, R.: Convergence and duality, Appl. Categorical Structures. 10 (2002), [9] FRIČ, R.: Nonelementary notes on elementary events, In: Matematyka IX, Prace Naukowe, Wyzsa Szkola Pedagogyczna w Czestochowie, , [10] FRIČ, R.: Duality for generalized events, Math. Slovaca 54 (2004), [11] FRIČ, R.: Remarks on statistical maps and fuzzy (operational) random variables, Tatra Mountains Math. Publ... (2004),..... [12] GUDDER, S.: Fuzzy probability theory, Demonstratio Math. 31 (1998), [13] GUDDER, S.: Combinations of observables, Internat. J. Theoret. 31 (2002), [14] KÔPKA, F. CHOVANEC, F.: D-posets, Math. Slovaca 44 (1994), [15] PAPČO, M.: On measurable spaces and measurable maps, Tatra Mountains Math. Publ... (2004),..... Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, Košice, Slovak Republic address: martin.papco@murk.sk