Fuzzy and Non-deterministic Automata Ji Mo ko January 29, 1998 Abstract An existence of an isomorphism between a category of fuzzy automata and a cate

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1 University of Ostrava Institute for Research and Applications of Fuzzy Modeling Fuzzy and Non-deterministic Automata Ji Mo ko Research report No. 8 November 6, 1997 Submitted/to appear: { Supported by: Grant 201/96/0985 of GA R University of Ostrava Institute for Research and Applications of Fuzzy Modeling Br fova 7, Ostrava 1, Czech Republic tel.: fax: Mockor@osu.cz

2 Fuzzy and Non-deterministic Automata Ji Mo ko January 29, 1998 Abstract An existence of an isomorphism between a category of fuzzy automata and a category of chains of non-deterministic automata is proved and some relationships between output fuzzy sets of these systems are investigated. 1 Introduction A notion of a fuzzy automaton was introduced by several authors, see e.g. [1], [2], [3], [4]. Analogously as in a theory of classical automata there are several denitions of a fuzzy automaton and, hence, there are several categories of these fuzzy automata with possible dierent properties. Analogously a notion of non-deterministic automaton was introduced which seems to be similar (in some aspect) to the notion of a fuzzy automaton, although a denition of this notion is very far from a fuzzy set theory. In this paper we will be dealing with two categories - a category of fuzzy automata with fuzzy morphisms and with a category of chains of non-deterministic automata with some specic morphims. The principal result will be a theorem which states that these two categories are isomorphic. Moreover, we will be dealing with output fuzzy sets of these two kinds of automata and we prove that there are very specic relationships between these fuzzy sets. Hence, a conclusion of this paper is a fact that instead of a fuzzy automaton we can deal equivalently with a chain of non-deterministic automata. This idea is then an analogy of a well known relation between fuzzy set A and a chain of its -level sets fa : 2 (0; 1]g. 2 Types of fuzzy automata To unify various existing denitions of fuzzy automata, the following elementary form of a fuzzy automaton can be identied as a common generic form in a fuzzy automaton theory. Denition 2.1 Let (M; ) be a semigroup (called sequences of input alpha- bet). A fuzzy automaton over M is then a system A = (S; F ), where S is a set (called a set of states) and F is a fuzzy set in S M S (called a fuzzy transition function) which has to satisfy the following conditions 1; 1. F (s; 1 M ; t) = if s = t 0; if s 6= t 2. F (s; m n; t) = W z2s F (s; m; z) ^ F (z; n; t) for all s; t 2 S. This denition is correct since for m 2 M we have (8s; t 2 S)F (s; 1 M m; t) = u2s F (s; 1M ; u) ^ F (u; m; t) = = F (s; 1 M ; s) ^ F (s; m; t) = F (s; m; t) Partially supported by the grant 201/96/0985 of GA R 2

3 Fuzzy and non-deterministic automata 3 and analogously for F (s; m 1 M ; t). We will now recall a notion of a fuzzy commutativity of a diagram which is a fuzzy analogy of a classical commutativity in a category of fuzzy sets. This denition was rstly introduced in [4]. Denition 2.2 Let A; B be sets and let f A; g B. Then a diagram A? fy u! B? y g (0; 1] (0; 1] is called to fuzzy commute, if the following condition holds. 8b 2 u(a) g(b) = a2a u(a)=b Sometimes it is useful to use instead of this condition the following system of two conditions which are equivalent to the original one: 1. (8a 2 A)f(a) gu(a) 2. 8b 2 u(a) g(b) W u(a)=b f(a) Then the following denition recalls a principal category of fuzzy automata we will be dealing with which was introduced in [4]. Let (M; ) be a semigroup. By F M we denote a category of fuzzy automata the objects of which are fuzzy automata over M dened in Denition 2.1 and morphisms between fuzzy automata A = (S; F ); B = (T; G) are maps : S! T such that the following diagram fuzzy commutes. f(a) 1M S M S! T M T?? Fy yg (0; 1] (0; 1] (1) The following simple lemma then states that a "global" notion of a morphism in a category F M can be substituted by a system of "local" conditions. Lemma 2.3 The diagram (1) fuzzy commutes if and only if for any m 2 M the following diagram fuzzy commutes: S S! T T?? F?y?y m G m (0; 1] (0; 1] where F m (s; s 0 ) = F (s; m; s 0 ) and G m (t; t 0 ) = G(t; m; t 0 ) for all s; s 0 2 S; t; t 0 2 T. The proof of this lemma is trivial and will be omitted. Sometimes fuzzy automata were dened over a set of input signals only instead over a semigroup (see e.g. [4], [5]). In this case a fuzzy automaton is only a system A = (S; F ), where F is a fuzzy set in S S (without any additional conditions required). Even in this case we can extend this simplied version of a fuzzy automaton onto the previous one, but dened over a free semigroup (M; ) over. In fact, a function F can be extended onto a fuzzy set F in S M S such that for m = 1 k 2 M we put (8s; t 2 S)F (s; m; t) = (F 1 F k )(s; t); where for any 2, a fuzzy relation F S S is dened such that F (s; s 0 ) = F (s; ; s 0 ) for any s; s 0 2 S and \" is a composition of fuzzy relations in S S.

4 4 Ji Mo ko In this case (S; F ) is a fuzzy automaton over M in an above sense. In fact, let m = 1 g ; n = 1 h 2 M. Then for any s; t 2 S we have F (s; m n; t) = (F 1 F g F 1 F h )(s; t) = = r2s (F1 F g )(s; r) ^ (F 1 F h )(r; t) = = r2s F (s; m; r) ^ F (r; n; t) : Recall that a classical automaton over a semigroup (M; ) is a system B = (S; d) such that d : M S! S is a map (called a transition function) satisfying the conditions 1. (8s 2 S)d(1 M ; s) = s 2. (8m; n 2 M; s 2 S)d(m n; s) = d(n; d(m; s)) Then it is clear that any classical automaton (S; d) over M is a fuzzy automaton (S; F ) as well, where we set 1; if t = d(m; s) F (s; m; t) = 0; otherwise Further, we will introduce a notion of a general non-deterministic automaton over a semigroup (M; ). Recall that by 2 S we understand the set of all subsets of S. Denition 2.4 A system (S; d) is called a non-deterministic automaton over a semigroup (M; ) (in abbreviation : ND-automaton), if S is a set (of states) and d : S M! 2 S is a (non-deterministic transition) function such that 1. (8s 2 S)d(s; 1 M ) = fsg, 2. (8m; n 2 M; s 2 S)d(s; m n) = S t2d(s;m) d(t; n). As in a case of fuzzy automata, even ND-automata are sometimes dened only for an abstract set instead of a semigroup. Hence, instead of a ND-automaton over a semigroup M we can consider a ND-automaton (S; d) over a set, where d : S! 2 S is a map. Then the following extension principle holds for these ND-automata. Proposition 2.5 Let (S; d) be a ND-automaton over a set. Then a map d can be extended onto a map d : S M! 2 S, where (M; ) is a free semigroup over in such a way that (S; d) is a ND-automaton over a semigroup (M; ) dened in 2.4. Proof. An extension d will be dened by induction principle applied on the length jmj of elements m 2 M in the following way. 1. Let m 2 M be such that jmj = 1 (hence, m 2 ). Then we set d(s; m) = d(s; m) for any s 2 S. 2. Let d be dened for any m 2 M such that jmj a 2 N. Let m 2 M; 2 be such that jmj a. Then we set (8s 2 S) d(s; [ m ) = d(t; ): 3. (8s 2 S) d(s; 1 M ) = fsg. We show that then d satises the condition t2 d(s;m) (8s 2 S)(8m; n 2 M) d(s; m n) = [ t2 d(s;m) d(t; n):

5 Fuzzy and non-deterministic automata 5 The proof will be done by induction principle applied on the length a = jm nj. For a = 1 we have either m n = m 1 M or m n = 1 M n, where m; n 2. Then in the rst case we have [ d(t; [ 1 M ) = ftg = d(s; m) = d(s; m 1 M ) t2 d(s;m) and in the other case we have [ t2 d(s;1m ) t2d(s;m) d(t; m) = [ t=s d(t; m) = d(s; m) = d(s; 1 M m): Let us assume that this proposition holds for any m; n 2 M such that jm nj a. Let m; n 2 M be such that jm nj = a + 1, i.e. there exist 2 ; n 0 2 M such that m n = m n 0. Then according the induction assumption we have d(s; m n) = d(s; (m n 0 ) ) = = [ t2 d(s;mn 0 ) d(t; ) = [ t2[ v2 d(s;m) d(v;n 0 ) d(t; ) (2) Let x 2 d(s; m n). According to (2) there exist v 2 d(s; m) and t 2 d(v; n 0 ) such that x 2 d(t; ). Then according to the induction assumption we have [ x 2 d(t; ) = d(v; n 0 ) = d(v; [ n) d(r; n): t2 d(v;n 0 ) r2 d(s;m) Conversely, let x 2 S t2d(s;m) d(t; n). Then there exists t 2 d(s; m) such that according to the induction assumption we have x 2 d(t; n) = d(t; [ n 0 ) = d(r; ): r2 d(t;n 0 ) Then according to (2) we have x 2 d(s; m n). 2 We now introduce a category N D M of ND-automaton the object of which are ND-automata over a semigroup M and : (S; d)! (T; f) is morphism in N D M if : S! T is a map such that (8s 2 S)(8m 2 [ M) d(s 0 ; m) = f (s); m \ (S): s 0 2S (s 0 )=(s) Proposition 2.6 There exist a functor A : N D M! F M. Proof. Let A = (S; d) 2 N D M. We set A (A) = (S; F ), where F S M S be dened such that 1; if t 2 d(s; m) (8s; t 2 S)(8m 2 M)F (s; m; t) = 0; if t 62 d(s; m) Then A (A) 2 F M. In fact, a condition (1) from a denition 2.1 is clearly satised. By a simple computation can be veried that also the condition (2) is satised. Now, let : A = (S; d)! B = (T; f) be a morphism in N D M. Then for A (A) = (S; F ); A (B) = (T; G) also is a morphism from (S; F ) into (T; G) in F M. To prove it, we have to verify that for any t; t 0 2 (S), G m (t; t 0 ) = F m (s; s 0 ): (3) (s;s 0 )2SS (s)=t;(s 0 )=t 0 Let G m (t; t 0 ) = 1. Then t 0 2 f(t; m) \ (S), t = (s), and there exists s 00 2 d(r; m), (r) = (s), such that (s 00 ) = t 0. Then F m (r; s 00 ) = 1, (r) = (s) = t,(s 00 ) = t 0 and it follows that (3) holds. For the case G m (t; t 0 ) = 0 the proof can be done analogously. 2 Using a generic form of a fuzzy automaton which was introduced in 2.1, we can dene a fuzzy automaton with initial and nal fuzzy states.

6 6 Ji Mo ko Denition 2.7 A fuzzy automaton over a semigroup M with initial and nal fuzzy states is a system (S; F; G; P ) such that 1. (S; F ) is a fuzzy automaton over M, 2. G and P are fuzzy sets in S called an initial and nal fuzzy states, respectively. By FM 0 we denote the category of fuzzy automata over a semigroup M with initial and nal fuzzy states (sometimes we call these systems shortly fuzzy automata as well) and with as a morphism between automaton (S 1 ; F 1 ; G 1 ; P 1 ) and automaton (S 2 ; F 2 ; G 2 ; P 2 ), if 1. is a morphism between (S 1 ; F 1 ) and (S 2 ; F 2 ) in a category F M, 2. the following diagramms fuzzy commute S 1! S 2 S 1! S 2 G 1??y G 2??y??y P 1??y P 2 (0; 1] (0; 1](0; 1] (0; 1] An analogical form of these automata we can introduce for ND-automata as well. Denition 2.8 A non-deterministic automaton over a semigroup M with initial and nal sets of states is a system (S; d; P; G) such that 1. (S; d) is a ND-automaton over M, 2. P; G S. By N D 0 M we denote a category of these ND-automata with initial and nal sets of states where : (S 1 ; d 1 ; P 1 ; G 1 )! (S 2 ; d 2 ; P 2 ; G 2 ) is a morphism if 1. : (S 1 ; d 1 )! (S 2 ; d 2 ) is a morphism in N D M, 2. (P 1 ) = P 2 \ (S 1 ); (G 1 ) = G 2 \ (S 2 ). Recall that any fuzzy automaton A = (S; F; P; G) 2 F 0 M denes an output fuzzy set (A) M such that (8m 2 M)(A)(m) = P F m G: Analogously, any ND-automaton B = (S; d; P; G) 2 N D 0 M denes an output set (B) M such that m 2 (B) () (9s 2 P )d(s; m) \ G 6= ;: Proposition 2.9 There exist a functor B : N D 0 M! F 0 M such that if A 2 N D0 M, then (A) = (B (A)). The proof follows directly from a Proposition 2.6 and from denitions by simple computation and it will be omitted. 3 Fuzzy automata and nets of ND-automata Although it is not necessary for proofs of all propositions in this section, we will consider all automata in this section to be nite, i.e. their sets of states are nite. We begin this section with an existence of a functor which is a partial converse of a functor B from proposition 2.9. The following lemma is then a non-deterministic analogy of a [4]; Theorem 2.4.

7 Fuzzy and non-deterministic automata 7 Lemma 3.1 Let 2 (0; 1]. 1. There exists a functor C : F M! N D M. 2. There exists a functor D : F 0 M! N D0 M such that (8A 2 F 0 M )(8m 2 M)(A)(m) () m 2 (D (A)): Proof. (1) For A = (S; F ) 2 F M and > 0 we set C (A) = (S; d ), where d (s; m) = ft 2 S : F (s; m; t) g. Then C (A) 2 N D 0 M. In fact, we have (8s 2 S)d (s; 1 M ) = ft 2 S : F (s; 1 M ; t) > 0g = fsg: By using only a simple computation and a fact that S is nite we can prove that (8m; n 2 M)(8s 2 S)d (s; m n) = [ t2d (s;m) d (t; n): We show that C is a functor. Let be a morphism (in F M ) from A 1 = (S 1 ; F 1 ) into A 2 = (S 2 ; F 2 ) and let C (A i ) = (S i ; d i ). Then we have (8s; s 0 2 S 1 )(8m 2 M)F 2 (s); m; (s 0 ) = F 1 (t; m; t 0 ): (t)=(s) (t 0 )=(s 0 ) Using this equation and a fact that S 1 is nite, by simple computation we can prove that (8s 2 S 1 )(8m 2 M) S (s 0 )=(s) d1 (s 0 ; m) = d 2 ((s); m) \ (S 1 ). Hence, is a morphism in N D M. (2) For A = (S; F; P; G) 2 F 0 M and > 0 we set D (A) = (S; d ; P ; G ), where (S; d ) = C (S; F ) and P and G are -level sets of corresponding fuzzy sets. Then D (A) 2 N D 0 M as follows from a part (1). The required relationship between (A) and (D (A)) follows directly from the relation (8m 2 M)P F m G, (9s 2 P )(9t 2 G )t 2 G \ d (s; m): Finally, we show that D is a functor. Let be a morphism (in FM 0 ) from A1 = (S 1 ; F 1 ; P 1 ; G 1 ) into A 2 = (S 2 ; F 2 ; P 2 ; G 2 ) and let D (A i ) = (S i ; d i ; P i; Gi ). Then from a part (1) it follows that is a morphism from (S 1 ; d 1 ) to (S 2 ; d 2 ) and we have G 2 ((s)) = G 1 (t); P 2 ((s)) = P 1 (t): (t)=(s) (t)=(s) Using these equations and a fact that S 1 is nite, by simple computation we can prove that 1. (P 1 ) = P 2 \ (S 1 ), 2. (G 1 ) = G 2 \ (S 1 ). Hence, is a morphism in N D 0 M. 2 A functor D is only a partial inverse of a functor B since from D (A) we are not able to reconstruct completely an original fuzzy automaton A. On the other hand, it seem hopefull that we would be able to reconstruct A from a set fd (A) : 2 (0; 1]g as we are able to reconstruct a fuzzy set A from its -level sets A. In fact, the following simple lemma holds. Lemma 3.2 Let S be (in general nonnite) set and let A S. Let s 2 S and let = W s2a, where A = fx 2 S : A(x) g. Then s 2 A and A(s) =. The proof is straightforward and it will be omitted. Hence, we introduce another categories CN D M and CN D 0 M the objects of which will be some chains of ND-automata with some special morphisms between such chains.

8 8 Ji Mo ko Denition 3.3 Let CN D M be a category with objects all chains C = f(s; d ) : 2 (0; 1]g, where 1. (S; d ) 2 N D M for any 2 (0; 1], 2. (8; 2 (0; 1])(8s 2 S)(8m 2 M) =) d (s; m) d (s; m), 3. (8s; t 2 S)(8m 2 M) = W t2d (s;m) =) t 2 d (s; m): If C i = f(s i ; d i ) : 2 (0; 1]g 2 CN D M, i = 1; 2, then : C 1! C 2 is a morphism if : S 1! S 2 is a map such that it is a morphim from (S 1 ; d 1 ) into (S 2 ; d 2 ) in N D M for any. As we have mentioned above, the category CN D M could then serve as a reconstruction category for fuzzy automata. To prove this assertion we construct another two mutually inverse functors between categories F M and CN D M which then enable us to construct any fuzzy automaton from a sequence of ND-automata. Theorem 3.4 There exist functors J : F M! CN D M and K : CN D M! F M such that J and K mutually inverse isomorphisms. Hence, categories F M and CN D M are isomorphic. are Proof. Let A = (S; F ) 2 F M. We set J(A) = C (A) : 2 (0; 1] : If for A 2 F M and > 0 we have C (A) = (S; d ), then it is clear that d d for and the condition (2) from 3.3 holds. Moreover, since A = F (s; m; ) S is a fuzzy set such that A = d (s; m), then the condition (3) from 3.3 follows directly from a lemma 3.2. Hence, J(A) 2 CN D M. Since C is a functor, it is clear that if is a morphism in F M then is a morphism in CN D M as well. Hence, J is a functor. Conversely, for R = f(s; d ) : 2 (0; 1]g 2 CN D M we put K (R) = (S; F ), where (8s; t 2 S)(8m 2 M)F (s; m; t) = Then (S; F ) 2 F M. In fact, we have F (s; 1 M ; t) = = t2d (s;1m ) t2fsg t2d (s;m) : 1; if s = t = 0; if s 6= t Moreover, we prove that for any s; t 2 S; m 2 M, (a =) F (s; m n; t) = r2s F (s; m; r) ^ F (r; n; t) (= b): In fact, let be such that t 2 d (s; m n). According to 2.4(2), there exists r 2 d (s; m) such that t 2 d (r; n). Then F (s; m; r) ^ F (r; n; t) b and it follows that a b. Conversely, let r 2 S and let for example = F (s; m; r) F (r; n; t) =. Then according to 3.3(3), we have r 2 d (s; m) and t 2 d (r; n). Since, according to 3.3(2), we have t 2 d (r; n) d (r; n) and it follows that t 2 S r2d (s;m) d (r; n) = d (s; m n) according to 2.4(2). Hence, F (s; m n; t) = W t2d (s;mn) = F (s; m; r) ^ F (r; n; t) and it follows that a b. Therefore, (S; F ) 2 F M. Further, we show that K is a functor. In fact, let R i = f(s i ; d i ) : 2 (0; 1]g 2 CN D M, i = 1; 2 and let be a morphism R 1! R 2 in CN D M. Then for K (R i ) = (S i ; F i ), : (S 1 ; F 1 )! (S 2 ; F 2 ) is a morphism in F M, as well. In fact, according to the denition of a category F M we have to prove that for any m 2 M; s; t 2 S, F 2 ((s); m; (t)) = W (s 0 )=(s) F 1 (s 0 ; m; t 0 ) holds. Let us denote (t 0 )=(t) A = : (t) 2 d 2 (s); m ; B = : (9t 0 ; s 0 2 S 1 )(t 0 ) = (t); (s 0 ) = (s); t 0 2 d 1 (s 0 ; m) :

9 Fuzzy and non-deterministic automata 9 Let 2 A. Since is a morphism in CN D M, we have (t) 2 d 2 ((s); m) \ (S 1 ) = [ (s 0 )=(s) d 1 (s 0 ; m) and it follows that 2 B. Conversely, let 2 B, then for some s 0 ; t 0 2 S 1 such that (s 0 ) = (s); (t 0 ) = (t) we have (t) 2 (d 1 (s 0 ; m)) d 2 ((s); m) and it follows that 2 A. Therefore, the required equality holds. Hence, K is a functor. Finally, we show that J and K are mutually inverse. Let R = f(s; d ) : 2 (0; 1]g 2 CN D M and let K (R) = (S; F ), J((S; F )) = f(s; h ) : 2 (0; 1]g. Then h = d. In fact, let t 2 h (s; m). Then = F (s; m; t) = W t2d (s;m) and according to 3.3, we have t 2 d (s; m) d (s; m). Conversely, for t 2 d (s; m) we have F (s; m; t) = W t2d (s;m) and it follows that t 2 h (s; m). Hence J(K(R)) = R. Conversely, let A = (S; F ) 2 F M and let J(A) = f(s; d ) : 2 (0; 1]g, K (J(A)) = (S; G). Then G = F. In fact, let = G(s; m; t), then according to 3.3, t 2 d (s; m) and it follows that F (s; m; t). Conversely, let = F (s; m; t). Then t 2 d (s; m) and it follows that G(s; m; t). Hence, K (J(A)) = A. 2 A category CN D 0 M will be now dened as follows. Denition 3.5 By CN D 0 M we denote a category the object of which are chains f(s; d ; P ; G ) : 2 (0; 1]g such that 1. f(s; d ) : 2 (0; 1]g 2 CN D M, 2. (8 > 0)(S; d ; P ; G ) 2 N D 0 M, 3. (8; > 0)(8s 2 S) =) G G ; P P, 4. (8s 2 S) = W s2p =) s 2 P 5. (8s 2 S) = W s2g =) s 2 G. If C i = f(s i ; d i ; P i ; Gi ) : > 0g 2 CN D 0 M ; i = 1; 2, then : C1! C 2 is a morphism if 1. : f(s 1 ; d 1 ) : > 0g! f(s 2 ; d 2 ) : > 0g is a morphism in CN D M, 2. (8 > 0)(P 1 ) = P 2 \ (S 1); (G 1 ) = G 2 \ (S 1). Analogously as we did for ND-automata with initial and nal states, we can dene an output fuzzy set for automata from a category CN D 0 M. In fact, let R = fr = (S; d ; P ; G ) : 2 (0; 1]g 2 CN D 0 M. Then R denes an output fuzzy set (R) M such that (8m 2 M)(R)(m) = f : m 2 (R )g: Hence a "global" output fuzzy set (R) is dened by "local" output sets (R ), 0. Then the following theorem is an analogy of Theorem 3.4 and Proposition 2.9 for automata with with initial and nal states. Theorem 3.6 There exist functors U : F 0 M! CN D0 M and V : CN D 0 M! F 0 M such that 1. Functors U and V are mutually inverse isomorphisms. Hence, categories F 0 M and CN D0 M are isomorphic. 2. For any R 2 CN D 0 M we have (R) = (V(R)).

10 10 Ji Mo ko Proof. (1) A functor U is dened as follows. Let A = (S; F; P; G) 2 FM 0. Then we set U(A) = fd (A) : > 0g. Then according to the lemma 3.2 and lemma 3.1, U(A) 2 CN D 0 M. Let : A 1 = (S 1 ; F 1 ; P 1 ; G 1 )! A 2 = (S 2 ; F 2 ; P 2 ; G 2 ) be a morphism in FM 0. Then for any > 0 we have (P 1 ) = P 2 \ (S 1); (G 1 ) = G 2 \ (S 1) as follows directly from the fact that diagramms S 1! S 2 S 1??y? P 1 P 2y! S 2???yG yg 1 2 (0; 1] (0; 1](0; 1] (0; 1] fuzzy commute and from a fact that S 1 is a nite set. Hence, according to 3.1, : U(A 1 )! U(A 2 ) is a morphism in a category CN D 0 M and U is a functor. Further, let R = f(s; D ; P ; G ) : > 0g 2 CN D 0 M and let (S; F ) = K (f(s; d ) : > 0g). Let fuzzy sets P and G S be dened such that P (s) = W f : s 2 P g; G(s) = W f : s 2 G g for all s 2 S. Then we put V(R) = (S; F; P; G). From 3.5, it follows that V(R) 2 FM 0. Further, let R i = f(s i ; d i ; P i; Gi ) : > 0g; i = 1; 2, and let : R 1! R 2 be a morphism in CN D 0 M. Then : (S 1 ; F 1 )! (S 2 ; F 2 ) is a morphism in FM 0, where (S i; F i ) = K (f(s i ; d i ) : > 0g). Moreover, if V(R i ) = (S i ; F i ; P i ; G i ), then (8s 2 S 1 )P 2 ((s)) = P 1 (t); G 2 ((s)) = G 1 (t) (t)=(s) (t)=(s) as it follows from a fact that is a morphism in CN D 0 M. Hence, V is a functor. From 3.5, and by a simple computation we can also obtain that V and U are mutually inverse functors. (2) Let R = f(s; d ; P ; G ) : > 0g 2 CN D 0 M and let V(R) = (S; F; P; G), where according to the part (1), we have (8m 2 M)(8s; t 2 S)F (s; m; t) = P (s) = G(t) = Let us denote for any m 2 M t2d (s;m) ; s2p ; t2g : A 0 (m) = f 0 : (9s; t 2 S) = P (s) ^ F (s; m; t) ^ G(t)g A(m) = f 0 : (9s; t 2 S) P (s) ^ F (s; m; t) ^ G(t)g B(m) = f 0 : (9s 2 P )d (s; m) \ G 6= ;g: Then from denitions of and it follows that for any m 2 M we have (R)(m) = ; (V(R))(m) = = 2B(m) 2A 0(m) 2A(m) Let 2 B(m). Then there exist s 2 P and t 2 d (s; m)\g and it follows that P (s)^f (s; m; t)^g(t). Hence 2 A(m). Conversely, let 2 A 0 (m), i.e = P (s)^f (s; m; t)^g(t) for some s; t 2 S. Then P (s) = W s2p. If = W s2p, then according to 3.5, we have s 2 P. If < W s2p, then there exists > such that s 2 P P. Hence, s 2 P. Analogously we can prove that t 2 G. Since = F (s; m; t) = W t2d (s;m), we have t 2 d (s; m) d (s; m) and it follows that t 2 d (s; m) \ G. Hence, 2 B. Therefore, the part (2) is proved. 2 :

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