Associative and Fair Shuue of !-words

We introduce and investigate some sets of !-trajectories that have the following properties: each of them de nes an associative and commutative operation of parallel composition (shu e) of !-words and, moreover, each of them satis es a certain condition of fairness. The fairness conditions considered involve the operation of parallel composition of !-words. The interrelations between these sets of !-trajectories are studied as well as with other well-known classes of !-words, like in nite Sturmian words, periodic and ultimately periodic !-words. TUCS Research Group Mathematical Structures of Computer Science 1 Preliminaries Usually, the operation of parallel composition of processes, words or languages is modelled by the shu e operation or restrictions of this operation, such as literal shu e, insertion, left-merge, or shu e on trajectories. This paper continues our investigations on the operation of shu e on trajectories of !-words and !-languages, see [14] and [13]. We introduce and investigate several sets of !-trajectories such that each of these sets satis es two important features: the set is associative and the set ful ls a certain condition of fairness. Both conditions are important: associativity ensures that the set of processes has a structure of semiring (where the product is de ned by the shu e on this set of trajectories), whereas the fairness condition ensures good properties for practical use of this parallel composition operation in parallel computations. Therefore, we consider that such an approach is of some importance, both: theoretical and practical. In this paper a process is an in nite sequence of (atomic) actions, also referred as letters, i.e., a process is an !-word over some alphabet . A rst fairness condition on in nite processes was considered by D. Park, see [16]. This condition states that during the parallel composition of two processes, in nitely many actions from the rst process, as well as in nitely many actions from the second process should appear (be performed) in the resulting sequences. The set of all !-trajectories ful lling this condition is denoted in the sequel by V ! + and it is the largest set of !-trajectories that is studied in our paper. However, all other condition of fairness that we introduce and study here, are restrictions of the above, general condition. A rst restriction, referred to as bounded increase, (see Section 4), says that an !-trajectory satis es this condition i there are two constants k1 > 0 and k2 > 0 such that, during the parallel composition, after at most k1 occurrences of actions from the rst process, it occurs at least one occurrence of an action from the second process and, after at most k2 occurrences of actions from the second process, it occurs at least one occurrence of an action from the rst process. A second type of restrictions, asymptotic linear, (see Section 5), states that, in the pre x of length n of the parallel composition sequence, the value of the number of occurrences of actions from the rst process divided by the number of occurrences of actions from the second process, is a sequence of real numbers, say (xn)n>0, such that (xn)n>0 has a nite limit l, l > 0, when n ! +1. Intuitively, this means that the !-trajectory behaves at in nity stable, in such a way that it keeps some \balance" in performing actions from the rst/second process. This \balance" is de ned as being the limit l. 1 A third sort of restrictions, quasi linear, (see Section 6) has a very intuitively geometrical interpretation: an !-trajectory is in this class i there are two parallel lines such that the graph of the !-trajectory is contained between this two lines. It turns out that the trajectories satisfying the quasi linear restriction are exactly those trajectories that are quasi-Sturmian, see Section 6. A fourth type of fairness restriction is pre x bounded, (see Section 7). An !-trajectory satis es this condition i there are two constants a1 > 0 and a2 > 0 such that, in the pre x of length n of the parallel composition sequence, the value of the number of occurrences of actions from the rst process is at most a1-times the number of occurrences of actions from the second process, and symmetrically, the number of occurrences of actions from the second process is at most a2-times the number of occurrences of actions from the rst process. We show that all these conditions lead to associative and commutative sets of !-trajectories. Moreover, we prove the interrelations that exist between these classes, and also with other well-known classes of !-words, like in nite Sturmian words, periodic !-words, etc. Section 8 contains the main result that summarize our results from this paper. The fairness phenomenon was studied in various contexts, see for instance the monograph [4] and also [6, 10, 18, 19, 20]. However, in this paper the fairness concept is considered in connexion with the parallel composition (shu e) operation. The shu e-like operations considered below are de ned using the notion of the !-trajectory. An !-trajectory de nes the general strategy to switch from one !-word to another !-word. Roughly speaking, an !-trajectory is a line in plane, starting in the origin and continuing parallel with the axis Ox or Oy. The line can change its direction only in points with nonnegative integer coordinates. An !-trajectory de nes how to move from an !-word to another !-word when carrying out the shu e operation. Each set T of !-trajectories de nes in a natural way a shu e operation over T . Given a set T of !-trajectories the operation of shu e over T is not necessarily an associative operation. However, for each set T there exists a smallest set of trajectories T such that T contains T and, moreover, shu e over T is an associative operations. Such a set T is referred to as the associative closure of T . The set of nonnegative integers is denoted by !. If A is a set, then the set of all subsets of A is denoted by P(A). Let be an alphabet, i.e., a nite nonempty set of elements called letters. The free monoid generated by is denoted by . Elements in are referred to as words. The empty word is denoted by . 2 If w 2 , then jwj is the length of w. Note that j j = 0. If a 2 and w 2 , then jwja denotes the number of occurrences of the symbol a in the word w. The mirror of a word w = a1a2 : : : an, where ai are letters, 1 i n, is mi(w) = an : : : a2a1 and mi( ) = . A word w is a palindrome i mi(w) = w. Let be an alphabet. An !-word over is a function f : ! ! . Usually, the !-word de ned by f is denoted as the in nite sequence f(0)f(1)f(2) : : :. An !-word w is ultimately periodic i w = vvvvv : : :, where is a ( nite) word, possibly empty, and v is a nonempty word. In this case w is denoted as v!. The set of all ultimately periodic !-words over is denoted by UltPer( ) (or UltPer, if is understood from context). An !-word w is referred to as periodic i w = vvv : : : for some nonempty word v 2 . In this case w is denoted as v!. The set of all periodic !-words over is denoted by Per( ) (or Per, if is understood from context). Moreover, the set of all periodic !-words over that have a palindrome as their period is denoted by PalPer( ) (or PalPer, if is understood from context). The set of all !-words over is denoted by !. An !-language is a subset L of !, i.e., L !. The reader is referred to [2, 17, 23] for general results on !-words and to [7, 21] for notions and results in formal languages. The shu e operation, denoted by , is de ned recursively by: (au bv) = a(u bv) [ b(au v); and (u ) = ( u) = fug; where u ,v 2 and a ,b 2 . The above operation is extended in a natural way to languages: the shu e of two languages L1 and L2 is: L1 L2 = [ u2L1;v2L2 u v: The literal shu e, denoted by l, is de ned as: a1a2 : : : an l b1b2 : : : bm = a1b1a2b2 : : : anbnbn+1 : : : bm; if n m; a1b1a2b2 : : : ambmam+1 : : : an; if m < n; where ai; bj 2 : (u l ) = ( l u) = fug; where u 2 . 3 2 Shu e on !-trajectories In this section we introduce the notions of the !-trajectory and shu e on !-trajectories. The shu e of two !-words has a natural geometrical interpretation related to lattice points in the plane (points with nonnegative integer coordinates) and with a certain \walk" in the plane de ned by each !-trajectory. Let V = fr; ug be the set of versors in the plane: r stands for the right direction, whereas, u stands for the up direction. De nition 2.1 An !-trajectory is an element t, t 2 V !. A set T , T V !, is called a set of !-trajectories. Let be an alphabet and let t be a ( nite) trajectory, let d be a versor, d 2 V , let ; be two ( nite) words over . De nition 2.2 The shu e of with on the trajectory dt, denoted dt , is recursively de ned as follows: if = au and = bv, where a; b 2 and u; v 2 , then: au dt bv = a(u t bv); if d = r; b(au t v); if d = u: if = au and = , where a 2 and u 2 , then: au dt = a(u t ); if d = r; ;; if d = u: if = and = bv, where b 2 and v 2 , then: dt bv = ;; if d = r; b( t v); if d = u: Finally, t = ; if t = ; ;; otherwise: Comment. Note that if j j 6= jtjr or j j 6= jtju, then t = ;. Now we de ne the operation of shu e of !-words on !-trajectories. De nition 2.3 An !-trajectory is an !-word t over V , i.e., t 2 V !. 4 Let be an alphabet and let , be !-words over . Let t be an !-trajectory, t 2 V !. De nition 2.4 The shu e of with on the !-trajectory t is de ned as the limit of the sequence ( 0 t0 0)t02Pref(t), where 0 2 Pref( ), 0 2 Pref( ) such that j 0j = jt0jr and j 0j = jt0ju. One can easily verify that the sequence ( 0 t0 0)t02Pref(t) has always a limit. Observe that there is an important distinction between the nite case, i.e., the shu e on trajectories, and the in nite case, i.e., the shu e on !trajectories: sometimes the result of shu ing of two words and on a trajectory t can be empty whereas the shu e of two !-words over an !trajectory is always nonempty and consists of only one !-word. The shu e on ( nite) trajectories of ( nite) words is investigated in [14]. In this case a trajectory is an element t, t 2 V . Equivalent de nitions of shu e on nite and in nite trajectories can be done as it follows: Remark 2.1 Let be an alphabet and let t be a trajectory, t = t0t1 : : : tn, where ti 2 V; 1 i n. Let ; be two words over , = a0a1 : : : ap; = b0b1 : : : bq, where ai; bj 2 ; 0 i p and 0 j q. The shu e of with on the trajectory t, denoted t , is de ned as follows: if j j 6 = jtjr or j j 6= jtju, then t = ;, else t = c0c1c2 : : : cp+q+2, where, if jt0t1 : : : tijr = k1 and jt0t1 : : : tiju = k2, then ci = ak1 1; if ti = r; bk2 1; if ti = u: Let be an alphabet and let t be an !-trajectory, t = t0t1t2 : : :, where ti 2 V; i 0. Let ; be two !-words over , = a0a1a2 : : : ; = b0b1b2 : : :, where ai; bj 2 ; i; j 0. The shu e of with on the !-trajectory t, denoted t , is de ned as follows: t = c0c1c2 : : :, where, if jt0t1t2 : : : tijr = k1 and jt0t1t2 : : : tiju = k2, then ci = ak1 1; if ti = r; bk2 1; if ti = u: 5 If T is a set of !-trajectories, the shu e of with on the set T of !-trajectories, denoted T , is: T = [ t2T t : The above operation is extended to !-languages over , if L1; L2 !, then: L1 T L2 = [ 2L1; 2L2 T : Notation. If T is V ! then T is denoted by !. Example 2.1 Let and be the !-words = a0a1a2a3 : : :, = b0b1b2b3 : : : and assume that t = r2u2r3ur2uru : : :. The shu e of with on the trajectory t is: t = fa0a1b0b1a2a3a4b2a5a6b3a7b4 : : :g: The result has the following geometrical interpretation (see Figure 1): the trajectory t de nes a line starting in the origin and continuing one unit right or up, depending of the de nition of t. In our case, rst there are two units right, then two units up, then three units right, etc. Assign on the Ox axis and on the Oy axis of the plane. The result can be read following the line de ned by the trajectory t, that is, if being in a lattice point of the trajectory, (the corner of a unit square) and if the trajectory is going right, then one should pick up the corresponding letter from , otherwise, if the trajectory is going up, then one should add to the result the corresponding letter from . Hence, the trajectory t de nes a line in the plane, on which one has \to walk" starting from the origin O. In each lattice point one has to follow one of the versors r or u, according to the de nition of t. Assume now that t0 = ur5u4r3 : : : is another trajectory. Observe that: t0 = fb0a0a1a2a3a4b1b2b3b4a5a6a7 : : :g: Consider the set of trajectories, T = ft; t0g. The shu e of with on the set T of trajectories is: T = fa0a1b0b1a2a3a4b2a5a6b3a7b4 : : : ; b0a0a1a2a3a4b1b2b3b4a5a6a7 : : :g: 6 6 O x y a0 a1 a2 a3 t a4 a5 a6 a7 : : : b0 b1 b2 b3 b4 ... Figure 1 3 Associativity and commutativity The main results in this paper deal with associativity. After a few general remarks, we restrict the attention to the set V ! + of !-trajectories t such that both r and u occur in nitely often in t. (It will become apparent below why this restriction is important.) It turns out that associativity can be viewed as stability under four particular operations, referred to as 3-operations. De nition 3.1 A set T of !-trajectories is associative i the operation T is associative, i.e., ( T ) T = T ( T ); for all ; ; 2 !. A set T of !-trajectories is commutative i the operation T is commutative, i.e., T = T ; for all ; 2 !. The following sets of !-trajectories are associative: 1. T = fr; ug!. 7 2. T = ft 2 V ! j jtjr <1g. 3. T = f 0 0 1 1 : : : j i 2 r , i 2 u and, moreover, i and i are of even length, i 0g. Nonassociative sets of !-trajectories are for instance: 1. T = f(ru)!g. 2. T = ft 2 V ! j t is a Sturmian !-word g. 3. T = fw0w1w2 : : : j wi 2 Lg, where L = frnun j n 0g. Notation. Let A be the family of all associative sets of !-trajectories. Proposition 3.1 If (Ti)i2I is a family of associative sets of !-trajectories, then T 0 = \ i2I Ti , is an associative set of !-trajectories. De nition 3.2 Let T be an arbitrary set of !-trajectories. The associative closure of T , denoted T , is T = \ T T 0;T 02A T 0: Observe that for all T; T fr; ug , T is an associative set of !-trajectories and T is the smallest associative set of !-trajectories that contains T . Remark 3.1 The function , : P(V !) ! P(V !) de ned as above is a closure operator. Notation. Let V ! + be the set of all !-trajectories t 2 V ! such that t contains in nitely many occurrences both of r and of u. Now we give another characterization of an associative set of !-trajectories from V ! + . This is useful in nding an alternative de nition of the associative closure of a set of !-trajectories and also to prove some other properties related to associativity. However, this characterization is valid only for sets of !-trajectories from V ! + and not for the general case, i.e., not for sets of !-trajectories from V !. 8 De nition 3.3 Let W be the alphabet W = fx; y; zg and consider the following four morphisms, i, 1 i 4, where i : W ! V ! + , 1 i 4; and 1(x) = , 1(y) = r , 1(z) = u; 2(x) = r , 2(y) = u , 2(z) = u; 3(x) = r , 3(y) = u , 3(z) = ; 4(x) = r , 4(y) = r , 4(z) = u: Next, we consider four operations on the set of !-trajectories, V ! + . De nition 3.4 Let 3i, 1 i 4 be the following operations on V ! + . 3i : V ! + V ! + ! V ! + , 1 i 4; Let t; t0 be in V ! + . By de nition: 31(t; t0) = 1((x! t y!) t0 z!); 32(t; t0) = 2((x! t y!) t0 z!); 33(t0; t) = 3(x! t0 (y! t z!)); 34(t0; t) = 4(x! t0 (y! t z!)): De nition 3.5 A set T V ! + is stable under 3-operations i for all t1; t2 2 T , it follows that 3i(t1; t2) 2 T , 1 i 4. The following theorem is proved in [13]: Theorem 3.1 Let T be a set of !-trajectories, T V ! + . The following assertions are equivalent: (i) T is an associative set of !-trajectories. (ii) T is stable under 3-operations. 9 Remark 3.2 We restricted our attention only to the set V ! + and not to the general case V !. However, if T contains a trajectory t that is not in V ! + , then 31(t; t)is not necessarily in V !. For instance t = rur!, then 31(t; t) = ur 62 V !. Thus the operation 31 is not well de ned. A similar phenomenon happens with the operation 33. Comment. Observe that D = (P(V ! + ); (3i)1 i 4) is a universal algebra. If T is a set of !-trajectories, then denote by ~ T the union of all those sets of !-trajectories that are in subalgebra generated by T with respect to the algebra D. Proposition 3.2 Let T V ! + be a set of !-trajectories. (i) ~ T is an associative set of !-trajectories and, morover, (ii) ~ T = T , i.e., the associative closure of T is exactly the subalgebra generated by T in D. Proposition 3.3 Let T V ! + be a set of !-trajectories. (i) If each t 2 T is a periodic !-word, then the associative closure of T , T , has the same property, i.e., each !-trajectory in T is periodic. (ii) if additionally, each t 2 T has a palindrome as its period, then the associative closure of T , T , has the same property. (iii) If T is a set of ultimately periodic !-trajectories, then the associative closure of T , T , has the same property, i.e., each !-trajectory in T is ultimately periodic. The above proposition yields: Corollary 3.1 The following sets of !-trajectories are associative: (i) the set of all periodic !-trajectories from V ! + . (ii) the set of all periodic !-trajectories from V ! + that have as their period a palindrome. (iii) the set of all ultimately periodic !-trajectories from V ! + . In the sequel we show some interrelations between associativity and commutativity of shu e operation. 10 De nition 3.6 Let sym be the following mapping, sym : V ! V , sym(r) = u and sym(u) = r. Also consider the mapping ' : fx; y; zg ! fx; y; zg, '(x) = z, '(y) = y and '(z) = x. sym and ' are extended to !-words over V and, respectively over fx; y; zg. One can easily verify the following properties. Proposition 3.4 The following equalities are true: 1. sym(sym(t)) = t, for all t 2 V ! + . 2. '( t ) = '( ) t '( ), for all ; 2 fx; y; zg! and t 2 V ! + . 3. 3('( )) = sym( 1( )), 2 fx; y; zg!. 4. 4('( )) = sym( 2( )), 2 fx; y; zg!. 5. 33(t1; t2) = sym(31(sym(t2); sym(t1))), t1; t2 2 V ! + . 6. 34(t1; t2) = sym(32(sym(t2); sym(t1))), t1; t2 2 V ! + . Consequently, we obtain the following: Corollary 3.2 Let T V ! + . Then: 1. sym(T ) = sym(T ). 2. T is associative if and only if sym(T ) is associative. 3. If T is commutative, then T is commutative. The converse is not true. Remark 3.3 Let T be a commutative set of !-trajectories. From Proposition 3.4 (8 and 9), it follows that T is an associative set of !-trajectories if an only if T is stable under the operations 31 and 32 (or under the operations 33 and 34). 11 4 Bounded increase of !-trajectories De nition 4.1 An !-trajectory t has a bounded increase i there exist two constants c1 and c2 such that: B1 jprefp(t)ju jprefq(t)ju c1(jprefp(t)jr jprefq(t)jr), whenever jprefp(t)jr jprefq(t)jr > 0, and B2 jprefp(t)jr jprefq(t)jr c2(jprefp(t)ju jprefq(t)ju), whenever jprefp(t)ju jprefq(t)ju > 0. Notation. The set of all !-trajectories with bounded increase is denoted by BInc. Proposition 4.1 The trajectory t = ri1uj1ri2uj2:::rinujn ::: 2 V ! + has a bounded increase i the sequences fingn 1 and fjngn 1 are bounded, i.e. there are k1 and k2 such that in k1 and jn k2, for all n 1. Proof. Assume that t satis es B1. By taking p = srn + sun + 1 and q = srn + sun 1 it results that jprefp(t)jr jprefq(t)jr = srn + 1 srn = 1 and jprefp(t)ju jprefq(t)ju = sun sun 1 = jn. Now, by using, B1 it results jn c1. Now, assume that the sequence fjngn 1 is bounded, i.e. there exists k2 such that jn k2. In order to prove that B1 holds, note that this condition means that for any subword of t, jsub(t)ju k2jsub(t)jr whenever jsub(t)jr 6= 0. Let sub(t) = r 1u 1r 2u 2:::r mu m such that jsub(t)jr 6= 0. We have jsub(t)jr = 1 + 2 + :::+ m jsub(t)ju = 1 + 2 + :::+ m On the other hand, the above conditions on t show that: 1 + 2 + :::+ m k2m 1 + 2 + :::+ m m 1 It follows that 1 + 2 + :::+ m k2(1 + 1 + 2 + :::+ m) Note that 1+ 2 + :::+ m 1 and we conclude that the condition B1 is ful lled: 12 1 + 2 + :::+ m 2k2( 1 + 2 + :::+ m) The same conclusion holds for the sequence fingn 1 in connection with the condition B2. Comment. Note that an !-trajecory t is in BInc i there are two constants k1 > 0 and k2 > 0 such that, during the parallel composition of two processes on t, after at most k1 occurrences of actions from the rst process, it occurs at least one occurrence of an action from the second process and, after at most k2 occurrences of actions from the second process, it occurs at least one occurrence of an action from the rst process. Theorem 4.1 The set of all !-trajectories with bounded increase, i.e., BInc, is an associative and commutative set of !-trajectories. Proof. First we prove that BInc is a commutative set of !-trajectories. To see this let t be an !-trajectory and = sym(t). It is easy to observe that t satis es B1 i satis es B2 and t satis es B2 i satis es B1 which proves that BInc is a commutative set of !-trajectories. Let t1; t2 be two !-trajectories that satisfy the conditions B1 and B2. First we prove that 31(t1; t2) satis es also conditions B1 and B2. In the sequel we will use the notation of constants as c11, c12 for t1 and c21, c22 for t2. Step I Let be the value of 31(t1; t2). From the de nition of the operation 31, it follows that jpref( )jr = jprefjprefn(t2)jr(t1)ju and jpref( )ju = jprefn(t2)ju for some n. In order to prove that the condition B1 is ful led, assume that jprefjprefn(t2)jr(t1)ju > jprefjprefm(t2)jr(t1)ju (1) It results that jprefn(t2)jr > jprefm(t2)jr. Since t2 also satis es condition B1, it follows that: jprefn(t2)ju jprefm(t2)ju c21(jprefn(t2)jr jprefm(t2)jr) (2) Using again relation (1) and as a consequence that t1 also satis es B2, it results that: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr c12(jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju) 13 and by adding jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju we obtain: jprefn(t2)jr jprefm(t2)jr < (1+c12)(jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju) From (2) it follows that: jprefn(t2)ju jprefm(t2)ju c21(1 + c12)(jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju) Now consider condition B2. Assume that jprefn(t2)ju > jprefm(t2)ju. Since t2 satis es condition B2 it results that jprefn(t2)jr jprefm(t2)jr c22(jprefn(t2)ju jprefm(t2)ju) Note that the following relations hold: jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju jprefn(t2)jr jprefm(t2)jr From the above two inequalities we obtain that: jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju c22(jprefn(t2)ju jprefm(t2)ju) Therefore satis es the condition B2. Step II Now assume that = 32(t1; t2). From the de nition of the operation 32, it follows that jpref( )jr = jprefjprefn(t2)jr(t1)jr and jpref( )ju = jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju for some n. In order to prove that satis es the condition B1, assume that jprefjprefn(t2)jr(t1)jr > jprefjprefm(t2)jr(t1)jr This implies that jprefn(t2)jr > jprefm(t2)jr. Since t2 satis es the condition B1, we obtain that: jprefn(t2)ju jprefm(t2)ju c21(jprefn(t2)jr jprefm(t2)jr) (3) On the other hand t1 also satis es the condition B1 and thus: jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju (4) 14 c11(jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr) The following relation holds: jprefn(t2)jr jprefm(t2)jr = jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr+ jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju Combining the above relation with relation (4), it follows that: jprefn(t2)jr jprefm(t2)jr (1 + c11)(jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr) The relation (3) leads to: jprefn(t2)ju jprefm(t2)ju c21(1+c11)(jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr) By adding the above inequality with (4) we obtain: jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju (jprefjprefm(t2)jr(t1)ju + jprefm(t2)ju) (c11 + c21(1 + c11))(jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr) Therefore satis es the condition B1. In order to prove that satis es the condition B2, assume that jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju > jprefjprefm(t2)jr(t1)ju + jprefm(t2)ju: Note that there are only two possible cases: Case I. jprefjprefn(t2)jr(t1)ju > jprefjprefm(t2)jr(t1)ju and jprefn(t2)ju jprefm(t2)ju. Since t1 satis es the condition B2 it follows that: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr c12(jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju) Note that jprefn(t2)ju jprefm(t2)ju 0, and therefore: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr c12(jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju + jprefn(t2)ju jprefm(t2)ju) 15 Case II. jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju and jprefn(t2)ju > jprefm(t2)ju. Notice that t2 satis es the condition B2 and hence: jprefn(t2)jr jprefm(t2)jr c22(jprefn(t2)ju jprefm(t2)ju) On the other hand, jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr jprefn(t2)jr jprefm(t2)jr and therefore: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr c22(jprefn(t2)ju jprefm(t2)ju) From the inequality, jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju 0 it follows that: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr c22(jprefn(t2)ju jprefm(t2)ju + jprefjprefn(t2)jr(t1)ju jprefjprefm(t2)jr(t1)ju) From the above two cases we conclude that: jprefjprefn(t2)jr(t1)jr jprefjprefm(t2)jr(t1)jr min(c12; c22)(jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju (jprefjprefm(t2)jr(t1)ju + jprefm(t2)ju)) The closure under the operations 33 and 34 follows from Remark 3.3. 5 Asymptotic linear !-trajectories. De nition 5.1 An !-trajectory t is linear asymptotic i the following limit does exist and belongs to (0;+1) l = lim jprefn(t)ju jprefn(t)jr : Notation. The set of all asymptotic linear !-trajectories is denoted by ALin. Comment. Note that an !-trajectory t is in ALin i performing the parallel composition of two processes on t, in the pre x of length n of the resulting 16 sequence, the value of the number of occurrences of actions from the rst process divided by the number of occurrences of actions from the second process, is a sequence of real numbers, say (xn)n>0, such that (xn)n>0 has a nite limit l, l > 0, when n ! +1. Intuitively, this means that the !-trajectory behaves at in nity stable, in such a way that it keeps some \balance" in performing actions from the rst/second process. This \balance" is de ned as being the limit l. Theorem 5.1 The set of all asymptotic linear !-trajectories, i.e., ALin, is an associative and commutative set of !-trajectories. Proof. Let t be an !-trajectory and = sym(t). The limit lim jprefn(t)ju jprefn(t)jr exists and belongs to (0;+1) i the limit lim jprefn(t)jr jprefn(t)ju exists and belongs to (0;+1). Therefore t 2 ALin i 2 ALin which means that ALin is a commutative set of !-trajectories. Let t1, t2 be two trajectories in V ! + such that l1 = lim jprefn(t1)ju jprefn(t1)jr ; l1 2 (0;+1) l2 = lim jprefn(t2)ju jprefn(t2jr l2 2 (0;+1) We compute the limits coresponding to the operations 3i, 1 i 4. The operation 31(t1; t2). From the de niton of 31(t1; t2) it follows that: jprefm(31(t1; t2)jr = jprefjprefn(t2)jr(t1)ju; and jprefm(31(t1; t2)ju = jprefn(t2)ju; where n is the number of items used in t2 to obtain a result of length m in 31(t1; t2). Therefore, jprefn(t2)ju jprefjprefn(t2)jr(t1)ju = jprefn(t2)ju jprefn(t2)jr jprefn(t2)jr jprefjprefn(t2)jr(t1)ju = = jprefn(t2)ju jprefn(t2)jr jprefjprefn(t2)jr(t1)jr + jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)ju = jprefn(t2)ju jprefn(t2)jr ( jprefjprefn(t2)jr(t1)jr jprefjprefn(t2)jr(t1)ju + 1) 17 Passing at limit with n ! +1, it follows that the value of the limit corresponding to 31(t1; t2) is: l2( 1 l1 + 1) Similarly, we obtain the limit corresponding to 32(t1; t2), i.e., jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju jprefjprefn(t2)jr(t1)jr = jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr+ jprefn(t2)ju jprefjprefn(t2)jr(t1)jr = jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr + jprefn(t2)ju jprefn(t2)jr jprefn(t2)jr jprefjprefn(t2)jr(t1)jr = = jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr+ jprefn(t2)ju jprefn(t2)jr jprefjprefn(t2)jr(t1)jr + jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr = jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr + jprefn(t2)ju jprefn(t2)jr (1 + jprefjprefn(t2)jr(t1)ju jprefjprefn(t2)jr(t1)jr ) Passing at limit we obtain that the value of the limit corresponding to 32(t1; t2) is: l1 + l2(1 + l1) = l1 + l2 + l1l2 The limits for 33 and 34 can be easily obtained by using the "symmetric" relations, see Proposition 3.4. l1 1 + l2 l1l2 1 + l1 + l2 From the above proof, it follows that 3i(t1; t2) are linear asymptotic whenever t1 and t2 are linear asymptotic, where 1 i 4. This means that the set of asymptotic linear !-trajectories, ALin is an associative set of !-trajectories. Proposition 5.1 BInc ALin 6= ; Proof. Consider the notations, v1 = r2u, v2 = ru2 and de ne the following !-trajectory: t = v2v2 1v22 2 v23 1 :::v22k 2 v22k+1 1 ::: 18 By using Proposition 4.1 it results that t 2 BInc. Let N = 22k+2 3 . It is easy to see that: jv2v2 1v22 2 v23 1 :::v22k 2 ju = 5N 1 2 jv2v2 1v22 2 v23 1 :::v22k 2 jr = 2N 1 and hence, jv2v2 1v22 2 v23 1 :::v22k 2 ju jv2v2 1v22 2 v23 1 :::v22k 2 jr = 5N 1 4N 2 Passing at limit with N ! +1 the above limit is 5/4. On the other hand jv2v2 1v22 2 v23 1 :::v22k 2 v22k+1 1 ju = 4N jv2v2 1v22 2 v23 1 :::v22k 2 jr = 5N and therefore, jv2v2 1v22 2 v23 1 :::v22k 2 v22k+1 1 ju jv2v2 1v22 2 v23 1 :::v22k 2 v22k+1 1 jr = 4N 5N Passing at limit with N ! +1, the above limit is 4/5. It followss that there are two subsequences of jpref(t)ju jpref(t)jr having di erent limits. Hence, the above sequence cannot be convergent. Proposition 5.2 ALin BInc 6= ; Proof. Assume that t = rur2u2r3u3:::rmumrm+1um+1:::, which is not in the class BInc (see Proposition 4.1), and observe that: jrur2u2r3u3:::rmju = m(m 1) 2 jrur2u2r3u3:::rmjr = m(m+ 1) 2 jrur2u2r3u3:::rmj = m2 and jrur2u2r3u3:::rmumrm+1ju = m(m+ 1) 2 jrur2u2r3u3:::rmumrm+1jr = (m+ 1)(m+ 2) 2 19 jrur2u2r3u3:::rmumrm+1j = (m+ 1)2 It follows that for n 2 [m2; (m+ 1)2], m(m 1) 2 (m+1)(m+2) 2 jprefn(t)ju jprefn(t)jr m(m+1) 2 m(m+1) 2 Passing at limit with n ! +1 we obtain that lim jprefn(t)ju jprefn(t)jr = 1 which it means that t belongs to the class Alin. 6 Quasi linear !-trajectories De nition 6.1 We say that an !-trajectory t is quasi liniar i there exist two constants, c1; c2 > 0 such that jjprefn(t)ju c1jprefn(t)jrj c2: Notation. The set of all quasi linear !-trajectories is denoted by QLin. Comment. An !-trajectory t is in QLin i the graph of t is bounded by two parallel lines. Proposition 6.1 The set of all quasi linear !-trajectories, i.e., QLin, is included in both classes BInc and ALin. Proof. Assume that t is a quasi linear !-trajectory, i.e., there exist c1; c2 > 0 such that jjprefn(t)ju c1jprefn(t)jrj c2 By dividing the above inequality with jprefn(t)jr we obtain that: j jprefn(t)ju jprefn(t)jr c1j c2 jprefn(t)jr Passing at limit with n !1 it follows that limj jprefn(t)ju jprefn(t)jr = c1 and therefore t is a linear asymptotic !-trajectory, i.e., t 2 ALin. 20 In order to prove that t has bounded increase we start by using the following inequalities jprefn(t)ju c1jprefn(t)jr + c2 jprefm(t)ju c1jprefm(t)jr + c2 By adding the above inequalities we obtain: jprefn(t)ju jprefm(t)ju c1(jprefn(t)jr jprefm(t)jr) + 2c2 If jprefn(t)jr jprefm(t)jr > 0, i.e., jprefn(t)jr jprefm(t)jr 1, then it follows that: jprefn(t)ju jprefm(t)ju (c1 + 2c2)(jprefn(t)jr jprefm(t)jr): Hence t satis es condition B1. In the same way one can show that t satis es condition B2. Therefore, t 2 BInc. Remark 6.1 . From the above proposition, it follows that class QLin is included in the intersection of classes BInc and ALin, i.e., QLin BInc\ ALin. Proposition 6.2 BInc \ ALin QLin 6= ;: Proof. Assume that v = ru and t = rvrv2rv3r:::vm 1rvmr:::. Note that by using Proposition 4.1 it results that t is in BInc. Now, notice that: jrvrv2rv3r:::vm 1jr = m 1 + m(m 1) 2 jrvrv2rv3r:::vm 1ju = m(m 1) 2 jrvrv2rv3r:::vm 1j = m2 1 If n 2 [m2 1; (m+ 1)2 1], then jprefn(t)jr 2 [m 1 + m(m 1) 2 ;m+ m(m+ 1) 2 ] 21 and jprefn(t)ju 2 [m(m 1) 2 ; m(m+ 1) 2 ] It follows that:m(m 1) 2 m+ m(m+1) 2 jprefn(t)ju jprefn(t)jr m(m+1) 2 m 1 + m(m 1) 2 Passing at limit with m ! +1 we obtain that: lim jprefn(t)ju jprefn(t)jr = 1 Therefore t is in the class ALin. Assume now that t is in QLin, i.e. there exist two constants, c1; c2 > 0 such that jjprefn(t)ju c1jprefn(t)jrj c2 By dividing the above inequality by jprefn(t)jr we obtain that: j jprefn(t)ju jprefn(t)jr c1j c2 jprefn(t)jr Passing at limit and using the fact that lim jprefn(t)ju jprefn(t)jr = 1 it results that c1 = 1. Therefore: jjprefn(t)ju jprefn(t)jrj c2 From the following equalities jrvrv2rv3r:::vm 1jr = m 1 + m(m 1) 2 , jrvrv2rv3r:::vm 1ju = m(m 1) 2 it follows that jjrvrv2rv3r:::vm 1ju jrvrv2rv3r:::vm 1jrj = = jm(m 1) 2 (m 1 + m(m 1) 2 )j = m 1 which cannot be bounded. We conclude that t cannot be in QLin. Theorem 6.1 The set of all quasi linear !-trajectories, QLin, is an associative and commutative set of !-trajectories. 22 Proof. Let t be an !-trajectory and = sym(t). Note that jprefn(t)ju = jprefn( )jr and jprefn(t)jr = jprefn( )ju. Hence it follows that there are c1; c2 > 0 such that jjprefn(t)ju c1jprefn(t)jrj c2 i jjprefn( )jr c1jprefn( )juj c2 i jjprefn( )ju 1 c1 jprefn( )jrj c2 c1 which means that t 2 QLin i 2 QLin. Consequently QLin is a commutative set of !-trajectories. Let t1 and t2 be two !-trajectories which satisfy: jjprefn(t1)ju c11jprefn(t1)jrj c12 jjprefn(t2)ju c21jprefn(t2)jrj c22 Thus, by using the above proposition, c11 = lim jprefn(t1)ju jprefn(t1)jr and c21 = lim jprefn(t2)ju jprefn(t2)jr Firstly, assume that: = 31(t1; t2). jjprefn(t2)ju c21(1 + 1=c11)jprefjprefn(t2)jr(t1)juj = jjprefn(t2)ju c21jprefjprefn(t2)jr(t1)ju c21 c11 jprefjprefn(t2)jr(t1)juj = jjprefn(t2)ju c21jprefn(t2)jr+c21jprefjprefn(t2)jr(t1)jr c21 c11 jprefjprefn(t2)jr(t1)juj jjprefn(t2)ju c21jprefn(t2)jrj+c21 c11 jc11jprefjprefn(t2)jr(t1)jr jprefjprefn(t2)jr(t1)juj c22 + c21 c11c12 Therefore is a quasi linear !-trajectory. Now, assume that = 32(t1; t2), jjprefjprefn(t2)jr(t1)ju + jprefn(t2)ju (c11 + c21 + c11c21)jprefjprefn(t2)jr(t1)jrj jjprefjprefn(t2)jr(t1)ju c11jprefjprefn(t2)jr(t1)jrj+ +jjprefn(t2)ju (c21 + c11c21)jprefjprefn(t2)jr(t1)jrj c12 + jjprefn(t2)ju (c21 + c11c21)jprefjprefn(t2)jr(t1)jrj = 23 c12 + jjprefn(t2)ju (c21 + c11c21)(jprefn(t2)jr jprefjprefn(t2)jr(t1)ju)j = c12 + jjprefn(t2)ju c21jprefn(t2)jr c11c21jprefn(t2)jr+ +c21jprefjprefn(t2)jr(t1)ju + c11c21jprefjprefn(t2)jr(t1)juj c12 + jjprefn(t2)ju c21jprefn(t2)jrj +c21jjprefjprefn(t2)jr(t1)ju + c11jprefjprefn(t2)jr(t1)ju c11jprefn(t2)jrj c12 + c22 + c21jjprefjprefn(t2)jr(t1)ju c11(jprefn(t2)jr jprefjprefn(t2)jr(t1)ju)j = c12 + c22 + c21jjprefjprefn(t2)jr(t1)ju c11jprefjprefn(t2)jr(t1)jrj c12 + c22 + c21c12 For the operations 33 and 34 we obtain the same conclusion by using the Remark 3.3. Proposition 6.3 UltPer is included in QLin and the inclusion is strict. Proof. Assume that t = wv!; jvju = p; jvjr = q. Then, for n 2 N; prefn(t) = wvm v. Let k1 = jwju; k2 = jwjr; s1 = j vju; s2 = j vjr; 0 s1 + s2 < m and note that: jprefn(t)ju = k1 +mp+ s1 and jprefn(t)jr = k2 +mq + s2 It follows that: jprefn(t)ju pq jprefn(t)jr = k1+mp+s1 k2pq mp s2pq = k1 k2pq+s1 s2pq and (jwj+ jvj)pq jprefn(t)ju pq jprefn(t)jr jwj+ jvj Thereforejjprefn(t)ju pq jprefn(t)jrj maxfjwj+ jvj; (jwj+ jvj)pqg which means t 2 QLin. Next theorem gives a charecterization of those !-trajectories that are in QLin. The theorem is based on an extension of the notion of a Sturmian word, see [1], pp.14-15. For general results on Sturmian words the reader is also referred to [12, 15]. 24 De nition 6.2 The !-trajectory t is quasi-Sturmian i there is c > 0 such that: jjvju jwjuj c;8v;w 2 Sub(t); jvj = jwj The set of all quasi-Sturmian !-trajectories is denoted by QSturm. Let t 2 V ! + . Consider the following function: t : ! ! !, t(n) = jprefn(t)ju Remark 6.2 The !-trajectory t is quasi-Sturmian i : j t(n+ p) t(n) ( t(m+ p) t(m))j c;8n;m; p 2 ! Proposition 6.4 If t 2 V ! + , t = ri1uj1ri2uj2:::rinujn ::: is a quasi-Sturmian !-trajectory, then the sequence fjngn is bounded. Proof. Let q be such that i1 + i2 + :::+ iq > c, and de ne p0 = i1 + i2 + :::+ iq + j1 + j2 + :::+ jq. The !-tarjectory has a bounded balance then, by taking m = 0 and p = p0 we obtain: 0 t(n+ p0) t(n) t(p0) + c But, t(p0)+c = j1+j2+:::+jq+c < i1+i2+:::+iq+j1+j2+:::+jq = p0, and results: 0 t(n+ p0) t(n) < p0;8n which means that every subword of t, having the length equal to p0, has at least one r. Consequently the sequence fjngn is bounded. Theorem 6.2 QSturm = QLin. Proof. Firstly, we prove that QLin is a subset of QSturm. Let t 2 QLin, i.e., there are d; c > 0 such that: j t(n) dnj c or dn c t(n) dn+ c By using the above relation for n + p and for n we have: 25 d(n + p) c t(n+ p) d(n + p) + c dn c t(n) dn + c and adding these relations it follows: dp 2c t(n+ p) t(n) dp + 2c dp 2c ( t(m+ p) t(m)) dp + 2c and adding these inequalities: 4c t(n+ p) t(n) ( t(m+ p) t(m)) 4c j t(n + p) t(n) ( t(m+ p) t(m))j 4c This means that t 2 QSturm. Now we prove the contrary inclusion, i.e., QSturm QLin. Claim I. There exists lim t(n) n and, moreover, its value is in (0; 1). Proof Claim I. Let t be in QSturm. Note that 0 < t(n) n < 1 and therefore there exist lim inf t(n) n and lim sup t(n) n and, 0 lim inf t(n) n lim sup t(n) n 1 Let xn and yn be two sequences such that: lim t(xn) xn = lim inf t(n) n lim t(yn) yn = lim sup t(n) n Since t 2 QSturm, it follows that: j t(n+ p) t(n) t(p)j c;8n; p 2 ! or t(n) + t(p) c t(n+ p) t(n) + t(p) + c;8n; p 2 ! By iterating the right part it results: t(nm) n t(m) + (n 1)c By iterating the left part it results: m t(n) (m 1)c t(nm) 26 Therefore: m t(n) (m 1)c n t(m) + (n 1)c Using the above relation for yn and xn instead of n and m it results: xn t(yn) (xn 1)c yn t(xn) + (yn 1)c and dividing by xnyn it results: t(yn) yn (xn 1)c xnyn t(xn) xn + (yn 1)c xnyn Passing at limit we obtain: lim sup t(n) n lim inf t(n) n : This means that the limit of t(n) n does exist. In order to prove that lim t(n) n 2 (0; 1) let consider ri1uj1ri2uj2:::riqujq be a pre x of t, having the length n. Since t 2 QSturm, i.e., there are c1; c2 > 0 such that in c1 and jn c2, 8n 2 ! and the minimum value of t(n) n is q 1 qc1+q 1 . Therefore lim t(n) n 1 1+c1 . Similarly, it results that lim t(n) n c2 1+c2 which complete the proof of the Claim I. We continue our proof by considering t 2 QSturm and l = lim t(n) n . From the above proof it follows that: m t(n) (m 1)c n t(m) + (n 1)c and hence: m t(n) n t(m) (n 1)c+ (m 1)c By dividing with m we obtain: t(n) n t(m) m (n 1)c + (m 1)c m and, passing at limit with m ! +1 it results: t(n) ln c Dividing the same relation by n it follows that: m t(n) n t(m) (n 1)c + (m 1)c n 27 and passing at limit with n ! +1 it results: lm t(m) c Using the last two relations we obtain: j t(n) lnj c Therefore t 2 QLin. Corollary 6.1 The set of all Sturmian !-words is included in QLin and the inclusion is strict. Remark 6.3 Note that, as a consequence that there are nondenumarable many Sturmian !-words, it follows that the QLin is a nondenumerable set, too. 7 Pre x bounded !-trajectories De nition 7.1 An !-trajectory t is referred as pre x bounded i there exist a1; a2 > 0 such that: P1 jprefn(t)ju a1jprefn(t)jr, whenever jprefn(t)jr > 0: and P2 jprefn(t)jr a2jprefn(t)ju, whenever jprefn(t)ju > 0: Notation. The set of all pre x bounded !-trajectories is denoted by PrefB. Comment. An !-trajectory is in PrefB i performing the parallel composition of two processes on t there are two constants a1 > 0 and a2 > 0 such that, in the pre x of length n of the resulting sequence, the value of the number of occurrences of actions from the rst process is at most a2-times the number of occurrences of actions from the second process, and, moreover, the number of occurrences of actions from the second process is at most a1-times the number of occurrences of actions from the rst process. Remark 7.1 It is easy to see that, if an !-trajectory t is in PrefB, then: P'1 n (1 + a1)jprefn(t)jr; wheneverjprefn(t)jr > 0 P'2 n (1 + a2)jprefn(t)ju; wheneverjprefn(t)ju > 0 28 Theorem 7.1 The set of pre x bounded !-trajectories, PrefB, is an associative and commutative set of !-trajectories. Proof. First we prove that PrefB is a commutative set of !-trajectories. To see this let t be an !-trajectory and = sym(t). It is easy to observe that t satis es P1 i satis es P2 and t satis es P2 i satis es P1 which proves that PrefB is a commutative set of !-trajectories. Let t1; t2 be two !-trajectories in PrefB. Denote by a11; a12 and a21; a22 the constants corresponding to these trajectories. Assume that = 31(t1; t2). Note that jpref( )jr = jprefjprefn(t2)jr(t1)ju and jpref( )ju = jprefn(t2)ju for some n. jprefjprefn(t2)jr(t1)ju jprefn(t2)jr a22jprefn(t2)ju Therefore satis es condition P1. jprefn(t2)ju a21jprefn(t2)jr a21(1 + a12)jprefjprefn(t2)jr(t1)ju Hence also satis es condition P2. Now assume = 32(t1; t2). Note that jpref( )jr = jprefjprefn(t2)jr(t1)jr and jpref( )ju = jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju for some n. jprefjprefn(t2)jr(t1)jr a12jprefjprefn(t2)jr(t1)ju a12(jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju) Therefore satis es condition P1. jprefjprefn(t2)jr(t1)ju + jprefn(t2)ju a11jprefjprefn(t2)jr(t1)jr + a21jprefn(t2)jr a11jprefjprefn(t2)jr(t1)jr + a21(1 + a11)jprefjprefn(t2)jr(t1)jr = = (a11 + a21(1 + a11))jprefjprefn(t2)jr(t1)jr Thus also satis es condition P2. For the operations 33 and 34 the conclusion follows from Remark 3.3. Proposition 7.1 The set PrefB includes the set BInc. Proof. It is easy to observe that the condition B1 becomes P1, whenever m = 0 and the condition B2 becomes P2, whenever m = 0. 29 Proposition 7.2 The set PrefB includes the set ALin. Proof. Let t be an !-trajectory in ALin. From Proposition 2.1 it follows that the sequence f jprefn(t)ju jprefn(t)jr g has a limit in (0;+1). Therefore, both f jprefn(t)ju jprefn(t)jr g and f jprefn(t)jr jprefn(t)jug have limits in (0;+1) and consequently are in PrefB. Remark 7.2 Since the sets BInc and ALin are incomparable (see Proposition 5.2 and Proposition 5.1) it follows that the inclusions from the above two propositions are strict. 8 Main Result Combaining our results from the previous sections, we obtain the following: Theorem 8.1 All sets of !-trajectories depicted in Figure 2 are associative, commutative and fair (with respect to a certain type of fairness). The arrows from Figure 2 denote strict inclusions. 30 V ! + ? PrefB HHHHjALin BInc \ALin HHHHHj BInc ? QLin = QSturm ? UltPer ? Per ? PalPer Figure 2 Notation. Let Sturm be the set of all in nite Sturmian words over the alphabt V = fr; ug. Denote by ASturm the associative closure of the set Sturm, i.e., ASturm = Sturm. Remark 8.1 From Propositon 6.1 it follows that Sturm QLin. Note that QLin is an associative set of !-trajectories, see Theorem 6.1. Therefore, ASturm QLin. However, it is an open problem whether this inclusion is strict or not. From the above Remark and fromTheorem 8.1 we deduce some properties of !-trajectories that are in ASturm. 31 Theorem 8.2 If t is an !-trajectory in ASturm, then1. t 2 QLin, i.e., there exist two constants, c1; c2 > 0 such thatjjprefn(t)ju c1jprefn(t)jrj c22. t 2 QSturm, i.e., there exist c > 0 such that:jjvju jwjuj c;8v;w 2 Sub(t); jvj = jwj3. t 2 BInc, i.e., there exist two constants c1 and c2 such that:jprefp(t)ju jprefq(t)ju c1(jprefp(t)jr jprefq(t)jr);whenever jprefp(t)jr jprefq(t)jr > 0 andjprefp(t)jr jprefq(t)jr c2(jprefp(t)ju jprefq(t)ju);whenever jprefp(t)ju jprefq(t)ju > 0.4. t 2 ALin, i.e., the following limit does exist and belongs to (0;+1)l = lim jprefn(t)jujprefn(t)jr :5. t 2 PrefB, i.e., there exist two constants a1; a2 > 0 such that:jprefn(t)ju a1jprefn(t)jr;whenever jprefn(t)jr > 0 andjprefn(t)jr a2jprefn(t)juwhenever jprefn(t)ju > 0:9 ConclusionTen sets of !-trajectories, all of them associative, commutative and ful l-ing a certain fairness condition were introduced and their interrelations wereestablished. 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