ideas Suffix semifilter-congruences Southwest Univ. Southwest Univ. Hongkong Univ. July 5 9, 2010, Nankai, China. Prefixsuffix
Contents ideas 1 2 ideas 3 Suffix- 4 Prefix-suffix- Suffix Prefixsuffix
ideas Suffix The theory of congruences is one of the key parts of the theory of universal algebras. In particular, the theory of congruences on free semigroups which are 2-algebras is crucial in the study of combinatorial semigroups. The present survey is concerning the important role of the theory of congruences on free semigroups in characterizing. This talk is devoted to characterizing semifilter-congruences. Some recent progress results in this aspect are surveyed. Furthermore, some are proposed. Prefixsuffix
Introduction ideas Suffix Prefixsuffix In the whole talk, A is a finite nonempty set that is called a finite alphabet in which any element is called a letter over A, A always denotes the free monoid generated A. Moreover, w A L A are called a word a language over A, respectively. In particular, the identity of A is called the empty word over A denoted 1. We also let A + = A {1}. It is well known that are important in theoretical computer science. From the theory of theoretical computer science, a language over A is if it can be accepted a finite states automaton. Furthermore, also have a lot of remarkable algebraic properties. In particular, can be characterized using the principal congruences (principal left congruences, principal right congruences) determined themselves.
Introduction ideas Suffix Prefixsuffix Since can be characterized using their corresponding principal congruences (principal left congruences, principal right congruences), we may generalize means of principal congruences (principal left congruences, principal right congruences). This idea is realized firstly Prodinger in 1980, he explored a generalization model of principal right congruences using so-called semifilters. By applying these kinds of generalized principal right congruences, Prodinger defined investigated some classes of generalized. Following Prodinger, several papers are also devoted to this topic. Recently, we continued to pay attention to this topic obtained some new results. In this talk, we shall survey the results obtained in this line. Moreover, some are proposed.
ideas ideas Suffix Prefixsuffix Before stating ideas, we first recall the concept of principal congruences, principal left congruences principal right congruences determined. For any L A, define three relations P (r) L, P (l) L P L on A as follows: where xp (r) L y if only if G L(x, y) = A, xp (l) L y if only if H L(x, y) = A, xp L y if only if P L (x, y) = A A, G L (x, y) = {v A xv L if only if yv L}, H L (x, y) = {u A ux L if only if uy L}, P L (x, y) = {(u, v) A A uxv L if only if uyv L}.
ideas ideas Suffix Prefixsuffix It can be proved that P L (P (r) L, P (l) L ) is a congruence (right congruence, left congruence) on A, which is called the principal congruence (principal right congruence, principal left congruence) on A determined L. Now, we can give the following well-known characterizations of over A. Theorem 2.1 ([3]) For L A, the following statements are equivalent: (1) L is ; (2) The index of P L is finite, that is, the number of P L -classes of A is finite; (3) The index of P (r) L is finite; (4) The index of P (l) L is finite.
ideas ideas Suffix Prefixsuffix In order to state idea, we need some concepts notations. For a monoid M N M, we use N to denote the complement of N in M. Now, let M be a monoid, L, K M x, y, z M. Denote z 1 L = {w M zw L}, Lz 1 = {w M wz L}, x 1 Ly 1 = {w M xwy L}. Furthermore, we also denote the set of finite subsets of M F(M), the complement of the symmetric difference of L K L K, respectively. Formally, we have F(M) = {F M F is finite}, L K = (L K) L K.
ideas ideas Suffix Prefixsuffix Now, we give the following concept of semifilters, which plays an important role in ideas. Definition 2.2 ([4]) Let M be a monoid, 2 M be the power set of M L 2 M. Then, L is called a semifilter (resp. left divisible semifilter, resp. right divisible semifilter) on M if the following (1),(2)(resp.(1),(2),(3) l ; resp.(1),(2),(3) r ) hold: (1) M L ; (2) ( X, Y L ) X Y L ; (3) l ( X L )( z M) z 1 X L ; (3) r ( X L )( z M) Xz 1 L.
ideas ideas Suffix Prefixsuffix Example 2.3 ([4]) The followings are some examples of semifilters on a monoid M. (1) {M}. (2) C (M) = {L M L is finite }, C (M) is called the cofinite semifilter on M. It is not difficult to see that both {A } C (A ) are left right divisible semifilters over A. Remark: Observe that the symmetry of left divisibility right divisibility of semifilters, we only consider left divisible semifilters in the sequel.
ideas ideas Suffix Prefixsuffix At this stage, we can state ideas. Let L be a left divisible semifilter on A L A. Define xp (r) L,L y if only if G L(x, y) L, where G L (x, y) = {v A xv L if only if yv L}. Then we have Theorem 2.4 ([4]) Let L be a left divisible semifilter on A L A. Then P (r) L,L is a right congruence on A. Since the above right congruence P (r) L,L is related to the left divisible semifilter L, the class of right congruences obtained in this way will be called semifilter-congruences in the sequel.
ideas ideas Suffix Prefixsuffix Let L be a left divisible semifilter on A. It can be showed that P (r) L = P (r) {A },L P (r) L,L for any L A. Thus, the principal right congruences determined are generalized using the left divisible semifilters. Furthermore, if we denote R (r) L = {L A the index of P (r) L,L is finite}, then Theorem 2.1, R (r) L contains the class of over A as a subclass. Thus, are generalized this method. It is reasonable to call them generalized determined semifilter-congruences.
ideas ideas Suffix Prefixsuffix It is known that the class of over A forms a Boolean algebra with the operations of union, intersection complement. For generalized determined semifilter-congruences, Prodinger obtained the similar result. Theorem 2.5 ([4]) Let L be a left divisible semifilter on A. Then R (r) L forms a Boolean algebra with the operations of union, intersection complement. In 1980, Prodinger pointed out that the cofinite semifilter C (A ) on A seemed to be the most interesting semifilter on A proposed the following problem. For convenience, we denote the class of over A R(A), or R if no confusion arises.
ideas ideas Suffix Prefixsuffix Problem 2.6 ([4]) Does the equality R (r) C (A ) = R hold? In 1983, Yuqi Guo, Shuiting Lian Li answered Problem 2.6 positively. That is, they obtained the following result. Theorem 2.7 ([1]) R (r) C (A ) = R. From Theorem 2.7, it follows that the class of over A can be characterized using the left divisible semifilter C (A ) as well as the left divisible semifilter {A } on A.
Suffix- ideas Suffix Prefixsuffix In order to obtain a class of generalized determined some left divisible semifilter on A which contains the class of over A properly, Shuhua Zhang, Yuqi Guo Liang Zhang introduced a kind of left divisible semifilters L S = {X A A S X} in 1987, where S S(A), the set of all suffix-free over A. For any S S(A), R (r) L S is called the class of S- over A. In the sequel, we denote R (r) L S R S for any S S(A). Furthermore, the class R S = S S(A) R S is called the class of suffix- over A. Theorem 3.1 ([5]) Let S S(A) L A. Then L R S if only if (1) Ls 1 R for all s S; (2) The set {Ls 1 s S} is finite.
Suffix- ideas Suffix Prefixsuffix Let K, L A. We denote K 1 L = u 1 L LK 1 = Lu 1. u K u K The theorem below investigates some closure properties of R S for S S(A). Theorem 3.2 ([5]) Let S S(A). Then the following statements hold. (1) If K A L R S, then K 1 L R S ; (2) If K R L R S, then KL R S ; (3) R S is closed under concatenation if only if R S = R; (4) R S is closed under the operation of Kleene s closure if only if R S = R.
Suffix- ideas Suffix Prefixsuffix The following theorem gives another characterization of. Theorem 3.3 ([5]) Let S S(A). Then R S = R if only if S is finite maximal in S(A) with respect to the usual inclusion order. Theorem 3.4 ([5]) Let S, T S(A). Then R S = R T if only if (1) For any w in A + (S T ), S {w} S(A) if only if T {w} S(A); (2) For any x S(resp. x T ), there exist y T (resp. y S) z A such that y = zx or x = zy; (3) Both ST 1 T S 1 are finite.
Suffix- ideas Suffix Recently, we introduced a new kind of semifilters using S(A). Let S S(A) L F,S = {L A ( F F(A )) F A S L}. It can be proved that L F,S is a left divisible semifilter on A L S L F,S. The following result indicates that S- can also be described the semifilter L F,S for any S S(A). Theorem 3.5 Let S S(A). Then R (r) L F,S = R S. Prefixsuffix
Suffix- ideas Suffix Prefixsuffix It is worth to remark that Theorem 2.7 can be obtained as a corollary of Theorem 3.5. In fact, if S = A, then L S = {A +, A } L F,S = C (A ). Observe that P (r) L = P (r) L S,L L for each L A, where L= {(x, y) A A x L if only if y L}. By Theorem 3.5, R (r) C (A ) = R S = R (r) {A } = R. On the class of suffix- over A, we have Proposition 3.6 ([6]) Let L A. Then L R S if only if Lt 1 R for some t A.
Suffix- ideas Suffix By using Proposition 3.6, the relationship between Chomsky hierarchy suffix- are also investigated Shuhua Zhang Yuqi Guo in 1988. Remark: In the above discussions, we have considered suffix-. In fact, symmetry, for any P P(A), the set of all prefix-free over A, we can also define the class R P of P - over A. Furthermore, we can also define investigate the class R P of prefix- over A. Prefixsuffix
Divisible semifilters on A A ideas Suffix Prefixsuffix In order to generalize principal congruences, we investigated divisible semifilters on the monoid product A A recently. Definition 4.1 A semifilter L on the monoid product A A is called divisible if ( N L )( z A ) z\n, N/z L, where N/z = {(x, v) A A (xz, v) N}, z\n = {(u, x) A A (u, zx) N}. It is easy to see that both C (A A ) {A A } are divisible semifilters on the monoid product A A.
Divisible semifilters on A A ideas Suffix Prefixsuffix Similar to the case of left divisible semifilters, for a divisible semifilter L on A A L A, we can define a relation on A where xp L,L y if only if P L (x, y) L, P L (x, y) = {(u, v) A A uxv L if only if uyv L}. It can be proved that P L,L is a congruence on A P L P L,L. We also regard P L,L as a semifilter-congruence. Denote R L = {L A the index of P L,L is finite}. Since P L P L,L, we have R R L. Thus, R L is also a kind of generalized determined.
Divisible semifilters on A A ideas Suffix Now, we shall introduce investigate two special kinds of divisible semifilters on A A. To give these semifilters, we need the following notations concepts. Let A A. Denote T ( ) = {(sx, yt) A A (s, t), x, y A, xy 1}. The following concept is basic in our discussion. Definition 4.2 Let A A. Then is called a prefix-suffix subset of A A if T ( ) =. The set of prefix-suffix subsets of A A is denoted PS(A). Prefixsuffix
Divisible semifilters on A A ideas Suffix Now, let PS(A) Define Ω = {(sx, yt) (s, t), x, y A }. L = {N A A Ω N}, L F, = {N A A ( F F(A A ))F Ω N}. Proposition 4.3 Let PS(A). Then both L L F, are divisible semifilters on A A. Prefixsuffix
Divisible semifilters on A A ideas Suffix Prefixsuffix Contrast to Theorem 3.5, we have Proposition 4.4 Let PS(A). In general, R L R L F,. The above Proposition 4.4 motivates the following natural problem. Problem 4.5 Let PS(A). How to characterize R L = R L F,? Finally, we can formulate another problem whose solutions may be helpful for solving Problem 4.5. Problem 4.6 Characterize R L R L F, for PS(A).
- ideas Suffix In the following, we shall investigate R L for PS(A). This work solves Problem 4.6 partially. For any PS(A), R L will be denoted R called the class of - over A in the sequel. Now, we consider - over A for a fixed PS(A). As consequences, some characterizations of are obtained. For PS(A) L A, we denote L( ) = s(s 1 Lt 1 )t, N( ) = (s,t) It is easy to see that L( ) = L N( ). (s,t) sa t. Prefixsuffix
- ideas Suffix Prefixsuffix Firstly, we give a characterization of -. Theorem 4.7 Let L A PS(A). Then L R if only if (1) s 1 Lt 1 R for all (s, t) ; (2) The set {s 1 Lt 1 (s, t) } is finite. Observe that {1} S PS(A) for any S S(A). Then we have Corollary 4.8 ([5]) Let L A S S(A). Then the following are equivalent: (1) L R {1} S ; (2) Lt 1 R for all t S the set {Lt 1 t S} is finite; (3) L R S.
- ideas Suffix Prefixsuffix Some characterizations of are given in the theorem below. For A A, we denote l = {u A (u, v) }, r = {v A (u, v) }. Theorem 4.9 Let PS(A) with l P(A) r S(A). Then the following are equivalent: (1) R = R; (2) N( ) is cofinite; (3) is finite R is closed under the operations of left quotient right quotient; (4) R is closed under the operation of concatenation; (5) R is closed under the operation of Kleene s closure.
- ideas Suffix Prefixsuffix The following result explores some closure properties of - under the operation of concatenation. Proposition 4.10 Let PS(A) with l P(A) r S(A). Then (1) RR R if only if R = R r ; (2) R R R if only if R = R l. Now, we give a sufficient necessary condition such that R 1 = R 2 with 1, 2 PS(A) 1, 2 <. Proposition 4.11 Let 1, 2 PS(A) 1, 2 <. Then R 1 = R 2 if only if the symmetric difference of N( 1 ) N( 2 ) is finite.
- ideas Suffix Prefixsuffix In the above Proposition 4.11, the finiteness of 1 2 is necessary, which can be illustrated the following example. Example 4.12 Let A = {a, b}, S 1 = {ba n n 1}, S 2 = {b 2 a n n 1} {aba n n 1} P P(A). Denote 1 = P S 1 2 = P S 2. Then R 1 = R 2. However, the symmetric difference of N( 1 ) N( 2 ) is infinite. Now, the following problem is natural. Problem 4.13 Let 1, 2 PS(A). How to characterize R 1 = R 2?
Prefix-suffix- ideas Suffix Similar to suffix- prefix-, we call R PS = R. PS(A) prefix-suffix- over A. In the sequel, some algebraic properties of prefix-suffix- are considered. Firstly, we give a characterization of prefix-suffix. Proposition 4.14 Let L A. Then, L R PS if only if s 1 Lt 1 R for some s, t A. Prefixsuffix
Prefix-suffix- ideas Suffix Prefixsuffix Corollary 4.15 ([6]) Let L A. Then L R S if only if Lt 1 R for some t A. For prefix-, suffix- prefix-suffix- over A, we have Proposition 4.16 (1) If A = 1, then R P = R S = R PS = R. (2) If A > 1, then R R P R S R S (resp. R P ) R P R S R PS 2 A. The following proposition gives some relationships among R P, R S, R PS, the class of context-free L 2 the class of context-sensitive L 1. Customarily, we let R = L 3.
Prefix-suffix- ideas Suffix Prefixsuffix Proposition 4.17 Let A 2. Then (1) L 2 R PS ; (2) L 1 L 2 R PS ; (3) (L i L i+1 ) (R PS R P R S ), i = 1, 2. Our final result considers the closure properties of prefixsuffix- under several operations. Proposition 4.18 Let A 2. Then (1) R PS is closed under the operation of complementation; (2) R PS is not closed under the operations of union, intersection, left quotient, right quotient, Kleene s closure concatenation.
Final comments ideas Suffix It would be interesting to investigate generalized determined semifilter-congruences. However, it seems difficult to generalize the previous results to general semifilters. So the strategy for studying this question is finding various kinds of semifilters investigating the corresponding semifilter-congruences generalized. The results surveyed in this paper seem to be only an starting point to deal with the questions. The authors believe that some progress will be made solving the raised in this talk. Prefixsuffix
References I ideas Suffix Prefixsuffix [1] Guo, Y. Q.,, S. T. Li, L., The Prodinger problem the Schreier method of language, Acta. Math. Sinica(in Chinese), 26(3), 1983: 332-340. [2] Howie, J. M., Automata, Clarendon Press, Oxford, 1991. [3] Lallement, G., Semigroup combinatorical applications, John Wiley, New York, 1979. [4] Prodinger, H., Congruences defined filters, Inform. Contr., 44: 1(1980), 36-46. [5] Zhang, S. H., Guo, Y. Q. Zhang, L., On S-, Acta. Math. Sinica(in Chinese), 30(2), 1987: 168-178. [6] Zhang, S. H. Guo, Y. Q., Two notes on S-, Acta. Math. Sinica(in Chinese), 31(1), 1988: 125-130.
Thank You! ideas Suffix Prefixsuffix