A. Lentin and M. P. Schutzenberger, A combinatorial problem in the theory of free monoids, In: R. C. Bose and T. E. Dowling (eds.), Combinatorial Mathematics, North Carolina Press, 112-144, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with a single associative function symbol. String unification has first been presented by Markov [15] in 1954 and is called Markov s problem by mathematicians in eastern countries. It is called Lob s problem by mathematicians in western countries, for example by A. Lentin and M.P. Schutzenberger [11]. A solution to the string unification problem was found by Makanin [13] in 1977. Subsequent papers on this topic [18, 9, 23, 10] were concerned with finding a better description of Makanin s algorithm, closing small gaps in the proof of correctness, and studying its complexity. Context ....

A. Lentin and M. Schutzenberger. A combinatorial problem in the theory of free monoids. Conference on Combinatorical Mathematics and its Applications., 1969.

....where t is primitive and i 2. Now if i is even, then immediately ff = fi, which is a contradiction. For odd i = 2n 1 we have ff = t n p, fi = qt n , where t = pq. But ff and fi are conjugates and so, by Lemma 6, we have a contradiction. In fact Corollary 2 is a special case of the claim in [LeS] which states under the additional assumption that ff; fi are primitive, that fffi m is primitive for all natural numbers m. The proof is not difficult, but we need only this special case to prove the next result. 11 Corollary 3. Consider set X = fff; fig with ff; fi 2 Sigma . Let w be a ....

Lentin, A., Schutzenberger, M.P., A combinatorial problem in the theory of free monoids, in: R.C. Bose and T.W. Dowling (eds), Combinatorial Mathematics and its Applications, Univ. North Carolina Press, 128--144, 1969.

....strings and it is left to the reader. Proposition 11 Two conjugate strings have conjugate roots and equal exponents. 15 5.2.1 The conjugacy equation As in the case of free monoids there is an alternative definition to the notion of conjugacy. The following result is a direct extension of [14], see also [15, p. 8] Proposition 12 The strings x; y; z 2 Sigma Ord , y 6= 1 satisfy the condition xz = zy if and only if there exist u; v 2 Sigma Ord and n such that x = uv, y = vu and z = uv) n u Observe that the equality xx = x 1 holds. Therefore the condition y 6= 1 is ....

A. Lentin and M. P. Schutzenberger. A combinatorial problem in the theory of free monoids. In R. C. Bose and T. E. Bowlings, editors, Combinatorial Mathematics, pages 112--144. North Carolina Press, Chapel Hill, N. C., 1967.

....equations, preserving solvability in both directions. He hoped to obtain a proof for the unsolvability of Hilbert s tenth problem by showing that solvability of word equations is an undecidable problem (see [Ma81] for more details) Approximately at the same time Lentin and Schutzenberger [LeSc67] independently considered word equations. In the following period, in the western countries the main attention was given to the (relatively simple) problem to enumerate in a compact form the set of all solutions of a word equation. Plotkin [Pl72] in the context of resolution based theorem ....

A. Lentin, M.P. Schutzenberger, "A Combinatorial Problem in the Theory of Free Monoids", in Proceedings of the University of North Carolina, (1967) pp. 67-85.

....journals he used. Later many of his results were discovered several times in different connections. The modern systematic research on words, in particular words as elements of free monoids, was initiated by M.P. Schutzenberger in the sixties. Two influencial papers of that time are [LySc] and [LeSc]. This research created also 2 C. Choffrut and J. Karhumaki the first monograph on words, namely [Len] which, however, never became widely used. Year 1983 was important to the theory: the first book Combinatorics on Words [Lo] covering major parts on combinatorial problems of words appeared. ....

....F semigroups having a prefix code as the minimal generating set. Such semigroups are often called right unitary. The second condition, which is often referred to as the stability condition, characterizes those F semigroups which are free, i.e. have a code as the minimal generating set, cf. [LeSc] or [BePe] The third condition, which differs from the others in the sense that it depends also on X, is introduced here mainly to stress the diversified nature of the defect theorem. As shown in [HK1] it characterizes those F semigroups, where X factorizes uniquely. For the sake of ....

A. Lentin and M.P. Schutzenberger, A combinatorial problem in the theory of free monoids, in: R.C. Bose and T.E. Dowling (eds.), Combinatorial Mathematics, North Carolina Press, Chapel Hill, 112--144, 1967.

....is, foe; ig is not a tree code, then, by lemma 11 either oe is suffix of i or i is suffix of oe We now prove that all solutions of tree equations in two indeterminates have a particular structure. This result generalizes the analogous result for equations on words with two indeterminates (cf. [6], 7] 8] Theorem 13. Let (oe; i) be the solution of a tree equation on two indeterminates. Then there exists a tree such that oe; i 2 # , that is, oe and i are powers of the same tree . Proof. The proof is by induction on joej jij. Initialization step: if joej jij = 2 then the ....

....is given by the following pair of binary trees: a a a a It remains open the problem to give a complete characterization of tree codes with two elements. 5 Applications and conclusions As an application of the result in section 4, we extend a well known combinatorial property of words (cf. [6] and [7] to trees. Theorem 14. Let be a labeled tree. Then there exists a unique primitive tree 0 such that 2 # 0 . Proof. By induction on the size of the tree. Initialization step: If is a punctual tree a, then is a primitive tree and we are done. Induction step: Let us suppose ....

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A. Lentin, M. P. Schutzenberger. "A combinatorial problem in the theory of free monoids". Proc. of the University of North Carolina ; pagg. 67--85, 1967.

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A. Lentin and M. P. Schutzenberger, A combinatorial problem in the theory of free monoids, In: R. C. Bose and T. E. Dowling (eds.), Combinatorial Mathematics, North Carolina Press, 112-144, 1967.

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A. Lentin and M. P. Schutzenberger. A combinatorial problem in the theory of free monoids. In Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967.

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A. Lentin, M. P. Schutzenberger. A combinatorial problem in the theory of free monoids. Proc. of the University of North Carolina; pagg. 67--85, 1967. BIBLIOGRAPHY 55

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A. Lentin, M. P. Schutzenberger. A combinatorial problem in the theory of free monoids. Proc. of the University of North Carolina , pagg. 67--85, 1967.

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