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Finite complete rewriting systems for the Jantzen monoid and the Greendlinger group. (English) Zbl 0555.20036
Author’s abstract: ”It is shown that for the presentation (a,b; $$abbaab=\lambda)$$ of the Jantzen monoid J no finite complete rewriting system exists that is based on a Knuth-Bendix ordering. However, a finite complete rewriting system is given for a different presentation of J that has four generators. Further, a finite complete rewriting system is given for the presentation (a,b,c; $$abc=cba)$$ of the Greendlinger group G. This system induces a polynomial-time algorithm for the word problem for G.”
Reviewer: B.Pondelíček

##### MSC:
 20M05 Free semigroups, generators and relations, word problems 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups
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##### References:
 [1] Avenhaus, J.; Madlener, K., Subrekursive komplexität bei gruppen, I. gruppen mit vorgechriebener komplexität, Acta inform., 9, 87-104, (1977) · Zbl 0371.02019 [2] Avenhaus, J.; Book, R.; Squier, C., On expressing commutativity by finite church-rosser presentations: A note on commutative monoids, RAIRO inform. theor., 18, 47-52, (1984) · Zbl 0542.20038 [3] Bauer, G., Zur darstellung von monoiden durch konfluente regelsysteme, () [4] Book, R.; Ó’Dúnlaing, C., Testing for the church-rosser property, Theoret. comput. sci., 16, 223-229, (1981) · Zbl 0479.68035 [5] Book, R., Confluent and other types of thue systems, J. assoc. comput. Mach., 29, 171-182, (1982) · Zbl 0478.68032 [6] Book, R., When is a monoid a group? the church-rosser case is tractable, Theoret. comput. sci., 18, 325-331, (1982) · Zbl 0489.68021 [7] Book, R., The power of the church-rosser property in string rewriting systems, Proc. 6th conf. on automated deduction, 360-368, (1982) [8] Buchberger, B.; Loos, R., Algebraic simplification, () · Zbl 0494.68045 [9] Dershowitz, N., Orderings for term-rewriting systems, Theoret. comput. sci., 17, 279-301, (1982) · Zbl 0525.68054 [10] Dershowitz, N., Applications of the Knuth-bendix completion procedure, () [11] Greendlinger, M., Dehn’s algorithm for the word problem, Comm. pure appl. math., 13, 67-83, (1960) · Zbl 0104.01903 [12] Huet, G., Confluent reductions: abstract properties and applications to term rewriting systems, J. assoc. comput. Mach., 27, 797-821, (1980) · Zbl 0458.68007 [13] Huet, G.; Lankford, D.S., On the uniform halting problem for term rewriting systems, () [14] Huet, G.; Oppen, D., Equations and rewrite rules—a survey, () [15] Jantzen, M., On a special monoid with a single defining relation, Theoret. comput. sci., 16, 61-73, (1981) · Zbl 0482.20043 [16] Jantzen, M., Semi-thue systems and generalised church-rosser properties, (), 60-75 [17] Kemmerich, S., Unendliche reduktionssysteme, Dissertation, (1983), TH Aachen · Zbl 0619.68034 [18] Knuth, D.; Bendix, P., Simple word problems in universal algebra, () · Zbl 0188.04902 [19] Lallement, G., Semigroups and combinatorial applications, (1979), Wiley New York · Zbl 0421.20025 [20] Lescanne, P., Analysis of data structures with non-distinct keys, (), Note · Zbl 0516.68020 [21] Lyndon, R.; Schupp, P., Combinatorial group theory, (1977), Springer Berlin · Zbl 0368.20023 [22] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial group theory, (1976), Dover New York [23] Narendran, P.; McNaughton, R., The undecidability of the preperfectness of thue systems, Theoret. comput. sci., 31, 1, 2, 165-174, (1984) · Zbl 0545.03022 [24] Nivat, M., Congruences parfaites et quasi-parfaites, Séminaire dubreuil, 7, (1971-1972) · Zbl 0338.02018 [25] Potts, D., Remarks on an example of jantzen, Theoret. comput. sci., 29, 3, 277-284, (1984) · Zbl 0538.03034 [26] Squier, C.; Wrathall, C., A note on representations of a certain monoid, Theoret. comput. sci., 17, 229-231, (1982) · Zbl 0473.20044
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