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Groups presented by finite two-monadic Church-Rosser Thue systems
Authors:
J. Avenhaus, K. Madlener and F. Otto
Journal:
Trans. Amer. Math. Soc. 297 (1986), 427-443
MSC:
Primary 20F10; Secondary 03D03, 03D40, 20F05
MathSciNet review:
854076
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Abstract: It is shown that a group can be defined by a monoid-presentation of the form , where is a finite two-monadic Church-Rosser Thue system over , if and only if is isomorphic to the free product of a finitely generated free group with a finite number of finite groups.
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- [1]
- J. Avenhaus and K. Madlener, Subrekursive Komplexität bei Gruppen I. Gruppen mit vorgeschriebener Komplexität, Acta Inf. 9 (1977), 87-104. MR 0498062 (58:16239)
- [2]
- -, On groups defined by monadic Thue systems, Algebra, Combinatorics, and Logic in Computer Science, Colloq. Math. Soc. János Bolyai, vol. 42, 1983, pp. 63-71. MR 875854 (88f:20054)
- [3]
- J. Avenhaus, R. Book and C. Squier, On expressing commutativity by finite Church-Rosser presentations: a note on commutative monoids, RAIRO Inform. Théor. 18 (1984), 47-52. MR 750450 (85j:03057)
- [4]
- R. Book, Confluent and other types of Thue systems, J. Assoc. Comput. Mach. 29 (1982), 171-182. MR 662617 (84c:03076)
- [5]
- -, The power of the Church-Rosser property in string rewriting systems, Proc. 6th Conf. on Automated Deduction, 1982, pp. 360-368.
- [6]
- -, When is a monoid a group? The Church-Rosser case is tractable, Theoret. Comput. Sci. 18 (1982), 325-331. MR 662676 (84b:20067)
- [7]
- R. Book, M. Jantzen and C. Wrathall, Monadic Thue systems, Theoret. Comput. Sci. 19 (1982), 231-251. MR 671869 (84d:03048)
- [8]
- W. W. Boone, Certain simple unsolvable problems of group theory. I, II, III, IV, V, VI, Nederl. Akad. Wetensch. Proc. Ser. A 57 (1954), 231-237; 57 (1954), 492-497; 58 (1955), 252-256, 58 (1955); 571-577; 60 (1957), 22-27; 60 (1957), 227-232. MR 0066372 (16:564d)
- [9]
- B. Buchberger and R. Loos, Algebraic simplification, Computer Algebra (Symbolic and Algebraic Computation), (B. Buchberger, G. E. Collins, and R. Loos, eds). Computing Supplementum 4, Springer-Verlag, Wien/New York, 1982. MR 728962
- [10]
- F. B. Cannonito, Hierarchies of computable groups and the word problem, J. Symbolic Logic 31 (1966), 376-392. MR 0224471 (37:70)
- [11]
- Y. Cochet, Church-Rosser congruences on free semigroups, Algebraic Theory of Semigroups, Colloq. Math. Soc. Jànos Bolyai, North-Holland, Amsterdam, vol. 20, 1976, pp. 51-60. MR 541109 (80i:03051)
- [12]
- M. Dehn, Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911), 116-144. MR 1511645
- [13]
- R. H. Haring-Smith, Groups and simple languages, Trans. Amer. Math. Soc. 279 (1983), 337-356. MR 704619 (85b:20045)
- [14]
- D. Kapur and P. Narendran, The Knuth-Bendix completion procedure and Thue systems, Proc. 3rd Conf. Foundations of Computer Science and Software Engineering (Bangalore, India, 1983). MR 743114 (85e:68040)
- [15]
- G. Lallement, Semigroups and combinatorial applications, Wiley-Interscience, New York, 1979. MR 530552 (81j:20082)
- [16]
- R. Lyndon and P. Schupp, Combinatorial group theory, Springer-Verlag, New York, 1977. MR 0577064 (58:28182)
- [17]
- W. Magnus, Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann. 106 (1932), 295-307. MR 1512760
- [18]
- W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, 2nd rev. ed., Dover, New York, 1976. MR 0422434 (54:10423)
- [19]
- D. E. Muller and P. E. Schupp, Groups, the theory of ends, and context-free languages, J. Comput. System Sci. 26 (1983), 295-310. MR 710250 (84k:20016)
- [20]
- P. S. Novikov, On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44 (1955). MR 0075197 (17:706b)
- [21]
- F. Otto, Conjugacy in monoids with a special Church-Rosser presentation is decidable, Semigroup Forum 29 (1984), 223-240. MR 742135 (87h:20107a)
- [22]
- M. O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172-194. MR 0110743 (22:1611)
- [23]
- J. R. Stallings, Group theory and three-dimensional manifolds, Yale Univ. Press, New Haven, Conn., 1971. MR 0415622 (54:3705)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1986-0854076-5
PII:
S 0002-9947(1986)0854076-5
Keywords:
Group,
monoid-presentation,
Church-Rosser property,
monadic system,
context-free group,
free product of groups
Article copyright:
© Copyright 1986 American Mathematical Society
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