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Groups with a complemented presentation. (English) Zbl 0870.20023
A presentation of a group is said to be right complemented if for any two generators \(x\) and \(y\), there exists one relation of the form \(xu=yv\), where \(u\) and \(v\) are finite products of generators, and furthermore any relation in the presentation is such a relation. Artin’s braid groups are an important example. The author investigates such groups and presents a simple quadratic algorithm for solving their word problem.

MSC:
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F36 Braid groups; Artin groups
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[1] Birman, J., Braids, links, and mapping class groups, ()
[2] S. Burckel, The well-ordering of positive braids, J. Pure Appl. Algebra, to appear.
[3] Clifford, A.M.; Preston, G.B., The algebraic theory of semigroups, AMS surveys 7, Vol. 1, (1961) · Zbl 0111.03403
[4] Dehornoy, P., Deux propriétés des groupes de tresses, Note C. R. acad. sci. Paris, 315, 633-638, (1992) · Zbl 0790.20056
[5] Dehornoy, P., Braid groups and left distributive operations, Trans. amer. math. soc., 345, 1, 115-151, (1994) · Zbl 0837.20048
[6] Dehornoy, P., The structure group for the associativity identity, J. pure appl. algebra, 111, 1-3, 59-82, (1996) · Zbl 0859.55009
[7] Dehornoy, P., Construction of left distributive operations, (1994), preprint
[8] Dershowitz, N.; Jouannaud, J.P., Rewrite systems, (), Chapter 6 · Zbl 0900.68283
[9] Elrifai, E.A.; Morton, H.R., Algorithms for positive braids, Quart. J. math. Oxford, 45, 2, 479-497, (1994) · Zbl 0839.20051
[10] Epstein, D., Word processing in groups, (1992), Jones & Barlett Publ · Zbl 0764.20017
[11] Gabriel, P.; Zisman, M., Calculus of fractions and homotopy theory, (1967), Springer Berlin · Zbl 0186.56802
[12] Garside, F.A., The braid group and other groups, Quart. J. math. Oxford, 20, 78, 235-254, (1969) · Zbl 0194.03303
[13] Laver, R., Braid group actions on left distributive structures and well orderings in the braid groups, J. pure appl. algebra, 108, 1, 81-98, (1996) · Zbl 0859.20029
[14] Tatsuoka, K., An isometric equality for Artin groups of finite type, Trans. amer. math. soc., 339, 2, 537-551, (1993) · Zbl 0798.20030
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