zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.