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The structure group for the associativity identity. (English) Zbl 0859.55009
Summary: A group of elementary associativity operators is introduced so that the bracketing graphs which are the skeletons of Stasheff’s associahedra become orbits and can be constructed as subgraphs of the Cayley graph of this group. A very simple proof of Mac Lane’s coherence theorem is given, as well as an oriented version of this result. We also sketch a more general theory and compare the cases of associativity and left self-distributivity.

##### MSC:
 55P45 $$H$$-spaces and duals
##### Keywords:
associativity operators; associahedra; Cayley graph
Full Text:
##### References:
 [1] Cartier, P., (), exposé [2] Dehornoy, P., Free distributive groupoids, J. pure appl. algebra, 61, 123-146, (1989) · Zbl 0686.20041 [3] Dehornoy, P., Structural monoids associated to equational varieties, (), 293-304 · Zbl 0776.08006 [4] Dehornoy, P., Braid groups and left distributive operations, Trans. amer. math. soc., 345, 115-151, (1994) · Zbl 0837.20048 [5] Dehornoy, P., From large cardinals to braids via distributive algebra, J. knot theory and ramifications, 4, 33-79, (1995) · Zbl 0873.20030 [6] P. Dehornoy, Groups with a complemented presentation, J. Pure Appl. Algebra, to appear. · Zbl 0870.20023 [7] Epstein, D.B., Word processing in groups, (1992), Jones and Barlett · Zbl 0764.20017 [8] Kapranov, M.M., The permutoassociahedron, mac Lane’s coherence theorem and asymptotic zones for the KZ equation, J. pure appl. algebra, 85, 119-142, (1993) · Zbl 0812.18003 [9] Lascoux, A.; Schützenberger, M.P., Symmetry and flag manifolds, (), 118-144 [10] Laver, R., The left distributive law and the freeness of an algebra of elementary embeddings, Adv. math., 91, 209-231, (1992) · Zbl 0822.03030 [11] Lane, S.Mac, Natural associativity and commutativity, Rice univ. studies, 49, 28-46, (1963) · Zbl 0244.18008 [12] Stasheff, J.D., Homotopy associativity of H-spaces I, Trans. amer. math. soc., 108, 275-292, (1963) · Zbl 0114.39402
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