The Structure Group for the Associativity Identity

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 selfdistributivity. AMS Classification: 08A05, 20L10, 20M50. The general purpose of this paper can be summarized as the introduction of some algebraic structure on the faces of Stasheff’s associahedra which are CW-complexes whose faces correspond to the complete bracketings of a given string (see [12]). We introduce a ‘structure group of associativity’ G̃A so that the (skeletons of the) associahedra become orbits for some natural action of G̃A – exactly like the usual regular polyhedra are orbits for the action of (the finite subgroups of) the orthogonal groups O(n). The main point is that the group G̃A shares many algebraic properties with Artin’s braid groups Bn, a similarity which actually extends in part to the general case where associativity is replaced by any another algebraic identity. In former papers ([2], [4], [6]) we have developed an analysis of the left distributivity identity in terms of a structure group that captures the geometry of this particular identity. This analysis was used