Version of 8-25-2000 ACHIEVABLE RELATORS

The principal result of this paper is a reduction of the word problem for one relator semigroups. Crudely, a relator (w,w′) on the alphabet X is achievable if the letters in the relator can actively participate in sequences of transitions for the semigroup presentation 〈X ; (w,w′)〉. (A precise definition of achievable relators is given in section 3.) Empirically, achievable relators are rare. We prove that the word problem for a one relator semigroup presentation can be reduced to the word problem for a presentation where the relator is achievable. 1 Background and Preliminaries A directed planar map (or simply map) is a finite collection of vertices, edges, and regions in the Euclidean plane, together with an orientation for the set of edges. Here we will require that each edge has two distinct vertices for its endpoints. We also make the usual requirements that vertices, edges, and regions are pairwise disjoint, that regions are homeomorphic to the unit disk, and that each region has a connected boundary which is a union of edges and their endpoints. The orientation of the edges distinguishes an initial endpoint and a terminal endpoint for each edge. Two maps are the same if one can be mapped onto the other by a homeomorphism of the plane which induces a one-to-one function preserving vertices, edges, regions, incidence, and orientation. Let D be a region of any map or let M be a connected and simply connected map. A boundary walk for D or for M is a closed walk of minimal length which includes all of the edges on the topological boundary of D or of M. For our purposes, we make the additional requirement that all boundary walks must be oriented counterclockwise in the plane. It follows that if π is any boundary walk for a fixed map or region, then all other boundary walks for this map or region can be regarded as cyclic permutations of the edges of π. A map or region Q is two-sided if it has a boundary walk of the form αQ(ωQ) −1 where αQ and ωQ are positive walks, the respective bottom and top sides of Q. We define W1 to be the map consisting of a single directed edge e1 together with its initial vertex v0 and its terminal vertex v1. For k > 1, the map Wk is defined inductively by Wk = Wk−1 ∪ {vk, ek} where vk is a vertex and ek is an edges with initial vertex vk−1 and terminal vertex vk. It is easy to see that each Wk is two-sided and a tree. Let F0 = {Wk|k ≥ 1}. Assume, for induction, that a set Fj of two-sided maps having two-sided regions has been defined and that each map in Fj has j regions. Let Ij be an index set of triples (k,M, σ) where k is a natural number, M ∈ Fj , and σ is a nontrivial segment of the top side of M having initial endpoint u0 and terminal endpoint ut. Given such a triple, we define a map F(k,M, σ) to be M∪ {v1, v2, . . . , vk−1, e1, . . . , ek, Dj+1}