A graphic theory of associativity and wordchain patterns

Abstract

The problem of deciding whether a partial binary operation, a "bin", can be embedded in a semigroup is the associativity problem (for general bins). It is known that it is equivalent to the word problem for (semi)groups and thus unsolvable, even for the class of finite bins. This paper establishes a close association between bins and their wordchains and 3-connected 3-regular planar graphs, or, equivalently convex 3-regular polyhedral nets (skeletons). This permits a constructive approach revealing the combinatorial depth of the associativity problem in detail and leads to a naturally enumerable hierarchy of standard wordchain patterns, of universal bins, and of associative laws. Each bin is a superposition of homomorphic images, i.e. "colourings" of edges, of universal bins. One side result is a purely algebraic equivalent of the 4-colour-theorem. The obtained results open further ways for an efficient search by computer for simplest non-associativity contradictions. It is hoped that they lead to solutions of the associativity problem for further subclasses of bins, further insight into the structure of partial binary operations and of polyhedra and will yield precise measures of presentations for associative systems and their classifications.

The term bin has been proposed by K. Osondu in his thesis (Buffalo N.Y., 1974) and can be used, when wanted, with "partial" or "full", similarly to "partial" or "linear" order.