Thickness of Feathers

Briefly, a feather is a semigroup derivation diagram with the labels on the edges removed. In this paper, we are concerned with possible definitions for the thickness of a feather. A major open problem is whether the word problem is solvable for semigroup presentations with one defining relation. It is known that word problems for semigroup presentations are solvable if the number of regions in minimal derivation diagrams is bounded. For some definitions for thickness, the number of regions in a derivation diagram over a presentation with one relation will be bounded if the thickness of the diagram is bounded. 1. Background and preliminaries This article is a consequence of a 2010 letter from the second author to the first author. The main content of that letter was a collection of interesting semigroup derivation diagrams. See Figure 4 below for a typeset illustration of one of these diagrams. These diagrams were of interest because they contained regions which were interior regions in the sense that the edges and vertices of the regions were not on the boundary of the diagram and because the diagrams were likely minimal derivation diagrams. Because the diagrams contained interior regions, we could consider these diagrams to be thick diagrams rather than thin diagrams. It was then natural to attempt to quantify the thickness of these diagrams. (See also Peifer, [17], who introduced a definition of thin for certain group diagrams having two-sided regions. This work of Peifer might have been an influence on the second author.) One major goal of this article is to introduce six different ways in which we might quantify the thickness of a semigroup derivation diagram. Secondly, we establish that these six definitions satisfy some of the properties that we believe should be satisfied by any definition for thickness. Our work is very much motivated by the thickness of derivation diagrams over finite presentations for semigroups and monoids and the possible use of these diagrams to solve word problems. At the same time, the thickness of a diagram is intuitively a geometric property that does not depend upon the labels on the edges. Because of this, except for some examples, our concern will be with feathers rather than with diagrams. 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 2000 Mathematics Subject Classification. Primary: 20M05;Secondary:20F06.