Axioms for Thickness of Feathers

We approach thickness of feathers from an axiomatic point of view. We show that our axioms are independent and we use our axioms to establish general properties for arbitrary definitions of thickness which satisfy these axioms. 1. Axioms for Thickness Informally, a feather is a semigroup derivation diagram with the labels on the edges removed. See [2] for a discussion of semigroup derivation diagrams and see [1] for a more formal definition of feathers. If M is a feather, we will use the notation M̂ for the reflection of M across some line in the plane and the notation ←− M for the feather obtained by reversing the direction of all of the edges in M. A medial vertex for a two-sided region D is a vertex of D which is neither the initial vertex nor the terminal vertex for D. A region D of M is a strongly interior region of M if no edge of D and no medial vertex of D occurs on the boundary of M. Let M be a feather with bottom side αM, top side ωM, initial vertex vι and terminal vertex vτ . A meridian in M is a positive path in M from vι to vτ . For example, αM and ωM are meridians. We can define a partial order on the meridians in M by μ ≤ ν if μν−1 is the counterclockwise boundary of a feathery submap N of M. We call such a feathery submap a layer. Here, we allow that μ = ν and that N is a feather without regions. A nontrivial layering of M with k layers is a sequence of meridians L = {μ0 = αM, μ1, μ2, . . . , μk = ωM} such that for 1 ≤ j ≤ k, the walk μj−1μ j is the counterclockwise boundary of a feathery submap Nj of M where each Nj has at least one region. The submaps N1,N2 . . . ,Nk are the layers of the layering. A vertex v in the feather M is a cut vertex in M if M− v is disconnected. Every cut vertex in M must be on both the top and bottom side of M. A feather is nonseparable if it has no cut vertices. A block of a feather is a maximal nonseparable submap. A block of a feather is itself a feather. A block is nontrivial if it contains at least one region. We call a block without any regions a trivial block. For any feather M, we may write M = M1M2 . . .Mt where the Mj are all of the blocks of M and their common vertices are all of the cut vertices of M. A region D of the map M is appended on the top side of M if ωD is a segment of ωM and is appended on the bottom side of M if αD is a segment of αM. A region D is an appended region of M if it is appended on either side of M. 2000 Mathematics Subject Classification. Primary: 20M05;Secondary:20F06.