Trace Diagram - Formal Definition

Formal Definition

Let V be a vector space of dimension n over a field F (with n≥2), and let Fun(V,V) denote the linear transformations on V. An n-trace diagram is a graph, where the sets V_i (i = 1, 2, n) are composed of vertices of degree i, together with the following additional structures:

• a ciliation at each vertex in the graph, which is an explicit ordering of the adjacent edges at that vertex;
• a labeling V₂ → Fun(V,V) associating each degree-2 vertex to a linear transformation.

Note that V₂ and V_n should be considered as distinct sets in the case n = 2. A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices V₁ into two disjoint ordered collections called the inputs and the outputs.

The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph:

• Loops are permitted (a loop is an edges that connects a vertex to itself).
• Edges that have no vertices are permitted, and are represented by small circles.
• Multiple edges between the same two vertices are permitted.