Diagram

A diagram: a set partition of points, a top row and a bottom row , written as blocks of signed labels. The basis element of every algebra below.

Diagram([[1,-1],[2,-2]])blocks of signed labels — positive on the top row, negative on the bottom

Domain: Diagram algebras

Details

Multiplication is geometric: stack above , glue 's bottom row to 's top, keep what stays connected between the outer rows and discard the middle
A block left entirely in the discarded middle was a closed loop, and contributes one factor of the loop parameter — so these algebras live over
Use the ORDERED product (NonCommutativeMultiply, infix ): a diagram algebra is not commutative
Diagrams normalise, so a written diagram and a computed one are the same expression
Every point must appear exactly once; a diagram missing one is malformed and is left as written

Examples