Presentations for the ghost algebra and the label algebra

The ghost algebra is a two-boundary extension of the Temperley-Lieb algebra, constructed recently via a diagrammatic presentation. The existing two-boundary Temperley-Lieb algebra has a basis of two-boundary string diagrams, where the number of strings connected to each boundary must be even. The ghost algebra is similar, but allows this number to be odd, using bookkeeping dots called ghosts to assign a consistent parity to each string endpoint on each boundary. Equivalently, one can discard the ghosts and label each string endpoint with its parity; the resulting algebra is readily generalised to allow any number of possible labels, instead of just odd or even. We call the generalisation the label algebra, and establish a non-diagrammatic presentation for it. A similar presentation for the ghost algebra follows from this.