The partition category admits a faithful involutive tensor representation by minimal power-of-2 dimensional zero-one matrices over additively idempotent semirings, with extensions to twisted variants and smaller Fibonacci-dimensional representations for the Temperley-Lieb category.
Auinger, Pseudovarieties generated by Brauer type monoids.Forum Mathematicum26, 1–24 (2014)
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Faithful linear and relational representations of diagram categories and monoids
The partition category admits a faithful involutive tensor representation by minimal power-of-2 dimensional zero-one matrices over additively idempotent semirings, with extensions to twisted variants and smaller Fibonacci-dimensional representations for the Temperley-Lieb category.