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.
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Lower bounds of 4, 3, and 8 additional relations for T_n, I_n, PT_n beyond the symmetric group, with matching explicit presentations for n large enough.
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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.
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Short presentations for transformation monoids
Lower bounds of 4, 3, and 8 additional relations for T_n, I_n, PT_n beyond the symmetric group, with matching explicit presentations for n large enough.