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A sequent calculus for a semi-associative law

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The classical Tamari order is the partial order on fully-bracketed words obtained by assuming that multiplication is only “semi-associative” (ab)c <= a(bc). In the talk I will describe a simple proof-theoretic characterization of the Tamari order as a sequent calculus with a weakened version of Lambek’s left rule for the non-commutative product. The main result is a “focusing” property (a strengthening of cut-elimination) for this sequent calculus, which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One novel combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice. Depending on the interests of the audience and if time permits, I may also discuss how some of these ideas relate to a natural notion of “left representable” multicategory, considered independently in recent work of Bourke and Lack on skew monoidal categories. (Based on a paper presented at FSCD 2017 .)

This talk is part of the Lab Lunch series.

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