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Xuexing Lu (Jul 19 2025 at 00:55):Hello, everyone. Recently, we have finished a paper, titled a composition theory for upward planar orders. You can find it at https://arxiv.org/abs/2505.13865, or https://www.researchgate.net/publication/391911196_A_composition_theory_for_upward_planar_orders.
In this paper, we provide a combinatorial formulation for Joyal-Street's graphical calculus for monoidal categories.
The abstract is as follows.
An upward planar order on an acyclic directed graph G is a special linear extension of the edge poset of G that satisfies the nesting condition. This order was introduced to combinatorially characterize upward plane graphs and progressive plane graphs (commonly known as plane string diagrams). In this paper, motivated by the theory of graphical calculus for monoidal categories, we establish a composition theory for upward planar orders. The main result is that the composition of upward planar orders is an upward planar order. This theory provides a practical method to calculate the upward planar order of a progressive plane graph or an upward plane graph.