Corollary 13.1.9. The poset \(\Tw ^r([n])\) is an adequate triple, with \(\Tw ^r([n])_L\) consisting of morphisms of the form \((i \leq j) \leq (i \leq l)\) and \(\Tw ^r([n])_R\) consisting of morphisms of the form \((i \leq j) \leq (k \leq j)\).
Proof. The two classes are closed under composition, and every cospan consisting of a morphism in \(\Tw ^r([n])_L\) and a morphism in \(\Tw ^r([n])_R\) has the form treated in Lemma 13.1.8. □
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