Products in the category of forests and p-morphisms via Delannoy paths on Cartesian products

In [2], the authors introduce a technique to compute finite coproducts of finite G\uf6del algebras, i.e. Heyting algebras satisfying the prelinearity axiom (\u3b1 \u2192 \u3b2) 28(\u3b2 \u2192 \u3b1). To do so, they investigate the product in the category opposite to finite G\uf6del algebras: the category of forests and open order-preserving maps, alias p-morphisms, which we denote by F. (A forest is a partially ordered set F such that, for every x in F, the set of lower bounds of x forms a chain, when endowed with the order inherited from F.) To achieve their result, the authors make use of ordered partitions of finite sets and of a specific operation \u2014 called merged-shuffle \u2014 on ordered partitions. In [1, Section 4.2], the authors present an alternative, recursive construction of finite products in the category of forests and open order-preserving maps. In the present work we introduce a further construction of the same finite products, based on products of posets along with a generalization of the combinatorial notion of Delannoy path. The new and most interesting aspect of our construction is that, dually, it uncovers a key relationship between the coproducts of finite G\uf6del algebras and the coproducts in the category of finite distributive lattices. Our main result explains the former coproducts in terms of a construction on the latter; the construction itself is currently best understood via duality using a generalisation of the Delannoy paths. 1. Stefano Aguzzoli, Simone Bova, and Brunella Gerla. Chapter IX: Free algebras and functional representation for fuzzy logics. In Handbook of mathematical fuzzy logic. Volume 2, volume 38 of Stud. Log. (Lond.), pages 713\u2013791. Coll. Publ., London, 2011. 2. Ottavio M. D\u2019Antona and Vincenzo Marra. Computing coproducts of finitely presented G\uf6del algebras. Ann. Pure Appl. Logic, 142(1-3):202\u2013211, 2006