Combinatorial descriptions of products in the category of forests and open order-preserving maps

In [1], the authors introduce a technique to compute finite coproducts of finite Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom $(\alpha \rightarrow \beta)\vee(\beta \rightarrow \alpha)$. To do so, they investigate the product in the category opposite to finite Gödel algebras: the category of forests and open order-preserving maps. (A forest is a partially ordered set $F$ such that, for every $x\in F$, the downset of $x$ is 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 -- called merged-shuffle -- on ordered partitions. In this talk, besides recalling the aforementioned construction of the product, we show that, from an enumerative point of view, such a product can be simply described in terms of Delannoy coefficients, of bipartite graphs, and of product of matrices. Bibliography [1] D’Antona, Ottavio M. and Marra, Vincenzo: Computing coproducts of finitely presented Gödel algebras, Ann. Pure Appl. Logic, 142 (2006), 202–211.