Computing coproducts of finitely presented Gödel algebras

We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom (α→β)∨(β→α)=1. (Since Gödel algebras are locally finite, ‘finitely generated’, ‘finitely presented’, and ‘finite’ have identical meaning in this paper.) We achieve this result using ordered partitions of finite sets as a key tool to investigate the category opposite to finitely generated Gödel algebras (forests and open order-preserving maps). We give two applications of our main result. We prove that finitely presented Gödel algebras have free products with amalgamation; and we easily obtain a recursive formula for the cardinality of the free Gödel algebra over a finite number of generators first established by A. Horn.