Recursive formulas to compute coproducts of finite Gödel algebras and related structures

Gödel logic and its algebraic semantics, namely, the variety of Gödel algebras, play a major rôle in mathematical fuzzy logic. The category of finite Gödel algebras and their homomorphisms is dually equivalent to the category FF of finite forests and order-preserving open maps. The combinatorial nature of FF allows to reduce the usually difficult problem of computing coproducts of algebras and their cardinalities to the combinatorial problem of computing products of finite forests. In this paper we propose a neat, purely combinatorial, recursive formula to compute the product objects. Further, we formulate a dual equivalence between finite Gödel∆-algebras and a category of finite multisets of finite chains, and we provide recursive formulas to compute coproducts, and their cardinalities, in the categories of finite Gödel hoops and of finite Gödel∆-algebras.