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

Godel logic and its algebraic semantics, namely, the variety of Godel algebras, play a major r ¨ ole in mathematical fuzzy logic. The category of finite Godel 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 Godel ¨ Δ-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 Godel hoops and of finite G ¨ odel ¨ Δ-algebras.