Spectra of Gödel Algebras

We exploit the duality between finite Gödel algebras and their homomorphisms, and the category of finite forests and open maps, to compute the duals of non-isomorphic k-element Gödel algebras, for k≥1k≥1. From this construction we obtain a recurrence formula to compute the fine spectrum of the variety of Gödel algebras GG. Using such a formula we easily compute the set of cardinalities of finite Gödel algebras (the spectrum of GG), that is equal to the set of positive integers N+N+. To complete the picture on spectra of Gödel algebras, we recall the well-known recurrence formula to compute the cardinality of every free k-generated Gödel algebra, that is the free spectrum of GG.