Computing coproducts of finitely presented Gödel algebras

Abstract

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.

Probability Measures in the Logic of Nilpotent Minimum

“…This fact often allows to transfer results concerning Gödel logic to NM. In particular in this work we shall use the definition of normal forms for NM formulas given in [3], and the techniques used to develop the spectral duality for finite Gödel algebras in [8] and for finite NM-algebras 1 in [1], to adapt the notion of finitely additive probability measure given in [6] for Gödel logic to the case of the logic of Nilpotent Minimum.…”

De Finetti’s No-Dutch-Book Criterion for Gödel logic

“…2 That is, without extra-logical axioms. 3 The function c(n) can be explicitly defined by a recurrence relation first discovered by Horn in [7, p. 479 The algebraic counterpart of truth-value assignments to n-variable formulae of Gödel logic are homomorphisms w: G n → [0, 1] of Gödel algebras. 4 As in the classical case, it is natural to think of such a function w ∈ [0, 1] c(n) as a possible world for Gödel logic.…”

The Automorphism Group of Finite Godel Algebras

“…Proof: The first statement can be found in [3], [7], [8]. The last statement follows from the first one, using the identities of Lemma 15, Corollary 2 and noticing that (H i ) ⊥ and (H j ) ⊥ are not isomorphic whenever i = j.…”

“…Direct computation (see [7], [8]) shows that J (G 1 ) is a forest of three elements {a, b, c} (see Fig. 2) equipped with the reflexive closure of b < c. It is easy to check that G 1 is rigid and …”