We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom (α→β)logical or(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.
Computing coproducts of finitely presented Gödel algebras / O. D'Antona, V. Marra. - In: ANNALS OF PURE AND APPLIED LOGIC. - ISSN 0168-0072. - 142:1-3(2006), pp. 202-211.
Computing coproducts of finitely presented Gödel algebras
O. D'AntonaPrimo
;V. MarraUltimo
2006
Abstract
We obtain an algorithm to compute finite coproducts of finitely generated Gödel algebras, i.e. Heyting algebras satisfying the prelinearity axiom (α→β)logical or(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.Pubblicazioni consigliate
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