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Embeddings of Heyting Algebras

Dick de Jongh, Albert Visser

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Abstract

Abstract In this paper we study embeddings ofHeyting algebras (Ha’s). Itis pointed out that such embeddings are naturally connected with Derived Rules and with propositional theories. We consider the Ha’s embeddable in the Ha of the Intuitionistic Propositional Calculus (IPC), i.e. the free Ha on N0 generators, those embeddable in the Ha of Heyting’s Arithmetic (HA) and those embeddable in the Ha of HA*, a ‘natural’ extension of HA. We prove the following theorems. The same Ha’s on finitely many generators are embeddable in the Ha of IPC and in the Ha of Boolean (or: Brouwerean) combinations of :>sentences of HA. The Ha’s on finitely many generators embeddable in the Ha ofIPC are finitely presented. There is a non-recursive Ha on three generators that can be embedded in the Ha of HA. Every recursively enumerable prime Ha is embeddable in the Ha of HA*.

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What this paper is about

Abstract In this paper we study embeddings ofHeyting algebras (Ha’s). Itis pointed out that such embeddings are naturally connected with Derived Rules and with propositional theories. We consider the Ha’s embeddable in the Ha of the Intuitionistic Propositional Calculus (IPC), i.e. the free Ha on N0 generators, those embeddable in the Ha of Heyting’s Arithmetic (HA) and those embeddable in the Ha of HA*, a ‘natural’ extension of HA. We prove the following theorems. The same Ha’s on finitely many generators are embeddable in the Ha of IPC and in the Ha of Boolean (or: Brouwerean) combinations of :>sentences of HA. The Ha’s on finitely many generators embeddable in the Ha ofIPC are finitely presented. There is a non-recursive Ha on three generators that can be embedded in the Ha of HA. Every recursively enumerable prime Ha is embeddable in the Ha of HA*.

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Available abstract

Abstract In this paper we study embeddings ofHeyting algebras (Ha’s). Itis pointed out that such embeddings are naturally connected with Derived Rules and with propositional theories. We consider the Ha’s embeddable in the Ha of the Intuitionistic Propositional Calculus (IPC), i.e. the free Ha on N0 generators, those embeddable in the Ha of Heyting’s Arithmetic (HA) and those embeddable in the Ha of HA*, a ‘natural’ extension of HA. We prove the following theorems. The same Ha’s on finitely many generators are embeddable in the Ha of IPC and in the Ha of Boolean (or: Brouwerean) combinations of :>sentences of HA. The Ha’s on finitely many generators embeddable in the Ha ofIPC are finitely presented. There is a non-recursive Ha on three generators that can be embedded in the Ha of HA. Every recursively enumerable prime Ha is embeddable in the Ha of HA*.

Key concepts: Heyting algebra, Recursively enumerable language, Mathematics, Propositional calculus, Extension (predicate logic), Intuitionistic logic, Algebra over a field, Pure mathematics

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