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Embeddings of Heyting Algebras

Dick de Jongh, Albert Visser

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Abstract

In this paper we study embeddings of Heyting Algebras. It is pointed out that such embeddings are naturally connected with Derived Rules. We compare the Heyting Algebras embeddable in the Heyting Algebra of the Intuitionistic Propositional Calculus (IPC), i.e. the free Heyting Algebra on countably infinitely many generators, and those embeddable in the Heyting Algebra of Heyting's Arithmetic (HA). A partial result is obtained. We show that every recursively enumerable prime Heyting Algebra is embeddable -in the Heyting Algebra of HA*, a ‘natural’ extension of HA.

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What this paper is about

In this paper we study embeddings of Heyting Algebras. It is pointed out that such embeddings are naturally connected with Derived Rules. We compare the Heyting Algebras embeddable in the Heyting Algebra of the Intuitionistic Propositional Calculus (IPC), i.e. the free Heyting Algebra on countably infinitely many generators, and those embeddable in the Heyting Algebra of Heyting's Arithmetic (HA). A partial result is obtained. We show that every recursively enumerable prime Heyting Algebra is embeddable -in the Heyting Algebra of HA*, a ‘natural’ extension of HA.

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Available abstract

In this paper we study embeddings of Heyting Algebras. It is pointed out that such embeddings are naturally connected with Derived Rules. We compare the Heyting Algebras embeddable in the Heyting Algebra of the Intuitionistic Propositional Calculus (IPC), i.e. the free Heyting Algebra on countably infinitely many generators, and those embeddable in the Heyting Algebra of Heyting's Arithmetic (HA). A partial result is obtained. We show that every recursively enumerable prime Heyting Algebra is embeddable -in the Heyting Algebra of HA*, a ‘natural’ extension of HA.

Key concepts: Recursively enumerable language, Mathematics, Propositional calculus, Discrete mathematics, Extension (predicate logic), Finitely-generated abelian group, Combinatorics, Prime (order theory)

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