### Finitely Presented Heyting Algebras

#### Abstract

In this paper we study the structure of finitely presented Heyting

algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact co- Heyting, improving on a result of Ghilardi who showed that Heyting algebras free on a finite set of generators are co-Heyting. Along the way we give a new and simple proof of the finite model property. Our main technical tool is a representation of finitely presented Heyting algebras in terms of a colimit of finite distributive lattices. As applications we construct explicitly the minimal join-irreducible elements (the atoms) and the maximal join-irreducible elements of a finitely presented Heyting algebra in terms of a given presentation. This gives as well a new proof of the disjunction property for intuitionistic propositional logic.

Unfortunately not very much is known about the structure of Heyting algebras, although it is understood that implication causes the complex structure of Heyting algebras. Just to name an example, the free Boolean algebra on one generator has four elements, the free Heyting algebra on one generator is infinite.

Our research was motivated a simple application of Pitts' uniform interpolation theorem [11]. Combining it with the old analysis of Heyting algebras free on a finite set of generators by Urquhart [13] we get a faithful functor J : HAop

f:p: ! PoSet; sending a finitely presented Heyting algebra to the partially ordered set of its join-irreducible elements, and a map between Heyting algebras to its leftadjoint

restricted to join-irreducible elements. We will explore on the induced duality more detailed in [5]. Let us briefly browse through the contents of this paper: The first section

recapitulates the basic notions, mainly that of the implicational degree of an element in a Heyting algebra. This is a notion relative to a given set of generators. In the next section we study nite Heyting algebras. Our contribution is a simple proof of the nite model property which names in particular a canonical family of nite Heyting algebras into which we can

embed a given finitely presented one.

In Section 3 we recapitulate the standard duality between nite distributive lattices and nite posets. The `new' feature here is a strict categorical

formulation which helps simplifying some proofs and avoiding calculations. In the following section we recapitulate the description given by Ghilardi [8]

on how to adjoin implications to a nite distributive lattice, thereby not destroying a given set of implications. This construction will be our major technical ingredient in Section 5 where we show that every nitely presented

Heyting algebra is co-Heyting, i.e., that the operation (−) n (−) dual to implication is dened. This result improves on Ghilardi's [8] that this is true

for Heyting algebras free on a finite set of generators. Then we go on analysing the structure of finitely presented Heyting algebras

in Section 6. We show that every element can be expressed as a finite join of join-irreducibles, and calculate explicitly the maximal join-irreducible elements in such a Heyting algebra (in terms of a given presentation). As a consequence we give a new proof of the disjunction property for propositional intuitionistic logic. As well, we calculate the minimal join-irreducible elements, which are nothing but the atoms of the Heyting algebra. Finally, we show how all this material can be used to express the category of finitely presented Heyting algebras as a category of fractions of a certain category with objects morphism between finite distributive lattices.

algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact co- Heyting, improving on a result of Ghilardi who showed that Heyting algebras free on a finite set of generators are co-Heyting. Along the way we give a new and simple proof of the finite model property. Our main technical tool is a representation of finitely presented Heyting algebras in terms of a colimit of finite distributive lattices. As applications we construct explicitly the minimal join-irreducible elements (the atoms) and the maximal join-irreducible elements of a finitely presented Heyting algebra in terms of a given presentation. This gives as well a new proof of the disjunction property for intuitionistic propositional logic.

Unfortunately not very much is known about the structure of Heyting algebras, although it is understood that implication causes the complex structure of Heyting algebras. Just to name an example, the free Boolean algebra on one generator has four elements, the free Heyting algebra on one generator is infinite.

Our research was motivated a simple application of Pitts' uniform interpolation theorem [11]. Combining it with the old analysis of Heyting algebras free on a finite set of generators by Urquhart [13] we get a faithful functor J : HAop

f:p: ! PoSet; sending a finitely presented Heyting algebra to the partially ordered set of its join-irreducible elements, and a map between Heyting algebras to its leftadjoint

restricted to join-irreducible elements. We will explore on the induced duality more detailed in [5]. Let us briefly browse through the contents of this paper: The first section

recapitulates the basic notions, mainly that of the implicational degree of an element in a Heyting algebra. This is a notion relative to a given set of generators. In the next section we study nite Heyting algebras. Our contribution is a simple proof of the nite model property which names in particular a canonical family of nite Heyting algebras into which we can

embed a given finitely presented one.

In Section 3 we recapitulate the standard duality between nite distributive lattices and nite posets. The `new' feature here is a strict categorical

formulation which helps simplifying some proofs and avoiding calculations. In the following section we recapitulate the description given by Ghilardi [8]

on how to adjoin implications to a nite distributive lattice, thereby not destroying a given set of implications. This construction will be our major technical ingredient in Section 5 where we show that every nitely presented

Heyting algebra is co-Heyting, i.e., that the operation (−) n (−) dual to implication is dened. This result improves on Ghilardi's [8] that this is true

for Heyting algebras free on a finite set of generators. Then we go on analysing the structure of finitely presented Heyting algebras

in Section 6. We show that every element can be expressed as a finite join of join-irreducibles, and calculate explicitly the maximal join-irreducible elements in such a Heyting algebra (in terms of a given presentation). As a consequence we give a new proof of the disjunction property for propositional intuitionistic logic. As well, we calculate the minimal join-irreducible elements, which are nothing but the atoms of the Heyting algebra. Finally, we show how all this material can be used to express the category of finitely presented Heyting algebras as a category of fractions of a certain category with objects morphism between finite distributive lattices.

#### Full Text:

PDFDOI: http://dx.doi.org/10.7146/brics.v5i30.19436

ISSN: 0909-0878

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