On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

Luck Darni` ere

D´epartement de math´ematiques Facult´e des sciences

Universit´e d’Angers, France

Markus Junker

Mathematisches Institut

Abteilung f¨ur mathematische Logik Universit¨at Freiburg, Deutschland

December 8, 2008

1 Introduction

Heyting algebras are a generalisation of Boolean algebras; the most typical example is the lattice of open sets of a topological space. Heyting algebras play the same rˆole for intuitionistic logic as Boolean algebras for classical logic. They are special distributive lattices, and they form a variety. They are mainly studied by universal algebraists and by logicians, hardly by model theorists. In contrast to Boolean algebras, finitely generated free Heyting algebras are infinite, as was shown in the first article on Heyting algebras by McKinsey and Tarski in the 1940s. For one generator, the free Heyting algebra is well understood, but from two generators on, the structure remains mysterious, though many properties are known. With the help of recursively described Kripke models, Bellissima has given a representation of the finitely generated free Heyting algebras F_n as sub-algebras of completions Fc_n of them. Essentially the same construction is due independently to Grigolia.

Our paper offers a concise and readable account of Bellissima’s construction and analyses the situation closer.

Our interest in Heyting algebras comes from model theory and geometry. Our initial questions concerned axiomatisability and decidability of structures like the lattice of Zariski closed subsets of Kⁿ for fields K, which led us rapidly to questions about Heyting algebras. One of the problems with Heyting algebras is that they touch many subjects: logic, topology, lattice theory, universal algebra, category theory, computer science. Therefore there are many different approaches and special languages, which often produce papers that are hard to read for non-insiders. An advantage of our article should be clear proofs and the use mainly of standard mathematical terminology. There is a bit of logic that one might skip if one believes in Bellissima’s theorem; and there are basic model theoretic notions involved in the section about model theoretic results.

Our paper is organised as follows: Section 2 introduces the definitions, basic prop- erties, and reference examples. Section 3 contains an account of Bellissima’s con- struction, a short proof, and results from his article that we are not going to prove.

∗The first author would like to thank the Universit¨at Freiburg for inviting him in July 2008.

∗∗The second author would like to thank the Universit´e d’Angers for supporting him as an invited professor in march 2005, when main parts of this work were done.

MSC 2000:06D20, 03C64, 06B23, 06B30, 08B20

Section 4 analyses the Heyting algebra constructed by Bellissima: we show that it is the profinite completion of the finitely generated free Heyting algebra as well as the metric completion for a naturally defined metric. In Section 5, we reconstruct the Kripke model as the principal ideal spectrum and prove that the Zariski topology on this spectrum is induced by the partial ordering. Section 6 shows the Kripke model to be first order interpretable, from which several model theoretic and al- gebraic properties for dense sub-algebras of Fc_n follow. For example, we show the set of generators to be ∅-definable, and we determine automorphism groups. We solve questions of elementary equivalence, e.g. we prove that no proper sub-algebra of F_n is elementarily equivalent toF_n, and thatF_n is an elementary substructure of Fc_n iff both algebras are elementarily equivalent. And we settle some questions about irreducible elements:F_n contains the same meet-irreducible elements asFc_n; we characterise the join-irreducible elements, we show that there are continuum many of them in Fc_n and that the join-irreducibles of F_n remain join-irreducible in Fc_n. Finally, Section 7 collects open problems and miscellaneous considerations.

Some of the properties we isolated were known before, some were published after we started this work in 2004. We added references where we were able to do so, and apologise for everything we have overlooked. Though not all the results are new, the proofs might be, and the way of looking at the problem is hopefully interesting.

This paper is closely related to [DJ2]; both complement each other. When we started to study finitely generated Heyting algebras, we did it in two ways: on the one hand by analysing Bellissima’s construction, on the other hand by analysing the notion of dimension and codimension in dual Heyting algebras. Many insights were obtained by both approaches, but some features are proper to the free Heyting algebras, others hold for a much wider class than just the finitely generated Heyting algebras. Therefore, we decided to write two papers: this one, which collects results that follow more or less directly from Bellissima’s construction, and the paper [DJ2], which analyses the structure of Heyting algebras from a more geometric point of view.

Acknowledgements

We would like to thank Guram Bezhanishvili for many helpful remarks.

2 Basic facts about Heyting algebras

2.1 Definitions and notations

Under a lattice we will always understand a distributive lattice with maximum 1 and minimum 0, join (union) and meet (intersection) being denoted by t and u respectively. A Heyting algebra is a lattice where for every a, b there exists an element

a→b := max x

xua=bua .

This is expressible as a universal theory, the theory THA of Heyting algebras, in the languageLHA={0,1,u,t,→}with constant symbols 0,1 and binary function symbolsu,t,→.

Note that everything is definable from the partial ordering

avb :⇐⇒ a=aub ⇐⇒ b=atb ⇐⇒ a→b= 1

In a poset (X,6), we callya successorofxifx < yand there is nozwithx < z < y.

Analogously forpredecessor.

If Λ = (Λ,0,1,u,t,v) is a lattice, then the dual lattice Λ^∗:= (Λ,1,0,t,u,w) is the lattice of the reversed ordering. Thus, in the dual of a Heyting algebra, for alla, b there exists a smallest elementb−awith the propertyat(b−a) =atb. Dual Hey- ting algebras are the older siblings of Heyting algebras; they were bornBrouwerian algebras in [McT], they appear under the nametopologically complemented lattices in [Da], and are often calledco-Heyting algebras nowadays. A lattice is abi-Heyting algebra (or double Brouwerian algebra in [McT]) if itself and its dual are Heyting algebras.

We definea↔bas (a→b)u(b→a), which is dual to the “symmetric difference”

aMb:= (a−b)t(b−a).

2.2 Examples

• The open sets of a topological spaceX form a complete¹ Heyting algebraO(X) with the operations suggested by the notations, i.e.aub=a∩b,atb=a∪b, avb ⇐⇒ a⊆b, and

a→b = a\b^{ = (a^{∪b)^◦.

(Here,^{ denotes the complement, topological closure and^◦ the interior.) We call such an algebra atopological Heyting algebra.

• If (X,6) is a partial ordering, then the increasing sets form a topology and hence a Heyting algebraO^↑(X,6), and the decreasing sets, which are the closed sets of O^↑(X,6), form a topology and Heyting algebra O_↓(X,6). Hence such an algebra is bi-Heyting, and both infinite distributive laws hold.

• The propositional formulae in κ propositional variables², up to equivalence in the intuitionistic propositional calculus, form a Heyting algebra IPLκ. It is freely generated by the (equivalence classes of the) propositional variables, hence iso- morphic to thefree Heyting algebraF_κ overκgenerators.

• Ifπ:H →H⁰ is a non-trivial epimorphism of Heyting algebras, then thekernel π⁻¹(1) is a filter. Conversely, if Φ is an arbitrary filter inH, then≡_Φdefined by x≡_Φy:⇐⇒ x↔y∈Φ is a congruence relation such that Φ =π⁻¹_Φ (1) for the canonical epimorphismπ_Φ:H→H/≡_Φ. In particular,≡_{1} is equality.

3 Bellissima’s construction

Bellissima in [Be] has constructed an embedding of the free Heyting algebra F_n into the Heyting algebra O_↓(Kn) for a “generic” Kripke model Kn of intuitionistic propositional logic in n propositional variables P1, . . . , Pn. We fix n > 0 and this fragment IPLn of intuitionistic logic, and we will give a short and concise account of Bellissima’s construction and proof.

We start with some terminology: we identify the set Val_n of all valuations (assign- ments) of the propositional variables with the power set of {P₁, . . . , P_n}, namely

1I.e.complete as a lattice. Note that in general only one of the infinite distributive laws holds in topological Heyting algebras, namelyauF

i∈Ibi=F

i∈I(aubi).

2For these “intuitionistic formulae”, the system of connectives{⊥,∧,∨,→} is used, and the following abbreviations:>:=⊥ → ⊥,¬A:=A→ ⊥,A↔B:= (A→B)∧(B→A).

a valuation with the set of variables to which it assigns “true”. A Kripke model K = (K,6,val) for IPLn consists of a reflexive partial order³ 6 on a set K and a function val : K → Valn satisfying the following monotonicity condition: If Pi ∈ val(w) for w ∈ K, written w Pi, and if w⁰ 6 w, then w⁰ Pi. Validity of formulae at a point w is then defined by induction in the classical way for the connectives ⊥, ∨ and ∧, and for ϕ → χ by the condition: w⁰ ϕ ⇒ w⁰ χ for all pointsw⁰∈Kwith w⁰6w. By induction, validity of all intuitionistic formulae obeys the monotonicity condition. It follows from this definition that the map

ϕ 7→ [[ϕ]] := {w∈K|wϕ}

induces a homomorphism of Heyting algebras from IPLn to O↓(K,6). The kernel of this morphism is thetheory of the model: all formulae valid at every point of the model. Thetheory of a point wconsists of all formulae valid atw. (See e.g. [Fi] for more details.)

A Kripke model is reduced if any two points differ either by their valuations or by some (third) point below. Precisely: there are no two distinct pointsw1, w2with the same valuation and (1) such thatw6w1 ⇐⇒ w6w2 for allw or (2) such that w₁ is the unique predecessor of w₂. (One can show that a finite model is reduced if and only if two distinct points have distinct theories.) One can reduce a finite model by applying the following two operations:

• identify points with same valuation and same points below;

• delete a point with only one predecessor, if both carry the same valuation;

and one can check by induction that these reductions do not change the theory of the model. (This is well known in modal logic: it is a special case of a bisimulation, see [BMV].) Therefore it follows:

Fact 3.1 (Lemma 2.3 of [Be]) For every finite model of IPLn, there is a reduced finite model with the same theory.

3.1 The construction of the generic Kripke model K

The idea of the construction ofK_nis to ensure that all finite reduced models embed as an initial segment. (Kn,6) will be a well-founded partial ordering of rankωand Kn will be an increasing union of Kripke models K^d_n = (K_n^d,6,val). We defineK^d_n by induction ondas follows (cf. Figure 1):

• We let K_n⁻¹ =∅. ThenK_n^d\K_n^d−1 consists of all possible elements w_β,Y such that:

◦ Y is a decreasing set inK_n^d−1 andY 6⊆K_n^d−2

(ford= 0 the last condition is empty, thereforeY =∅);

◦ β is a valuation in Val_n such thatβ⊆val(w⁰) for all pointsw⁰ ∈Y;

◦ ifY is the decreasing set generated by an elementw_β⁰_,Y⁰, thenβ6=β⁰.

• The valuation ofwβ,Y is defined to beβ.

• The partial ordering onK_n^d−1 is extended toK_n^d by w6w_β,Y :⇐⇒ (w∈Y orw=w_β,Y).

In particular, one sees that by construction everyK_n^d is finite,K_n^d is an initial part ofK_n^d+1, and K_n^d is the set of points of Kn of foundation rank6d. One can check that K^d_n is the maximal reduced Kripke model of foundation rankdfor IPLn.

3Note that the order is reversed with respect to the usual approach to Kripke models. This is for the sake of an easy description and to be in coherence with the order of the Heyting algebra, see Remark 4.4, and the order on the spectrum, compare with Fact 5.3.

D D D D D D D D D D D D D DD

... ...

K_n⁰ K_n¹\K_n⁰

K_n²\K_n¹ ...

K_n^d\K_n^d−1 K_n^d+1\K_n^d

...

K_n^d

w_∅,∅

r · · ·

wβ,Y

r

B

B B

B B

B B

B B B

B B

B B

B B

B B Y

Figure 1: The construction ofKn

Theorem 3.2 (Bellissima in [Be]) The map ϕ 7→ [[ϕ]] = {w ∈ K_n | w ϕ}

induces an embedding of IPL_n, and hence of the free Heyting algebra F_n with a fixed enumeration of nfree generators, into the Heyting algebra

Fc_n := O_↓(Kn,6).

This map identifies a set ofnfree generators ofF_n with the propositional variables P1, . . . , Pn that were used in the construction of Kn. We will see in Corollary 6.3 that there is only one set of free generators ofF_n, therefore the embedding is unique up to the action of Sym(n) on the free generators.

Proof⁴: We have to show that the homomorphism is injective, which amounts to show that the theory of Kn consists exactly of all intuitionistic tautologies. Each finite reduced Kripke model embeds by a straightforward induction onto an initial segment of (Kn,6). Because validity of formulae is preserved under “going down”

along6, the theory ofKn is contained in that of all finite reduced models. On the other hand, as intuitionistic logic has the finite model property, a non-tautology is already false in some finite model, hence also in some finite reduced model. Thus the theory ofKn consists exactly of the intuitionistic tautologies.

For Grigolia’s version of this construction see e.g. [Gr2] or the account in [Bz] which offers a wider context.

From now on, we will identify F_n with its image in Fc_n. Thus the free generators become [[P₁]], . . . ,[[P_n]], and the operations can be computed in the topological Heyt- ing algebra Fc_n as indicated in Example 2.2. Moreover, we speak offinite elements ofF_n or Fc_n meaning elements which are finite subsets ofKn.

3.2 Results from [Be]

In this section we collect all results from [Be] that we are going to use.

We call an element ain a latticet-irreducible if it is different from 0 and can not be written as a union b1 tb2 with bi 6= a. It is completely t-irreducible, or F

- irreducible for short, if it can not be written as any proper union of other elements

4The proof is essentially Bellissima’s: we have simplified notations, separated the general facts from the special situation and left out some detailed elaborations, e.g. a proof of Fact 3.1.

(possibly infinite, possibly empty), thus if and only if it has a unique predecessor a⁻ :=F

{x| x < a}. The dual notions apply for u. In particular, au-irreducible element is by convention different from 1, and ad

-irreducible has a unique successor a⁺:=d

{x|a < x}.

For X ⊆ Kn, we let X_↓ be the 6-decreasing set and X^↑ the 6-increasing set generated byX.⁵ThusX_↓andX^↑{are both open inO↓(K_n,6) and hence elements of Fc_n. (Note that X_↓ and X^↑ are the closures of X in the topologies O^↑(K_n,6) and O_↓(Kn,6) respectively.) We call a set{w}_↓ forw∈Kn aprincipal set and a set {w}^↑{ aco-principal set (see Figure 2).

D D D D D D D D D D D

. . . . . . . .

. . . . . ...

wr

--- -- -- -- -- -- -- -- -- -- --

A

A A

A A

A A

A A

A A

A {w}↓

D D D D D D D D D D D

. . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . ...

wr ..

.

...

---- ---- ---- ---- ---- ---- ---- ---- ----

--- --

-- -- -- -- -- -- -- -- -- -- -- --

---

D D D D D D D D D D D

A A A A A A A A

A A A A A A A A

{w}^↑{

Figure 2: A principle set{w}_↓ and a co-principle set{w}^↑{

The following theorem and its corollary are main results of Bellissima⁶. We follow his proof except that we simplify notations and arguments and that we get shorter formulae.⁷Of course, an empty disjunction stands for the formula⊥and an empty conjunction for>.

Theorem 3.3 For everyw∈Kn, there are formulaeψw andψ_w⁰ such that[[ψw]] = {w}↓ and[[ψ⁰_w]] ={w}^↑{. They can be defined by induction on the foundation rank of was follows. If Y_max denotes the maximal elements of Y, then

ψw_β,Y := _

w∈Y_max

ψ⁰_w ∨ _

Pi∈β/

Pi

→ _

w∈Y_max

ψw

∧ ^

P_i∈β

Pi

ψ_w⁰_β,Y := ψw_β,Y → _

w∈Y_max

ψw

It follows that if w has foundation rank d, then the implication depth of ψ_w is at most 2d+ 1 and of ψ_w⁰ at most 2d+ 2.

Proof: Let wβ,Y be of foundation rank d+ 1. We assume the proposition to be shown for all points of smaller foundation rank inKn, and conclude by induction.

By induction, [[W

w∈Y_maxψw]] =Y (ifd=−1, thenY =∅, and everything works as well). Nowψw_β,Y is intuitionistically equivalent to a conjunction of three formulae that define the following subsets of Kn:

5Because we are working here with the reversed order, the arrows are the other way round compared to Bellissima.

6Lemma 2.6, Theorem 2.7 and Corollary 2.8 in [Be]; our Corollary 3.4 (b) is implicit in Bellis- sima’s Lemma 2.6.

7This is mainly because Bellissima’sϕ1is implied by the second conjunct of hisϕ2.

A₁ := [[ _

w∈Y_max

ψ⁰_w → _

w∈Y_max

ψ_w]] = n w

∀v6w

v∈ \

z∈Y_max

[[ψ⁰_z]]^{ ∪ [

z∈Y_max

[[ψ_z]] o

=

w

∀v6w Y ⊆ {v}_↓ or v∈Y

⊆ B1 :=

w

{w}_↓∩K_n^d =Y ∪ Y A₂ := [[_

P_i∈β/

P_i → _

w∈Y_max

ψ_w]] = n w

∀v6w v∈ \

P_i∈β/

[[P_i]]^{ ∪ [

z∈Y_max

[[ψ_z]] o

=

w

∀v6w val(v)⊆β or v∈Y and A₃ := [[^

P_i∈β

P_i]] =

w∈K_n

β⊆val(w) .

One sees that Y ⊆A_i∩K_n^d for all i, andA₁∩K_n^d =Y. Thus [[ψ_wβ,Y]]∩K_n^d =Y. Moreover, if w∈ [[ψw_β,Y]]\K_n^d, then w has the following property: for all v 6w, either v ∈Y, or val(v) =β (from A₂ and A₃) and {v}_↓∩K_n^d =Y (fromB₁). By construction ofK_n, there is only one such point, namely w_β,Y.

Then [[ψ⁰_w

β,Y]] is by definition the largest decreasing set contained in [[ψ_wβ,Y]]^{∪ [[W

w∈Y_maxψ_w]] =K_n\ {w_β,Y}, which is exactly {w_β,Y}^↑{. Corollary 3.4

(a) The principal and the co-principle sets are in F_n, hence also all finite sets.

(b) Forw∈Kn, ifa={w}↓ andb={w}^↑{, thenb=a→a⁻. Proof:(b) follows because{w_β,Y}_↓⁻=Y = [[W

w∈Y_maxψ_w]].

From the construction in Theorem 3.2 we will mainly use two properties: The “fil- tration” of the Kripke model into levels of finite foundation rank. And the property that any finite set of at least two incomparable elements in Kn has a common successor without other predecessors. Moreover, we need the following result from [Be]:

Fact 3.5 (Theorem 3.0 in [Be]) (a) The principal sets are exactly the F

-irreducible elements of both algebras, F_n andFc_n.

(b) The co-principal sets are exactly theu-irreducible sets of both algebras.

The theorem in [Be] is formulated forF_n only, but the proof works as well forFc_n. For the sake of completeness, we add a sketch of the proof:

Proof:(a) ForX ∈Fc_nwe haveX =S

w∈X{w}↓. It follows thatXisF

-irreducible, if and only if there is a greatest elementw0 inX, if and only ifX ={w0}↓. (b) If there are two minimal elements w0, w1 ∈ Kn\X, then X = (X∪ {w0})∩ (X∪ {w1}) is notu-irreducible. Conversely,{w}^↑{ has a unique successor, namely {w}^↑{∪ {w}, hence{w}^↑{ isd

-irreducible.

Let F^{^}_n be the sub-Heyting algebra of F_n generated by all F-irreducible elements ofF_n. ThusF^{^}_n=B_n in Bellissima’s notation in [Be].

Fact 3.6 (Theorem 4.4 in [Be]) For n > 1, the algebra F^{^}_n is not finitely gen- erated, because the sub-algebra generated by all {w}_↓ forw of foundation rank 6d can’t separate points of Kn of higher foundation rank that differ only by their valu- ations.

In particular, it follows thatF^{^}_nis not isomorphic toF_nforn >1. To our knowledge, Grigolia has shown that no proper sub-algebra of F_n is isomorphic toF_n. Without being explicitly mentioned, it is clear in [Be] thatF^{^}₁=F₁=Fc₁and thatF_n 6=Fc_n forn >1.

Fact 3.7 (Lemma 4.1 in [Be]) Forn >1, there is an infinite antichain in K_n.

4 Some consequences

This section collects some rather immediate consequences from Bellissima’s con- struction that are not explicitly mentioned in Bellissima’s paper, and which we will use in our analysis of Bellissima’s setting. Other consequences are collected in section 7. Many of the results hold in a much wider context, see [DJ2].

4.1 More on irreducible elements

Forain a Heyting algebra, let us define thesupports supp^F(a) :=

xF-irreducible

xva , supp^d(a) :=

xd

-irreducible

avx .

supp^mind (a) := the minimal elements in supp^d(a) Lemma 4.1

(a) Theu-irreducible elements are d

-irreducible in both,F_n andFc_n. (b) For eacha∈Fc_n, we have

a = G

supp^F(a) = l

supp^mind (a), anda6=d

S for every proper subsetS⊂supp^mind (a).

It follows from part (a) of this Lemma and from Fact 3.5 (a) that the (well-founded) partial ordering of K_n equals the partial ordering of the d

-irreducibles, and that of the F-irreducibles. In particular, the (co-)foundation rank of w in K_n equals the (co-)foundation rank of {w}_↓ in the partial ordering of the F-irreducibles and also the (co-)foundation rank of{w}^↑{in the partial ordering of thed

-irreducibles.

Hence all these foundation ranks are finite, and all these co-foundation ranks equal to∞, due to the existence of an infinite chainw < w⁰ < w⁰⁰<· · · inKn. Foundation and co-foundation ranks are used to compute dimensions and co-dimensions, cf.

[DJ2] remark 6.9.

Proof:(a) has already be shown in the proof of Fact 3.5 (b).

(b) For any decreasing set a⊆Kn, we havea=S

w∈a{w}_↓ =T

w /∈a{w}^↑{, which proves the first equality and the second for the full supp^d. But because the order is well-founded on the d

-irreducible elements, it is enough to keep the minimal elements in the support. The last statement is clear by definition.

Corollary 4.2 (a) TheF

-irreducibles andd

-irreducibles of Fc_n and of F_n are the same.

(b) Any element ofFc_n is a (possibly infinite) union as well as a (possibly infinite) intersection of elements of F_n.

A characterisation of the t-irreducible elements can be found in Section 7.2.

Example 4.3 Lemma 4.1 doesn’t work with the maximal elements in supp^F, be- cause the order on the F-irreducible elements is not anti-well-founded. For exam- ple, ifn>2, then there are no maximal elements in supp^F [[Pi]]

. However, forFc_n Corollary 6.10 in [DJ2] provides a decomposition into t-irreducible components.

Proof:Sayi= 2. One can show that for eachk, there are at least two elements in Kn of foundation rank kand with P2in the valuation. For example because there is a copy of the Kripke modelK1inside the pointswofKn withP2∈val(w) —just add P2 to the valuations of the points of K1, i.e. start with the pointsw_{P2},∅ and w_{P1,P₂},∅— andK1is well known to have two points of each foundation rank. Now if w ∈ [[P2]], choose v ∈ [[P2]], v 6= w, of same foundation rank. Then the point w_{{P2},{v,w}↓} shows that {w}↓ is not maximal in supp^F [[P2]]

.

Remark 4.4 We can identify the underlying partially ordered set of the Kripke model Kn with either the F

-irreducibles via w7→ {w}_↓, or with thed

-irreducibles viaw7→ {w}^↑{, both with the order induced by the partial ordervof the Heyting algebraF_n(which is one of the reasons to work with the reversed order on the Kripke model). We can further identify thed

-irreducibles with the principal prime ideals of F_n that they generate, ordered by inclusion. By Lemma 4.1 (a), all principal prime ideals are of that form. The valuations of the Kripke model can also be recovered from F_n, as will be explained in Section 5.

A survey about the relationship between Kripke models and Heyting algebras can be found in the doctoral thesis of Nick Bezhanishvili [Bz].

Remark 4.5 The decomposition a=d

supp^mind (a) of Lemma 4.1 provides a sort of infinite (conjunctive) normal form for elements ofFc_n. Provided the partial order on the d

-irreducibles is known, the Heyting algebra operations can be computed from the supports in a first order way: First note that supp^mind and supp^d can be computed from each other, and then

supp^mind (atb) = the minimal elements in supp^d(a) ∩supp^d(b) supp^mind (aub) = the minimal elements in supp^d(a) ∪supp^d(b) supp^mind (a→b) = the minimal elements in supp^d(b)\ supp^d(a) Note that every set of pairwise incomparable d

-irreducible elements forms the supp^mind of an element ofFc_n, namely of its intersection.

Question 4.6 Can we characterise those sets that correspond to elements ofF_n? Because {w}_↓ ∈ supp^F(a) ⇐⇒ {w}^↑{ ∈/ supp^d(a), one can translate the above rules into computations of the Heyting algebra operations from theF-supports, but they are less nice.

4.2 The completion as a profinite limit

Consider in F_n for eachi the filter (K_nⁱ)^↑={a∈ F_n |K_nⁱ ⊆a} generated byK_nⁱ, and denote the corresponding congruence relation by≡i. Thus

a≡ib ⇐⇒ a∩K_nⁱ =b∩K_nⁱ.

We denote the quotient F_n/≡_i by F_nⁱ and the canonical epimorphism by π_i. It extends to an epimorphism bπ_i : Fc_n → F_nⁱ, where bπ_i⁻¹(1) ={a ∈Fc_n |K_nⁱ ⊆a} is the filter generated byK_nⁱ inFc_n. For simplicity, we denote it and the corresponding congruence relation again by (K_nⁱ)^↑and≡_i and we will identifyFc_n/≡_i withF_nⁱ. F_nⁱ is naturally isomorphic to the finite Heyting algebraO↓(K_nⁱ,6) via “truncation”

πi(a)7→ a∩K_nⁱ (the surjectivity needs part (a) of Corollary 4.2). Again, we will identify both without further mentioning.⁸

Remark 4.7 There is a natural notion of dimension (more precisely: dual codi- mension, see [DJ2] and [Da]) in lattices such that F_nⁱ is the free Heyting algebra over n generators of dimensioni. Among universal algebraists, it is known as the free algebra generated by n elements in the variety of Heyting algebras satisfying P_i+1= 1 (where P_i+1 is defined inductively asx_i+1t(x_i+1→P_i) withP₀= 0).

Proposition 4.8 F_n andFc_n are residually finite, i.e. for each elementa6= 1there is a homomorphism ψonto a finite Heyting algebra such that ψ(a)6= 1. Moreover, ψ can be chosen to be someπi orπbi respectively.

Proof:This is clear from the above considerations and the construction ofKn as

union of the K_nⁱ.

Because K_nⁱ ⊆ K_n^j for i < j, the morphism πi : F_n → F_nⁱ factors through F_n^j for i < j and yields an epimorphism πji : F_n^j → F_nⁱ. The system of maps πji

is compatible, hence the projective limit lim

←−F_nⁱ exists. Because of the universal property of the projective limit and because the system of morphismsπbi:Fc_n→ F_nⁱ is compatible as well, there is a natural morphism Fc_n →lim

←− F_nⁱ.

Fc_n ←- F_n

↓ · · · ^πi+1$

πi

% · · · lim←−F_nⁱ − · · · − F_nⁱ⁺¹ −−

πi+1,i

F_nⁱ − · · ·

Proposition 4.9 Fc_n is the projective limit of the finite Heyting algebras F_nⁱ, in symbols

Fc_n = lim←−

i∈N

F_nⁱ.

Any finite quotient of F_n factors through some F_nⁱ, hence Fc_n is the profinite com- pletion of F_n, i.e. the projective limit of all finite epimorphic images ofF_n. The content of this proposition or parts of it is, in various forms, contained in several articles. To our knowledge, Grigolia was the first to embed F_n into lim

←−F_nⁱ,

8There is a certain ambiguity here that should not harm: AsK_nⁱ is also an element ofF_n, there is a mapF_n→ F_n,a7→a∩Kⁱ_n. The image of this map is a sub-poset (but not a sub-algebra) of F_nisomorphic toF_nⁱ.

see [Gr1] or [Gr3]. More about profinite Heyting algebras can be found in [BGMM]

and [Bz²]. In the first of these two articles, the profinite completion of a Heyting algebra is embedded in the topological Heyting algebra of its dual space, i.e. its prime spectrum.

Proof:The natural morphism Fc_n →lim

←− F_nⁱ is in fact the mapa7→(πbi(a))_i∈N= (a∩K_nⁱ)_i∈N. It is injective by Proposition 4.8 and it is surjective because for any compatible system (ai)_i∈N of decreasing setsai ⊆K_nⁱ, the unionS

i∈Nai is a pre- image inFc_n.

Now let Φ be a filter such that F_n/≡_Φ is finite. Choose an i such that for all elements in F_n/≡Φthere is a pre-image inF_nwith pairwise distinct images inF_nⁱ.

ThenF_n→ F_n/≡Φfactors throughF_nⁱ.

The epimorphismπi+1,i:O_↓(K_nⁱ⁺¹,6)→ O_↓(K_nⁱ,6) is induced from the inclusion (K_nⁱ,6) ,→ (K_nⁱ⁺¹,6). The theorem shows that O_↓ behaves like a contravariant functor, i.e.

Fc_n = O_↓(K_n,6) = O_↓ lim

−→i∈N

(K_nⁱ,6)

= lim

←−i∈N

O_↓(K_nⁱ,6) = lim

←−i∈N

F_nⁱ

Note that any profinite lattice is a complete lattice as projective limit of complete lattices, and that any profinite structure is well known to be a compact Hausdorff topological space (the topology being the initial topology for the projections onto the finite quotients in the defining system, equipped with the discrete topology).

We are going to examine this topology a little further:

Definition Fora, b∈Fc_n, define their distance to be

d(a, b) := 2^{−min{i|bπi(a)6=πbi(b)}} = 2^{−min{i|a∩Kni6=b∩Kni}} with the convention 2^−min∅= 0.

Theorem 4.10 (Fc_n,d)is a compact Hausdorff metric space. The metric topology is the profinite topology, and the Heyting algebra operations are continuous. Fc_n is the metric completion of its dense subset F_n.

This is analogous to properties of thep-adic integersZpas a projective limit of the ringsZ/pⁿZ. See also Theorem 6.1 in [DJ2] for a generalisation to a wider class of Heyting algebras comprising the finitely presented ones.

Proof:It is straightforward to check that d is a metric: symmetry and the ultra- metric triangular inequality d(a, c)6max

d(a, b),d(b, c) are immediate from the definition, and d(a, b) = 0⇐⇒a=bholds by Proposition 4.8.

The profinite topology and the metric topology coincide because they have the same basis of (cl-)open sets πbi−1

(a) = {x| d(x, x0) <2⁻ⁱ⁺¹} for a ∈ F_nⁱ and x0 with πbi(x0) =a.

It is a standard result that a projective limit of compact Hausdorff spaces is again compact and Hausdorff. (Here, this is easily seen directly: Metric topologies are always Hausdorff. If a family of closed sets with the finite intersection property is given, we may suppose the closed sets to be of the form πbi−1(ai). The fip implies that everyiappears only once and that the ai are compatible. Hence (ai)_i∈ω is an element of lim

←−F_nⁱ in the intersection of the family and the topology is compact.) By definition of the profinite topology, the mapsπb_i are continuous. Then the con- tinuity of the Heyting algebra operations on the F_nⁱ (with discrete topology!) lifts

to Fc_n, e.g.u⁻¹ πbi−1

(a)

= (πbi×πbi)⁻¹ u⁻¹(a)

is open. An elementa∈Fc_n is a limit of the sequence (a∩K_nⁱ)i∈ωinF_n. ThereforeFc_n is the metric completion and

thusF_n is dense inFc_n.

As points are closed, we get that any term in the language of Heyting algebras (even with parameters) defines a continuous map.

Remark 4.11 There is a fundamental difference between the cases n = 1 and n >1. In casen= 1, the mapπi+1i:F₁ⁱ⁺¹→ F₁ⁱ has a kernel of 3 elements, but is otherwise injective. It follows that the mapπi:Fc₁→ F₁ⁱ is injective onFc₁\(K_nⁱ)^↑, and thatF₁=Fc₁. In casen >1, the size ofF_nⁱ is growing faster than exponentially withi. Moreover, the mapsπi+1i:F_nⁱ⁺¹→ F_nⁱ are non-injective enough to allow a tree of elementsas₁,...,s_i ∈ F_nⁱ withsj∈ {0,1}such thatπi+1i(as₁,...,s_i+1) =as₁,...,s_i. Hence Fc_n has size continuum and differs fromF_n.

4.3 Dense sub-algebras

Definition We say thatH isdense inFc_nifHis a dense Heyting sub-algebra ofFc_n in the metric topology. By Theorem 4.10, the metric topology equals the profinite topology, hence density of a sub-algebraH ofFc_n means that all the induced maps π_d:H → F_n^d are surjective.

Lemma 4.12 The isolated points in Fc_n are exactly the finite elements. They are dense in Fc_n.

Proof: If a is finite ⊆ K_nⁱ, then there is no other element in the 2⁻⁽ⁱ⁺²⁾-ball around a, hence a is isolated. If a is infinite, then a is the limit of a sequence of finite elements distinct froma, namelya= lim_i∈ω(a∩K_nⁱ). Henceais not isolated,

but a limit of isolated points.

Recall thatF^{^}_n is the sub-Heyting algebra ofF_n generated by allF-irreducible ele- ments ofF_n, i.e. the sub-algebra generated by all finite elements. Thus Lemma 4.12 immediately implies:

Proposition 4.13 F^{^}_n is the smallest dense sub-Heyting algebra ofF_n.

As we have mentioned in Fact 3.6, Bellissima has shown that F^{^}_n is not isomorphic to F_n for n>2. If one knew that F^{^}_n 6=F_n, then Proposition 4.13 would yield an easier proof, becauseF_n then has a proper dense sub-algebra, namelyF^{^}_n, whereas F^_n does not. We will see later thatF^{^}_n6≡ F_n forn>2.

Lemma 4.14 If H is dense in Fc_n, then H has the same F

-irreducibles and the same d

-irreducibles asFc_n.

Proof:Because of the density, all the finite sets are inH. Thus in particular all the principal sets are inH, and as they areF

-irreducible inFc_n, they remain irreducible inH. IfX ∈H is not principal, thenX is the proper union of its supp^F, hence not F-irreducible.

Because of Corollary 4.2 (b),H also contains all thed

-irreducibles ofFc_n, and they remain for trivial reasons d

-irreducible in H. With the same argument as above,

one sees that there are no other d

-irreducible elements in H, because any other

element if the proper intersection of its supp^d.

It follows thatF^{^}_nis also the sub-Heyting algebra generated by thed

-irreducible ele- ments ofFc_n, because any finite set is the intersection of finitely manyd

-irreducible elements, namely its supp^mind . See also section 7.3 for more on F^{^}_n, and Proposi- tion 6.12 of [DJ2] for a generalisation of F^{^}_n to, among others, finitely presented Heyting algebras.

Remark 4.15 Ifn >1 andb1, b2 are two incomparabled

-irreducible elements in F_n, then supp^mind (b1tb2) is infinite. Thus not all elements ofF^{^}_nhave finite supp^mind . This is a reason why an explicit description of F^{^}_n is not as easy as one could first think.

Proof:Ifb_i ={w_i}^↑{, then{w_{∅,{w1,w2,v}↓}}^↑{is in supp^mind (b₁tb₂) for anyv that is incomparable with w₁ or withw₂. Ifn >1, there are infinitely many suchv.

Remark 4.16 There are two canonical sections of the projection map π_i :Fc_n → F_nⁱ:

theminimal section σ^min_i :x 7→ l

y∈Fc_n

π_i(y) =x and themaximal section σ^max_i :x 7→ G

y∈Fc_n

πi(y) =x

In fact, both have images in F^{^}_n: eachσ_i^min(x) is a finite set, thus a finite union of F-irreducibles, and each σ^max_i (x) is what one might call a “co-finite set”, namely a finite intersection ofd

-irreducibles. If one seesx∈ F_nⁱ as a subset of the Kripke modelKⁱ_nforF_nⁱ, then the minimal section mapsxto itself but now seen as a subset of the Kripke modelKn. One can computeσ^min_i (x) =K_nⁱuσ^max_i (x) andσ_i^max(x) = (K_nⁱ →σ^min_i (x)), and one can check thatσ_i^minis a{0,u,t,v}-homomorphism and σ^max_i is a {0,1,u,→,v}-homomorphism.

The quotient F_n⁰ of F_n is isomorphic toP(K_n⁰), which, via the valuation, can be identified with the free Boolean algebraP(P(P₁, . . . , P_n)) over the free generators P1, . . . , Pn. The image of the maximal section ofπ0 consists exactly of the regu- lar elements of F_n. Thus the regular elements form (as a sub-poset, but not as a sub-algebra) a free Boolean algebra over n generators. (This is easy to see with Bellissima’s characterisation of regular elements in Corollary 2.8 of [Be], and much of it is already in [McT].)

5 Reconstructing the Kripke model

5.1 Another duality

The following might be well known in lattice theory. In a (sufficiently) complete lattice, define

a^u := G

{x|a6vx} and a^t := l

{x|x6va}

Lemma 5.1 SupposeΛis a complete lattice that satisfies both infinite distributive laws. Then the maps a 7→ a^u and b 7→ b^t are inverse order-preserving bijections between the F-irreducibles and thed

-irreducibles. Moreover,a6va^u andb^t6vb.

Proof:The maps are order-preserving by definition (and the transitivity ofv). Let abeF

-irreducible and supposeava^u. Thena=aua^u=F

{aux|a6vx}. Then theF

-irreducibility ofaimpliesa=auxfor somex6wa: contradiction. Hencea^u is the greatest element xwith the propertya6vx. It follows thata^u@ata^u, and ifa^u@b, then avb. Thusa^utais the unique successor ofa^u, which therefore is d-irreducible. Since duallya^ut is the minimal elementxwith the propertyx6va^u, and a 6va^u, we get a^ut va. If we had a6va^ut, then a^ut va^u by definition of the latter. But dually to the argument above, a^ut is the smallest element xwith x6va^u: contradiction anda=a^ut. The remaining parts are by duality.

One can reformulate Lemma 5.1 partially as O↓

l-irreducibles,v

∼= O↓

G-irreducibles,v .

Remark 5.2 (1) Asx /∈supp^d(a) ⇐⇒ a6vx ⇐⇒ x^t va by definition ofx^t, it follows that

supp^F(a) = x^t

xd

-irreducible, x /∈supp^d(a) ifa6= 0, and of course the dual statement also holds.

(2) Fc_n as a lattice of sets satisfies the hypotheses of Lemma 5.1. In that special situation, we have that any F

-irreducibleais of the form{w}_↓. The elementa^uis then the corresponding co-principal set{w}^↑{, and we have seen in Corollary 3.4 (b) thata^u=a→a⁻. For ad

-irreducibleahowever, we havea⁺→a=a. Considering Fc_n with its co-Heyting structure, we get the “dual” rulea^t=a⁺−a. (Recall from the Example 2.2 that all partial orders are bi-Heyting.)

5.2 The spectrum

IfH is a Heyting algebra, theideal spectrum Spec_↓(H) is the partially ordered set of all prime ideals ofH endowed with theZariski topology, a basis of which consists of the sets ¯I(a) :={i∈Spec_↓(H)|a /∈i}. The first part of the following fact goes back to Marshall Stone in [St].

Fact 5.3 The mapa7→I(a)¯ defines (functorially) an embedding of Heyting alge- bras

H ,→ O(Spec_↓(H)).

Moreover, ifH is generated byg₁, . . . , g_n, thenSpec_↓(H), partially ordered by inclu- sion, can be turned into a Kripke model ofIPLnby definingval(i) :={Pi|i∈I(g¯ i)}.

In the special case ofF_n, the Kripke modelKnconstructed by Bellissima is naturally isomorphic to the restriction of this construction to the principal ideal spectrum, as we will show in the remaining of this section.

Remark 5.4 There is a bijectioni7→i^{ between the set of prime ideals Spec_↓(H) and the set of prime filters Spec^↑(H) = Spec_↓(H^∗). Therefore there is also an embedding

H ,→ O Spec^↑(H)

a 7→F(a) := {p|pprime filter, a∈p}

where Spec^↑(H) denotes the space of prime filters endowed with the co-Zariski topology, a basis of open sets of which is given by the F(a)’s.

A principal ideal in a lattice is the decreasing set generated by an element x6= 1, which we denote by (x)_↓. A principal ideal is prime if and only if its generator isu- irreducible. Let theprincipal ideal spectrum Spec⁰_↓(H) be the space of all principal prime ideals endowed with the (trace of the) Zariski topology. The continuous in- clusion mapι: Spec⁰_↓(H)→Spec_↓(H) induces an epimorphism of Heyting algebras ι^∗:O(Spec_↓(H))→ O(Spec⁰_↓(H)), given byU 7→U∩Spec⁰_↓(H).

Definition We denote by ¯I0 the mapι^∗◦I¯: H → O Spec⁰_↓(H) . Proposition 5.5 ForH dense inFc_n, we have that

O Spec⁰_↓(H)

= O↓ Spec⁰_↓(H),⊆ ,

that is, the Zariski topology on the principal ideal spectrum is the topology of de- creasing sets.

Proof:By Lemma 4.14,H has the sameF - andd

-irreducibles asFc_n. As the latter satisfies the assumptions of Lemma 5.1, we may use it freely. In fact, the proposition holds more generally for a Heyting algebra satisfying the duality in Lemma 5.1.

By definition of the Zariski topology on the ideal spectrum, the inclusion “⊆” is clear. Conversely, letX be a decreasing set in Spec⁰_↓(F_n) with respect to inclusion.

We have to show that X is open in the Zariski topology. In fact, we are proving X =S I¯0(a^t)

(a)↓∈X :

From a^t 6v a (Lemma 5.1) it follows thata^t ∈/ (a)_↓, i.e. (a)_↓ ∈ I¯0(a^t) and thus

“⊆”. Conversely, let (b)_↓ ∈I¯0(a^t) for some (a)_↓ ∈ X, i.e. a^t ∈/ (b)_↓. This means a^t 6v b, whence (by definition of ^u, see Lemma 5.1) b v a^tu = a. This implies (b)_↓⊆(a)_↓, and becauseX is decreasing, (b)_↓∈X.

Remark 5.6 By Lemma 4.1, in F_n as well as in Fc_n, the u-irreducibles are d - irreducible, and they are the same. Therefore, and with Proposition 5.5,

O Spec⁰_↓(F_n)

= O↓ Spec⁰_↓(F_n),⊆ ∼=naturally O↓ d

-irreducibles ofF_n,v

||

O Spec⁰_↓(Fc_n)

= O_↓ Spec⁰_↓(cF_n),⊆ ∼=naturally O_↓ d

-irreducibles ofFc_n,v In particular, Spec⁰_↓(Fc_n) and Spec⁰_↓(F_n) are naturally homeomorphic. (Note that Spec_↓(cF_n) and Spec_↓(F_n) are not homeomorphic.)

Theorem 5.7 The generic Kripke model Kn is, as a partial order, the principal ideal spectrum of F_n (equivalently ofFc_n) with inclusion. The valuations are deter- mined by the images I¯0(gj) of the free generators gj ofF_n, namely Pj ∈val(i) for some principal ideal iiffg_j∈/i, or equivalently, i∈I¯₀(g_j).

Proof: By Fact 3.5 (a) and Lemma 4.1, an element of F_n or Fc_n is determined by its supp^d, that is by the d

-irreducibles of the algebra. It follows that the map ¯I0 is injective, Together with Proposition 5.5, we get an embedding F_n ,→ O↓ Spec⁰_↓(F_n),⊆

, and the right side is, by the previous remark, naturally isomor- phic toO↓ d

-irreducibles ofF_n,v

. Thed

-irreducibles are of the form{w}^↑{ for w ∈ Kn by Fact 3.5 (a), and v 6 w ⇐⇒ {v}^↑{ v {w}^↑{. This proves the first statement.

A principal prime ideal ofF_nis generated by some{w}^↑{forw∈Kn. As the unique successor of{w}^↑{ is{w}^↑{∪ {w}, an elementxofF_n is not in the ideal generated by{w}^↑{ iffw∈x. Thus the free generatorsgj are mapped on

I¯₀(g_j) = {i∈Spec⁰_↓(F_n)|g_j∈/i} =ˆ {w∈K_n |w∈g_j},

where “ ˆ=” stands for the image under the natural isomorphism. On the other hand the image of g_j is {w ∈ K_n | P_j ∈ val(w)}. This proves the second part of the

theorem.

Remark 5.8 Theorem 5.7 identifies an element a of F_n with supp^d(a), whereas Bellissima’s construction is more easily understood as identifying it with supp^F(a), that is with the underlying embedding

F_n ,→ O Spec^↑_c(F_n) ∼= O↓ F-irreducibles,v ,

where Spec^↑_c(F_n) is the space of “completely prime principal filters”. Algebraically, this space is less natural than the principal ideal spectrum — the lack of complete duality comes from the fact that not all t-irreducible elements are completelyt- irreducible. However, up to the duality of Lemma 5.1, the embedding is the same as in Theorem 5.7. In the light of Remark 5.6, one sees that this duality is nothing else than the order preserving homeomorphism between Spec⁰_↓(F_n) and Spec^↑_c(F_n) mapping a principal prime ideal on its complement, which is exactly the map (a)_↓7→

(a^t)^↑.

6 Some model theory of finitely generated free Heyting algebras

The basic model theoretic notions like elementary equivalence ≡, elementary sub- structure4, definability and interpretability, are explained in any newer model the- ory textbook, see for example [Ho]. “Definable” means definable with parameters, and “A-definable” with parameters in A.

The theory of Heyting algebras has a model completion (in [GhZ], as a consequence of a result by Pitts [Pi]), and there are some results about (un)decidability (see for example [Ry] and [Id]), but otherwise little seems to be known about the model theory of Heyting algebras.

6.1 First order definition of the Kripke model

Theorem 6.1 Fix free generators g1, . . . , gn of F_n. Let H be dense in Fc_n and containingg1, . . . , gn. Then the set{g1, . . . , gn} is∅-definable inH.

Proof: First we note that the partial order (Kn,6) of the Kripke model Kn is

∅-definable in H: the underlying set can be identified with the F-irreducibles of H by Lemma 4.14. It is ∅-definable as the set of those elements having a unique predecessor. They are ordered by the restriction of the partial order ofH. According to Remark 4.4, this order can be identified with (Kn,6).

In the sequel of the proof, we will simply write K_n for the definable set of F- irreducibles of H. We have then a∅-definable injectionH →P(Kn) that maps an