Bi-Gödel algebras and co-trees

(1)

Master Thesis

Bi-G ¨odel algebras and co-trees

Author: Supervisor:

Miguel Martins

dr. Nick Bezhanishvili

Examination date: Daily supervisor:

March 15, 2021

dr. Tommaso Moraschini

(University of Barcelona)

Korteweg-de Vries Institute for Mathematics

(2)

Abstract

Bi-Heyting algebras are algebraic models of the bi-intuitionistic propositional calculus

Bi-IPC. A bi-Heyting algebra is said to be a bi-G ödel algebra if it satisfies G ödel’s prelinearity axiom. The variety bG of all bi-G ödel algebras algebraizes the extension

Bi-LC of Bi-IPC obtained by adding G ödel’s axiom. Because of this, the lattices of axiomatic extensions of Bi-LC and that of subvarieties of bG are dually isomorphic. We study the latter, which, in turn, is amenable to the methods of universal algebra and duality theory. In particular, we prove that the variety bG has the FMP, it is a discriminator variety, and, unlike its HA-reduct, it fails to be locally finite. Moreover, it also has subvarieties lacking the FMP. We then develop theories of Jankov-style formulas for bi-G ödel algebras, and use them to characterize the splitting bi-G ödel algebras. We also show that the lattice of extensions of Bi-LC has the cardinality that of the continuum. We then utilize these formulas to uniformly axiomatize all consistent extensions of the logic Bi-LC, and to characterize the locally finite varieties of bi-G ödel algebras.

Title: Bi-G ¨odel algebras and co-trees

Author: Miguel Martins, miguelplmartins561@gmail.com, 11907304 Supervisor: dr. Nick Bezhanishvili

Second Examiner: dr. Benno van den Berg Examination date: March 15, 2021

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

(3)

First of all, I would like to thank my supervisors, Nick Bezhanishvili and Tommaso Moraschini, for their immense help and patience while guiding me through all the stages of this thesis. Their insightful comments and helpful ideas were always present, and I hope we can work together in the future.

I would also like to thank my family and friends, for all their support, all their love, and for being there when I needed them most.

(4)

CHAPTER

1 Introduction

Bi-Heyting algebras are Heyting algebras whose order duals are also Heyting al-gebras. Much in the same way that Heyting algebras are algebraic models of the intuitionistic propositional calculus IPC, bi-Heyting algebras are algebraic models of the bi-intuitionistic propositional calculus Bi-IPC. Bi-Heyting algebras were first intro-duced by Rauszer in 1971 [31], where they were termed semi-Boolean algebras. Other

terminologies include Heyting-Brouwer algebras, and double Heyting algebras. Sub-sequently, Rauszer devoloped in a series of publications [32,33,34,35,36,37,38,39]

the foundations of the theory of bi-Heyting algebras. In the years that followed, some notable contributions to this field were made. Esakia [16] proved the (essential to

this thesis) duality between bi-Heyting algebras and bi-Esakia spaces, while K ¨ohler [24] characterized the filters of a bi-Heyting algebra that give rise to its congruences,

and constructed a subdirectly irreducible bi-Heyting algebra which is not simple. Beazer [2] used K ¨ohler’s work to fully characterize the subdirectly irreducible

bi-Heyting algebras. Sankappanavar [41] generalized the notion of a bi-Heyting algebra

to that of a dually pseudocomplemented Heyting algebra, that is, a Heyting algebra equipped with a co-negation operation∼, the dual of the Heyting negation ¬. More recently, Bezhanishvili et al. [6] established new bi-topological and spectral dualities

for bi-Heyting algebras.

Bi-Heyting algebras also arise naturally in some non-algebraic fields of research. For example, the set of opens of an Alexandrov space can always be viewed as a bi-Heyting algebra (see Example 2.7.7), and so can the lattice of subgraphs of an

arbitrary graph (see, e.g., [43]). Many other examples can be found, especially in the

field of Category and Topos Theory: in [26,27], Lawvere highlights how the language

of bi-Heyting algebras can be used in the study of toposes; while in [40], Reyes and

Zolfaghari study bi-Heyting toposes, i.e., toposes in which lattices of subobjects are bi-Heyting algebras, and use both negations (¬ and ∼) to define modal operators on these lattices; and in [12], Doring shows how to associate a complete bi-Heyting

algebra to an arbitrary quantum system.

In the previously mentioned publications, Rauszer also introduced and studied

Bi-IPC, the logic algebraized by bi-Heyting algebras, then termed the Heyting-Brouwer logic. Bi-IPC is the conservative extension of IPC obtained by adding a new binary logical connective←to the language, called the co-implication (or exclusion,

(7)

or subtraction), which behaves dually to →. Hence Bi-IPC achieves a symmetry, which IPC lacks, between all its connectives∧,>,→and∨,⊥,←, respectively. The Kripke semantics of Bi-IPC, also developed by Rauszer [36], proved to be helpful in

the (intuitive) interpretation of the new connective←: given a Kripke model M, a point x in M, and formulas φ, ψ, then

M, x|= φ←ψ ⇐⇒ ∃y≤ x(M, y|=φand M, y6|=ψ).

Equipped with this new connective, the language of Bi-IPC achieves a far greater expressivity than that of IPC. For example, if the points of a Kripke frame are interpreted as states in time, then the semantics of Bi-IPC is strong enough to talk about the past, a feature that IPC lacks. With this example in mind, Wolter [44]

extends G ¨odel’s embedding of IPC into S4 to an embedding of Bi-IPC into tense-S4. In this paper, amongst other things, Wolter also proves a version of the Blok-Esakia Theorem (stating that the lattice of extensions of Bi-IPC is isomorphic to the lattice of extensions of tense-S4 containing Grz.t), that every variety of bi-Heyting algebras axiomatized by ∨-free formulas has the finite model property (FMP for short), and that the only splitting bi-Heyting algebras are the two-element and three-element chain.

For some recent developments in the study of Bi-IPC see, e.g., [1,19,20,42].

The latticeΛ(IPC)of intermediate logics (i.e., consistent axiomatic extensions of

IPC) has been thoroughly investigated (see, e.g, [10]). On the other hand, the lattice

Λ(Bi-IPC)of bi-intermediate logics (i.e., consistent extensions of Bi-IPC), lacks such an in-depth investigation. This is perhaps because Λ(Bi-IPC)has a far more complex structure than Λ(IPC). For example: unlike the Heyting case, the characterization of the subdirectly irreducible bi-Heyting algebras is more involved, and not very easy to work with. A usual approach to solve difficult general problems is to start with a less complex particular case, allowing one to develop tools and techniques in a simpler setting. This is exactly what we will do in this thesis. We shall study a simpler, yet non-trivial, sublattice of Λ(Bi-IPC): the lattice of (consistent) extensions of the bi-intuitionistic linear calculus (or the bi-G ¨odel-Dummett’s logic),

Bi-LC :=Bi-IPC+ (p→q) ∨ (q→ p).

The logic LC := IPC+ (p → q) ∨ (q → p) and its lattice of extensions are very well-understood, and Bi-LC appeared to be a good candidate for our starting point. This choice proved to be fruitful, as it gave rise to a lattice with a rich and complex, yet understandable, structure. Moreover, the logic Bi-LC has some characteristics that make it an interesting object of study by itself. We give two extra motivations for studying this system.

Firstly, the bi-Heyting algebras (respectively, the subdirectly irreducible bi-Heyting algebras) which satisfy the G ¨odel axiom (p→ q) ∨ (q→ p)are shown (in Chapter 3) to be exactly those whose bi-Esakia duals are co-forests (respectively, co-trees). Consequently, Bi-LC is the bi-intuitionistic logic of co-trees. Moreover, we prove that the bi-intuitionistic logic of chains is proper extension of Bi-LC. Compare this to the intuitionistic case, where the logic of chains, LC, coincides with that of co-trees. Thus, from the point of view of IPC, the class of co-trees is indistinguishable to that of chains (note that this suggests that the language of Bi-IPC is more appropriate to study tree-like structures).

(8)

Secondly, the logic Bi-LC admits a form of a classical inconsistency lemma. These types of lemmas were formalized and subsequently studied by Raftery in [30]. Given

a deductive system`, we say that`has a classical inconsistency lemma if, for each n ∈ ω, there exists a finite set of formulas on n-variables, Ψn, which satisfies the

equivalence

Σ∪Ψn(α1, . . . , αn)is inconsistent in ` ⇐⇒ Σ` {α1, . . . , αn},

for all sets of formulas Σ∪ {α1, . . . , αn}. For CPC, this lemma is usually written as:

(i) Σ∪ {α} `_CPC ⊥ ⇐⇒ Σ`_CPC ¬α;

(ii) Σ∪ {¬α} `CPC ⊥ ⇐⇒ Σ`CPC α.

While condition (i) holds for all (bi-)intermediate logics, this is not the case for (ii). Raftery proves that the only intermediate logic satisfying a condition of the form of (ii) is CPC. However, things change in the bi-intuitionistic case. While Bi-IPC still does not admit a lemma of this kind, there are some bi-intermediate logics, distinct from CPC, which do. Bi-LC is one of them, as shown by the following equivalence (which is proven in Chapter 4 of this thesis):

Σ∪ {∼ ¬ ∼α} `Bi-LC⊥ ⇐⇒ Σ`Bi-LC α.

In this thesis, we shall study co-trees, their bi-intuitionistic logic Bi-LC, and varieties of bi-G ¨odel algebras (the bi-Heyting algebras which algebraize this logic). Although they are axiomatized by the same axiom (over their respective sublogics), the more complex structure of the lattice of extensions of Bi-LC, compared to that of

LC, becomes apparent as soon as we start our study of the former. It is well-known [10] that a Heyting algebra satisfying LC is subdirectly irreducible if and only if its

Esakia dual is a strongly rooted bounded chain, while we show that the subdirectly irreducible bi-G ¨odel algebras are exactly those whose bi-Esakia duals are co-trees. Furthermore, while the variety axiomatized by LC is locally finite [21], hence all of

its subvarieties have the FMP, this is far from the case for bG; not only do we show that the variety of bi-G ¨odel algebras fails to be locally finite, but also that it admits subvarieties without the FMP. Finally, the lattice Λ(LC)is well-known [10] to be a

chain of order type ω+1 , whereas we prove thatΛ(Bi-LC)has the cardinality of the continuum, and that this lattice has a far more complicated structure than that of a chain.

Equipped with the algebraic properties of bi-G ¨odel algebras, the bi-Esakia duality, and the algebraic completeness of Bi-IPC, we will start by studying the variety

bG := {A: A|=Bi-LC}

(or, equivalently, the logic Bi-LC) and its subvarieties (or, equivalently, the extensions of Bi-LC). We prove the results stated above regarding this variety, and that bG is a discriminator variety and has the FMP. We also construct a particular subvariety of bG which lacks the FMP, and find an axiomatization of the variety generated by the linear Heyting algebras. We then define and study four types of Jankov-style formulas for bi-G ¨odel algebras, three of them algebraic in nature, the Jankov, the stable canonical, and the subframe formulas, and one using frame-theoretic methods,

(9)

the bi-subframe formulas. By adjusting the standard methods (as in [7]) to this

setting, we use Jankov formulas to characterize the splitting logics of the lattice of extensions of Bi-LC (equivalently, we characterize the splitting bi-G ¨odel algebras) and to determine the cardinality of said lattice, while the stable canonical formulas allow us to find a uniform axiomatization of all extensions of Bi-LC (equivalently, a uniform axiomatization of all subvarieties of bi-G ¨odel algebras). As for the subframe and bi-subframe formulas (which are proven to be Bi-LC-equivalent), we use them to axiomatize some particular extensions of Bi-LC (e.g., the logic of co-trees of depth less than n, for each positive n ∈ ω) and to establish a criterion for the existence

of an order-embedding from a given finite co-tree into an arbitrary Kripke frame. Lastly, we characterize the locally finite varieties of bi-G ¨odel algebras. To derive this characterization, we use the subframe formulas of a particular class of co-trees, the finite combs (finite posets whose shape resembles that of a comb), and two essentital results, which are proven in Chapter 5.

To summarize, the main contributions of this thesis to the theories of bi-Heyting and bi-G ¨odel algebras are:

• A characterization of a particular class of varieties of bi-Heyting algebras which have the FMP. Namely, the class of varieties of bi-Heyting algebras which are axiomatized by ←-free formulas, and whose Heyting algebra reducts are locally finite;

• The construction of a variety of bi-G ¨odel algebras which does not have FMP; • The development of the theory of Jankov, stable canonical, subframe, and

bi-subframe formulas of bi-G ¨odel algebras, and the subsequent use of these theories to:

(i) characterize the splitting logics ofΛ(Bi-LC); (ii) show that|Λ(Bi-LC)| =2ℵ0_;

(iii) derive a uniform axiomatization of all extensions of Bi-LC; (iv) characterize the locally finite varieties of bi-G ¨odel algebras.

This thesis is structured as follows. Chapter 2 is an overview of the necessary algebraic and logical background, also including a characterization of the bi-Esakia spaces. In Chapter 3, we study varieties of bi-Heyting and bi-G ¨odel algebras, their simple and subdirectly irreducible elements, and discuss some fundamental properties of these varieties. In Chapter 4, we develop the theory of Jankov, stable canonical, subframe, and bi-subframe formulas for bi-G ¨odel algebras, and present some uses of these formulas. (We note that the first four sections of this chapter follow closely the structure of [7], while Section 4.5 was heavily inspired by [8]). In Chapter 5, we

(10)

CHAPTER

2

Bi-Heyting algebras

In this chapter, we provide an overview of the basic concepts and results that we will need throughout the thesis. We define the bi-intuitionistic propositional calculus

Bi-IPCand its axiomatic extensions, and discuss how to interpret them using Kripke frames. We then give a brief introduction to universal algebra and to the theory of Heyting algebras. After this preliminary part, we finally introduce bi-Heyting algebras, present some of their algebraic properties, and we highlight how they algebraize the bi-intuitionistic propositional calculus. Because of this, the lattice of axiomatic extensions of Bi-IPC is dually isomorphic to that of varieties of bi-Heyting algebras. The chapter ends with a characterization of the order-topological duals of bi-Heyting algebras, the bi-Esakia spaces, and with some useful results that arise from this duality.

2.1 Notation and conventions

We assume that the reader is familiar with the very basics of Set Theory (basic operations on sets, the cardinality of a set, etc. See, e.g, [25]), Topology (what is a

basis for a topology, when is a topological space compact, etc. See, e.g., [14]), Category

Theory (what is a functor, what is a duality of categories, etc. See, e.g., [28]), and of

propositional and first order logic (how to define a formula, what are the subformulas of a formula, the Łos Theorem on ultraproducts, etc. See, e.g., [13]).

Let A and B be sets. We denote their disjoint union by A]B, and the powerset of A by P (A). Given a map f : A→ B and subsets A0 ⊆ A, B0 ⊆B, we write

f[A0]:= {b∈ B : ∃a∈ A0 (f(a) =b)}and f−1B0 := {a ∈ A : f(a) ∈B0}. If A0 6= A, A0 is said to be proper.

Given a binary relation R on A (i.e., R⊆ A2_{), we use aRb and}_¬(_aRb₎_{instead of}

(a, b) ∈ R and (a, b) ∈/ R, respectively. If R is a partial order (i.e., R is a reflexive, transitive, and anti-symmetric binary relation on A), the pair(A, R)is called a poset. We denote the set of maximal elements of a subset U⊆ A by

(11)

and if U has a maximum (i.e., a greatest element), we denote it by Max(U). Similarly, we define min(U)and Min(U). A point a∈ A is an upper bound of U if uRa, for all u∈U. If U has a least upper bound, we denote it byW

U and call it the supremum of U. Similarly, we say that b∈ A is a lower bound of U if bRu, for all u ∈U, and if there exists a greatest lower bound of U, we denote it byV

U and call it the infimum of U. We say that U is an upset of(A, R)if it contains all the points lying above it (i.e., for all u ∈U and a∈ A, uRa implies a∈U). We denote the upset generated by U (i.e., the least upset of(A, R)containing U) by

R[U]:= {a∈ A :∃u∈U (uRa)},

or by ↑U, when R is understood. If U = {u}is a singleton, we simply write R(u), or ↑u when R is understood, and we call ↑u a principal upset. Similarly, D ⊆ A is a downset of (A, R) if it contains all the points lying below it, and we denote the downset generated by D (i.e., the least downset of(A, R)containing D) by R−1[D]or by↓D. If D= {d}is a singleton, we simply write R−1(d)or↓d, and call↓d a principal downset. If D is both an upset and a downset, we call it an updownset. We denote the set of upsets of (A, R)by U p(A, R), the set of downsets by Do(A, R), and the set of updownsets by U pDo(A, R). Given two distinct points a, b ∈ A, we say that a is an immediate predecessor of b, denoted by a≺b, if aRb and no point of A lies between them, i.e., if c ∈ A is such that aRcRb, then either a = c or c=b. If this is the case, we call b an immediate successor of a.

We usually represent the set of natural numbersN= {0, 1, 2, . . .}by the ordinal

ω. Given n ∈ω, when we write i≤ n we always mean i∈ {0, . . . , n}. Let A be a set

(or a class). Given a map g : AI → A, where I is some index set, we say that A is closed under g, or that A is g-closed. When I is finite, g is said to be a finitary operation on A. In particular, when I =n, for n∈ω, we call g an n-ary operation on A. Given a

unary operation (i.e., a 1-ary operation) g on A, we define recursively, for all a∈ A: (i) g0(a):=a;

(ii) gm+1₍_a₎_:₌_{g g}m₍_a₎_{, for m}_∈_ω_.

LetX = (X, τ, R)be an ordered-topological space, i.e., a topological space equipped with a partial order. We denote the set of open sets, closed sets, and clopen sets of X by Op(X ), Cl(X ), and Cp(X ), respectively. Moreover, we use the notation CpU p(X )for the set of clopen upsets, CpDo(X )for the set of clopen downsets, and CpU pDo(X )for the set of clopen updownsets ofX. Similarly, we denote the set of closed upsets ofX by ClU p(X ), the set of closed downsets by ClDo(X ), and the set of closed updownsets by ClU pDo(X ).

2.2 Bi-intuitionistic propositional logic

Let Prop := {p0, p1, . . .} be a denumerable set of propositional variables, and L :=

{Prop;∨,∧,→,←;⊥}a propositional language, where{∨,∧,→,←}are logical connec-tives and ⊥ is a propositional constant. Form(L)(or simply Form) is the set of all well-formed formulas in the languageL, defined as usual. In what follows, we assume that p, q, r, . . . range over Prop, while φ, ψ, ϕ, . . . range over Form(L). We use the following abbreviations:

(12)

2.2. Bi-intuitionistic propositional logic

• ¬φ := φ→⊥,

• φ↔ψ := (φ→ψ) ∧ (ψ→φ),

• >:= ¬ ⊥, • ∼φ := > ← φ.

Definition 2.2.1. The bi-intuitionistic propositional calculus Bi-IPC is the smallest set of formulas containing the axioms:

1. (p→q) → (q→r) → (p →r), 2. p→ p∨q, 3. q→p∨q, 4. (p→q) → (r →q) → (p∨r→q), 5. p∧q→ p, 6. p∧q→q, 7. (p→q) → (p →r) → (p→q∧r), 8. p→ (q→r) → (p∧q→r), 9. (p∧q→r) → p → (q→r), 10. p∧q→ (¬q→ ¬p), 11. p→ q∨ (p←q), 12. (p←q) → ∼(p→q), 13. (p←q) ←r → (p←q∨r), 14. ¬(p←q) → (p→ q), 15. p→ (q←q)→ ¬p, 16. ¬p→ (p→ (q←q), 17. (p→ p) ←q → ∼q, 18. ∼q→ (p→ p) ←q,

and closed under the following inference rules: • Modus Ponens (MP): from φ and φ→ψ, infer ψ;

• Uniform Substitution (US): from φ(p1, . . . , pn)infer φ(ψ₁, . . . , ψn);

(13)

We define the intuitionistic propositinal calculus IPC as the subset of formulas of

Bi-IPCin the sublanguage L0 _:_{= {}_Prop;_∨_,_∧_,_→_;_⊥}_{. One way to define the classical}

propositional calculus CPC is to add the law of excluded middle (p∨ ¬p)to the list of axioms of Bi-IPC. Note that, in the setting of CPC, the new connective←can now be given in terms of the old ones, by

(φ←ψ) ↔ (φ∧ ¬ψ).

Moreover, the DN rule becomes superfluous in CPC, as it translates to “from φ infer

φ”. Note as well that we have Bi-IPC(CPC, by the definition of CPC and since, for example, (p∨ ¬p)∈/Bi-IPC.

The intuitionistic analogue of the following definition is well-known; just replace L by L0, disregard the pre-fix “bi-” everywhere, and only consider the first two inference rules.

Definition 2.2.2. A set of formulas L ⊆ Form(L)closed under the three inference rules is a bi-intermediate logic if Bi-IPC ⊆L⊆CPC. If we only have that Bi-IPC⊆ L, we call L a super-bi-intuitionistic logic.

Given a formula φ and a super-bi-intuitionistic logic L, we say that φ is a theorem of L, denoted by L `φ, if φ∈ L. If φ is not a theorem of L, we write L0φ. We call L

consistent if L 0 ⊥, and inconsistent otherwise. Given another super-bi-intuitionistic logic L0, we say that L0 is an extension of L if L⊆L0.

Finally, given a set of formulasΣ, we denote the least (with respect to inclusion) bi-intuitionistic logic containing L∪Σ (in other words, the smallest extension of L that contains Σ) by L+Σ. Such extension always exists. If Σ is a singleton {φ}, we

simply write L+φ. Given another formula ϕ, we say that φ and ϕ are L-equivalent if

L` φ↔ ϕ. Equivalently, L+φ=L+ϕ.

Let L be a super-bi-intuitionistic logic. Note that since⊥ → p∈Bi-IPC, it follows from the US rule that L is inconsistent iff L = Form(L). It can also be proven that if L is consistent, then it is a bi-intermediate logic. Since all bi-intermediate logics are super-bi-intuitionistic logics by definition, this discussion shows that the terms “consistent super-bi-intuitionistic logic” and “bi-intuitionistic intermediate logics” are

equivalent.

2.3 Kripke semantics of Bi-IPC

In what follows, we work mainly with Bi-IPC but most of the definitions and all the results we provide can be applied to IPC by disregarding everything concerning ←and only considering formulas in Form(L0). When this is not the case, we will explicitly say so, and provide the appropriate definition.

Definition 2.3.1. A Kripke frame is a non-empty poset F = (W, R). Given a map V : Prop → P (W), we call the pair M = (F, V)a Kripke model (on F), or simply a model (on F), if V satisfies the persistence condition:

∀p∈Prop,∀w∈W,∀v∈ ↑w w∈V(p) =⇒ v∈V(p).

In this case, V is said to be a valuation on F. We can rephrase the persistence condition as V[Prop] ⊆U p(F), or V : Prop→U p(F).

(14)

2.3. Kripke semantics of Bi-IPC

If M, w |= φ holds, we say that w satisfies φ, or φ is true at w, and we sometimes

simply write w|= φ, when M is understood. On the other hand, if M, w|=φdoes

not hold, we say that w does not satisfy φ, and denote this by M, w 6|=φ. Moreover, if

we have a set of formulasΣ such that M, w|=φfor all φ∈Σ, we write M, w|= Σ.

Definition 2.3.3. Let F be a Kripke frame, M a model on F, K a class of Kripke frames, and φ ∈Form(L). We say that:

• φ is true in M, denoted by M|= φ, if M, w|=φfor all w∈W;

• φ is valid in F, denoted by F |=φ, if(F, V) |=φfor all valuations V on F;

• φ is valid in K, denoted by K |=φ, if F|= φfor all F∈K.

Definition 2.3.4. Let K be a class of Kripke frames. For F∈ K, we denote the logic of Fby

Log(F):= {φ∈ Form(L): F|=φ},

and the logic of K by

Log(K):=\{Log(F): F∈K}.

Remark 2.3.5. It can be easily shown that for every Kripke frame F, we always have

Bi-IPC ⊆Log(F), and since F6|=⊥by the definition of |=, it follows that Log(F)is a bi-intermediate logic. Similarly, given a class of Kripke frames K, Log(K)is again a bi-intermediate logic.

Next we define the three fundamental methods of obtaining new Kripke frames and models from old ones, and present the truth-preserving properties of these operations.

Definition 2.3.6. Let F= (W, R) and F0 = (W0, R0)be Kripke frames. Then F0 is a bi-generated subframe of F, denoted by F0 F, if W0 is both an upset and a downset of F, and R0 = R∩W02. For the intuitionistic definition of a generated subframe, we drop the requirement “W0 is a downset of F”.

If M= (F, V)and M0 = (F0, V0)are models, then M0 is a (bi-)generated submodel of M, denoted by M0 M, if F0 is a (bi-)generated subframe of F and V0 = VF0,

(15)

Definition 2.3.7. Let F = (W, R) and F0 = (W0, R0) be Kripke frames. A map f : W →W0 is a bi-p-morphism from F to F0, denoted by f : F→F0, if it satisfies the following conditions:

• Order-preserving: ∀w, v∈W wRv =⇒ f(w)R0f(v);

• Forth: ∀w∈W,∀v0 ∈W0 f(w)R0v0 =⇒ ∃v∈ ↑w(f(v) =v0); • Back: ∀w∈W,∀u0 ∈W0 u0R0f(w) =⇒ ∃u∈ ↓w(f(u) =u0).

For the intuitionistic definition of a p-morphism, just disregard the back condition. If f is surjective, we write f : FF0, and call F0 a (bi-)p-morphic image of F, denoted by FF0.

Let M = (F, V) and M0 = (F0, V0) be Kripke models. A map f : W → W0 is called a (bi-)p-morphism from M to M0, denoted by f : M → M0, if f : F → F0 is a (bi-)p-morphism and we have

w∈V(p) ⇐⇒ f(w) ∈V0(p),

for all w ∈ W and p ∈ Prop. If f is surjective, we write f : M M0, and call M0 a (bi-)p-morphic image of M, denoted by MM0.

Definition 2.3.8. Let{F_i}_i∈I be a collection of Kripke frames, where Fi = (Wi, Ri), for

i∈ I. We define their disjoint union as the Kripke frameU

i∈IFi := (

U

i∈IWi, R), where

R is the relation given by

wRv ⇐⇒ ∃i∈ I w, v∈Wi and wRiv.

If, for each i ∈ I, we have a valuation Vi on Fi, we define the disjoint union of the

models Mi := (Fi, Vi)as the model

U

i∈IMi := (

U

i∈IFi, V), where V is the valuation

satisfying

w∈ V(p) ⇐⇒ ∃i∈ I w∈Wi and w ∈Vi(w),

for all p ∈Prop.

The following two results are well-known when restricting φ to Form(L0).

Theorem 2.3.9. Let{M, M0} ∪ {M_i}_i_∈_Ibe a collection of Kripke models and φ∈ Form(L). We have:

(i) If w0 ∈M0 and M0 M, then M0, w0 |=φ if and only if M, w0 |=φ.

(ii) If w∈Mand f : MM0, then M0, f(w) |=φ if and only if M, w|= φ.

(iii) If w∈M_jfor some j ∈ I, thenU

i∈IMi, w|= φ if and only if Mj, w|=φ.

Proof. (Sketch) All three proofs are done by a simple induction on the complexity of

φ. For (i) and (ii), the argument is the same as the one for the well-known analogous

result for intuitionistic propositional logic, except for the case involving ←. Here is exactly where the new conditions on the definitions of bi-generated subframes and bi-p-morphisms come into play. As for (iii), everything follows directly from the definition of the valuation of a disjoint union of models.

(16)

2.4. Universal algebra

Corollary 2.3.10. Let {F, F0} ∪ {F_i}_i∈I be a collection of Kripke frames and φ∈ Form(L).

We have:

(i) If F0 F, then F|= φ implies F0 |=φ.

(ii) If FF0, then F|= φ implies F0 |=φ.

(iii) U

i∈IFi |=φ if and only if Fj |=φ for all j ∈ I.

Proof. This is an immediate consequence of the the previous theorem, using the definition of validity in a Kripke frame.

2.4 Universal algebra

In this section, we provide an introduction to universal algebra. For a more in-depth look at this subject, and for the proofs of all the results in this section, which we omit, see [9].

Definition 2.4.1. A signature (or type) of algebras is a setF of function symbols, such that each element f ∈ F has an arity n∈ ω. If f ∈ F has arity n, we call it an n-ary

function symbol. In particular, if n=0, we call f a constant, and if n=1 or n =2, we call f a unary or binary function symbol, respectively.

An algebra A of type F is a pair (A, F), where A is a non-empty set (called the universe of A) and F is a collection of finitary operations on A indexed by F, such that for each n-ary function symbol f ∈ F, we have an n-ary operation fA _{on A}

corresponding to f (if there is no fear of confusion, sometimes we drop the superscript A). A is said to be finite if|A| ∈ω, and trivial if|A| =1.

Definition 2.4.2. LetF be a type of algebras and X a set of objects (disjoint fromF), which we call variables. We define the terms of typeF over X as the elements of T(X), the smallest set satisfying:

(i) X∪ {f ∈ F: f is a constant} ⊆T(X);

(ii) If t1, . . . , tn∈ T(X)and f ∈ F is n-ary, then f(t1, . . . , tn) ∈T(X).

We denote these terms by p(x1, . . . , xn)(where the variables occurring in our term

are among the x1, . . . , xn), or simply by p.

Definition 2.4.3. Given a term p(x1, . . . , xn)of typeF over X and an algebra A of

said type, the term function on A corresponding to the term p is the map pA_{: A}n _→ _A,

defined recursively as follows:

(i) if p is a variable xi ∈X, then for a1, . . . , an∈ A we set

pA(a1, . . . , an):= ai;

(ii) if p is of the form f p1(x1, . . . , xn), . . . , pm(x1, . . . , xn), for some m-ary f ∈ F,

then for a1, . . . , an∈ A we set

(17)

Definition 2.4.4. Let A be an algebra of type F and X a set of variables. A map v : X→ A is said to be a valuation on A. Moreover, any such valuation can be extended to one v : T(X) →A, by setting

v p(x1, . . . , xn) := pA v(x1), . . . , v(xn),

for all p(x1, . . . , xn) ∈T(X).

Definition 2.4.5. Given two terms p and q of typeF over X, we call an expression of the form p≈q an equation. If A is an algebra of typeF, we say that an equation p≈q is true in A, or p≈q holds in A, or A satisfies p≈ q, if for all valuations v on A, we have

v(p) =v(q).

We denote this by A|= p≈q. If p≈q is not true in A, we write A6|=p≈ q. Moreover, if Σ is a set of equations, we say A satisfies Σ if A satisfies every equation in Σ, in which case we write A|=Σ.

We can generalize the satisfiability relation|=and apply it to classes of algebras of the same type: given a class K and an equation p ≈q, K|= p≈q if p≈ q is true in every algebra in K, and for a set of equationsΣ, K |=Σ if all algebras in K satisfy Σ.

Definition 2.4.6. Let A and B be algebras of the same typeF. A map g : A→B is a homomorphism from A to B, denoted by g : A→B, if it preserves the operations of our type, i.e., for every n-ary f ∈ F and all a1, . . . , an∈ A, we have

g fA(a1, . . . , an)

= fB g(a1), . . . , g(an).

If g is injective, we write g : A,→Band call it an embedding. In this case, we say that Aembeds into B and write A,→B. If g is surjective, we write g : AB, and say that B a homomorphic image of A, denoted by A B. Finally, if g is bijective, we call it an isomorphism, denoted by g : A −→∼ B, and we say that A and B are isomorphic. We denote this by A∼=B.

Definition 2.4.7. Let A and B be algebras of the same type. A is a subalgebra of B if A⊆ B and for every function symbol f , fA_{is the restriction of f}B_{to A.}

If C ⊆ B, then hCi denotes, if it exists, the least subalgebra of B containing C, which we call the subalgebra of B generated by C. If C = {c1, . . . , cn}is finite, we

also use the notation hc1, . . . , cni and say that hCi is n-generated. If an algebra is

n-generated, for some n∈ω, we say that it is finitely generated.

Proposition 2.4.8. Let A and B be algebras of the same type. Then A embeds into B if and

only if A is isomorphic to a subalgebra of B.

Definition 2.4.9. Let A and B be algebras of the same type. If A embeds into B, we write A≤ B, and given an embedding g : A,→ B, we denote the subalgebra of B with universe g[A]by g[A].

Definition 2.4.10. Let A be an algebra of type F and θ an equivalence relation on A (i.e., a binary relation on A that is reflexive, symmetric, and transitive). Then θ is a congruence on A if for every n-ary f ∈ F and all a1, . . . , an, b1, . . . , bn ∈ A, the

following compatibility condition holds: ∀i∈ {1, . . . , n} aiθbi

(18)

2.4. Universal algebra

In other words, an equivalence relation θ is a congruence on A if it preserves fA_{, for}

all f ∈ F.

The quotient algebra A/θ is the algebra of typeF with universe A/θ := {_JaK_θ: a ∈ A},

the set of equivalence classes of θ (we sometimes omit the subscript and simply write JaK, when θ is understood), and whose operation are defined as follows: for every n-ary function symbol f ∈ F and all a1, . . . , an∈ A,

fA/θ _Ja1K, . . . , JanK :=q f

A₍_a

1, . . . , an)y.

Example 2.4.11. Every algebra A has two notable congruences: the trivial one,∇:= A2; and the identity, ∆ := {(a, a): a∈ A}. If these are the only congruences on A, and they are distinct, we say that A is simple.

Proposition 2.4.12. Let A and B be algebras of typeF. If θ is a congruence on A, then A/θ is a homomorphic image of A. Conversely, given a surjective homomorphism g : AB, then there is a congruence θg on A such that B∼=A/θg.

Remark 2.4.13. Note that the previous correspondence between the congruences of an algebra A and its homomorphic images is order-reversing, in the sense that if we have two congruences θ ⊆θ0 on A, then there exists a surjective homomorphism from

A/θ0 to A/θ. Conversely, if we have g : A B and h : B D(so D is “smaller” than B), then θg⊆θ_h◦g.

Proposition 2.4.14. Let Con(A)denote the set of all congruences of an algebra A. Then (Con(A),⊆)is a bounded lattice (see Definition2.6.1), called the lattice of congruences of A.

Definition 2.4.15. An algebra A is congruence-distributive if its lattice of congruences is a bounded distributive lattice (see Definition2.6.3).

Definition 2.4.16. Let{A_i}_i∈I be a set of algebras of type F. Their (direct) product

∏i∈IAi is the algebra of type F with universe ∏i∈IAi, and whose operations are

defined as follows: for every n-ary f ∈ F, i∈ I, and all a1, . . . , an∈∏i∈IAi,

f∏i∈IAi₍_a

1, . . . , an)(i):= fAi a1(i), . . . , an(i),

If I = {1, . . . , m}, for some m ∈ ω, we also write∏_i∈IAi = A1× · · · ×Am, and if

I = ∅, then ∏_i_∈_IA_i is the trivial algebra.

Definition 2.4.17. Let∏_i∈IAi be a product of algebras of the same type. For j∈ I, we

call the map

πj:

∏

i∈I

Ai → Aj

(ai)i∈I 7→ aj

the j-th projection map.

Proposition 2.4.18. Let ∏i∈IAi be a product of algebras of the same type. For every j∈ I,

(19)

Proposition 2.4.19. Let {A, B} ∪ {A_i}_i∈I be a set of algebras of type F, g : A → B a

homomorphism from A to B, θ ∈Con(A), p(x1, . . . , xn)and q(x1, . . . , xn)terms of typeF

over X, v a valuation on A, and a1, . . . , an, c1, . . . , cn∈ A. The following conditions hold:

(i) If a1θc1, . . . , anθcn, then pA(a1, . . . , an)θ pA(c1, . . . , cn);

(ii) g pA(a1, . . . , an)= pB g(a1), . . . , g(an);

(iii) g◦v is a valuation on B;

Remark 2.4.20. We will refer to (iv), (v), and (vi) in the previous proposition by “truth in an algebra is preserved under taking subalgebras, homomorphic images and products”.

Definition 2.4.21. Let I be a set and U⊆ P (I). Then U is an ultrafilter on I if U is a maximal ∩-closed proper upset of(P (I),⊆).

Given a set {A_i}_i∈I of algebras of the same type F and U an ultrafilter on I,

we define the ultraproduct A := _∏_i_∈_IA_i/U as the algebra of type F with universe ∏i∈IAi/U, where the equivalence relation U on∏i∈IAi (in fact, this is a congruence

on ∏_i∈IAi) is given by q (ai)i∈I y U = q (bi)i∈I y U ⇐⇒ {j∈ I : aj =bj} ∈U,

and whose operations are defined as follows: for every n-ary f ∈ F andJ(ai)i∈IK∈ ∏i∈IAi/U, fA J(ai)i∈IKU := q (fAi₍_a i))i∈I y U.

Definition 2.4.22. An algebra A is a subdirect product of a family of algebras{A_i}_i∈I

(all of the same type) if A is a subalgebra of ∏_i∈IAi, and for each i ∈ I, we have

πi[Ai] = A.

An embedding g : A,→∏_i_∈_IA_i is said to be subdirect if g[A]is a subdirect product of {A_i}_i∈I.

Definition 2.4.23. An algebra A is subdirectly irreducible (SI for short) if for every subdirect embedding g : A,→_∏_i_∈_IA_i, there exists a j∈ I such that πj◦g : A

∼

−→A_j is an isomorphism.

Remark 2.4.24. We note that since the trivial algebra is a subdirect product of the empty family, subdirectly irreducible algebras are non-trivial.

Theorem 2.4.25. Let A be an algebra. Then A is subdirectly irreducible if and only if

(Con(A)\{∆},⊆)has a least element. Consequently, simple algebras are always subdirectly irreducible.

Theorem 2.4.26. (Birkhoff) Every algebra A is isomorphic to a subdirect product of subdirectly

(20)

2.5. Varieties

2.5 Varieties

This section is a short summary of the concepts of the theory of varieties that we will need throughout the thesis. For a more in-depth look at this subject, and for the proofs of the results in this section (except for Theorem2.5.9), which we omit, see [9].

In this section, K always denotes a class of algebras of the same type.

Next we introduce some particular maps between classes of algebras (all of the same type), which we call class operators.

Definition 2.5.1.

• H(K):= {B: B is a homomorphic image of some A∈K}, • S(K):= {B: B is a subalgebra of some A∈K},

• P(K):= {B: B is a product of a family of algebras of K}, • I(K):= {B: B is isomorphic to some A∈ K},

• Pu(K):= {B: B is an ultraproduct of a family of algebras of K},

• KF := {B∈ K : B is finite},

• KSI := {B∈K : B is SI},

• KFSI := {B∈ K : B is finite and SI}.

Definition 2.5.2. A class V of algebras of the same type is called a variety if it closed under homomorphic images, subalgebras, and products (i.e.,H(V),S(V),P(V) ⊆V).

We denote the smallest variety containing K by Var(K), and call it the variety generated by K. If K = {A}, we simply write Var(A).

Theorem 2.5.3. (Tarski) The variety generated by K is given by Var(K) =HSP(K). The next theorem (which follows immediately from Theorem2.4.26) shows why

we are interested in characterizing the SI elements of a given variety.

Theorem 2.5.4. If V is a variety, then V is generated by its subdirectly irreducible elements,

i.e., V=Var(VSI).

Theorem 2.5.5. (Birkhoff) K is a variety if and only if it is equationally definable (i.e., there

exists a set of equationsΣ such that K= {A: A |=Σ}).

Definition 2.5.6. Let V be a variety. We say that:

• V is semi-simple if every subdirectly irreducible algebra in V is simple; • V is finitely generated if V =Var(K), for some finite set K of finite algebras; • V is locally finite if every finitely generated algebra in V is finite;

• V has the finite model property (FMP for short) if it is generated by its finite algebras, i.e., V=Var(VF);

(21)

• V is congruence-distributive if every algebra in V is so;

• V has the congruence extension property (CEP for short) if for every A ∈ V and every subalgebra B of A, if θ ∈ Con(B)then there exists θ0 ∈ Con(A)such that

θ = θ0∩B2. We also use this definition when V is an arbitrary class of algebras,

not necessarily a variety;

• V has equationally definable principal congruences (EDPC for short) if there exists a conjunction of finitely many equationsΦ(x, y, z, v)on four variables, such that for every algebra A in V and all a, b, c, d ∈ A, we have

cθa,bd ⇐⇒ A|= Φ(a, b, c, d),

where θa,b is the least congruence on A identifying a and b;

• V is a discriminator variety if there exists a discriminator term t(x, y, z)for V, i.e., a ternary term of the type of V such that for every A∈ VSI and all a, b, c ∈ A, we

have

tA(a, b, c) = (

c if a= b, a if a6=b.

Proposition 2.5.7. Let V be a variety and K a class of algebras. The following conditions

hold:

(i) If V is finitely generated, then it is locally finite; (ii) If V is locally finite, then it has the FMP; (iii) V has the FMP if and only if V=Var(VFSI);

(iv) If V is a discriminator variety, then it has EDPC and is semi-simple; (v) If V has EDPC, then it has the CEP;

(vi) If K has the CEP, thenHS(K) =SH(K);

(vii) If Var(K)is congruence-distributive, then Var(K) =HSP(K) =PHSPu(K).

Conse-quently, Var(K)_SI⊆HSPu(K).

Remark 2.5.8. We note that (vii) in the previous proposition is usually referred to as J ´onsson’s Lemma.

For a proof of the next theorem, which characterizes locally finite varieties of a finite type, see [3].

Theorem 2.5.9. If V is a variety of a finite type, then the following conditions are equivalent:

(i) V is locally finite;

(ii) ∀n∈ ω,∃m(n) ∈ω,∀A∈V A is n-generated =⇒ |A| ≤m(n);

(22)

2.6. Distributive lattices and (co-)Heyting algebras

2.6 Distributive lattices and (co-)Heyting algebras

This section is a short summary of the theory of bounded distributive lattices and (co-)Heyting algebras, focusing on some key results about this structures that we will need in what follows. For a more in-depth look at these algebras, see [11,17].

Definition 2.6.1. A bounded lattice is a poset(L,≤)with a greatest element 1, a least element 0, and such that every two elements a, b ∈ L have a least upper bound, denoted by a∨b, and a greatest lower bound, denoted by a∧b. We call∨ the join operation and a∨b the join of a and b, while∧is called meet operation and a∧b the meet of a and b.

Bounded lattices can be defined in a purely axiomatic way, as shown by the following proposition. For a proof, see [11].

Proposition 2.6.2. Let L= (L,∨,∧, 0, 1)be an algebra of typeF := (∨,∧,⊥,>), where ∧and∨ are binary function symbols, while⊥ and> are constants. Then L is a bounded lattice if and only if for all a, b, c∈ L, we have:

• Idempotency law: a∨a= a and a∧a =a,

• Commutativity law: a∨b=b∨a and a∧b= b∧a,

• Associativity law: a∨ (b∨c) = (a∨b) ∨c and a∧ (b∧c) = (a∧b) ∧c, • Absorption law: a∨ (b∧a) =a and a∧ (b∨a) =a,

• Neutral elements: a∨0=a and a∧1=a.

We use Hasse diagrams to depict bounded lattices (or posets), as exemplified in Figure 2.1. The elements of a lattice are depicted as points. If x<y, then x appears lower than y (e.g., 0<1) in our diagram. If x ≺y, we draw a line segment between x and y (e.g., 0 ≺a).

0

a b

1

Figure 2.1: A bounded lattice

Definition 2.6.3. A bounded lattice L is distributive if it satisfies the following distribu-tivity laws, for all a, b, c ∈L:

• a∨ (b∧c) = (a∨b) ∧ (a∨c), • a∧ (b∨c) = (a∧b) ∨ (a∧c).

We denote the category of bounded distributive lattices and their homomorphisms by DL.

(23)

Definition 2.6.4. Let L be a bounded distributive lattice and F a non-empty subset of L. We say that F is a filter of L if it is a∧-closed upset of L. We denote the set of filters of L by Filt(L). Moreover, if F 6=L and for all a, b∈ L, a∨b∈ F implies a∈ F or b∈ F, we call F a prime filter. We denote the set of prime filters of L by L∗.

Closely related to the concept of a filter is that of an ideal.

Definition 2.6.5. Let L be a bounded distributive lattice and I a non-empty subset of L. We say that I is an ideal of L if it is a∨-closed downset of L. We denote the set of ideals of L by Id(L). Moreover, if I 6=L and for all a, b∈ L, a∧b∈ I implies a∈ I or b∈ I, we call I a prime ideal.

Remark 2.6.6. Given a prime filter F of a bounded distributive lattice L, it is easy to see that the complement of F, L\F, is a prime ideal of L, and vice-versa.

The next theorem is well-known. For a proof, see [11]. Note that this proof makes

crucial use of Zorn’s Lemma.

Theorem 2.6.7. (Prime Filter Theorem) Let L be a bounded distributive lattice, F a filter, and

I an ideal. If F∩I = ∅, then there exists a prime filter F0 such that F⊆ F0 and F0∩I =∅.

Corollary 2.6.8. Let L be a bounded distributive lattice and a, b∈ L. If ab, then there exists a prime filter F such that a ∈F and b /∈ F.

Definition 2.6.9. LetX = (X, τ,≤)be an ordered-topological space. We say thatX is a Priestley space if it is compact and satisfies the Priestley Separation Axiom (PSA for short):

• PSA:∀x, y∈ X xy =⇒ ∃U ∈CpU p(X ) (x∈U and y /∈U).

We denote the category of Priestley spaces and Priestley morphisms, i.e., order-preserving continuous maps, by PRI.

Theorem 2.6.10. (Priestley) The categories DL and PRI are dually equivalent.

This is a well-known result. For a detailed proof, see, e.g. [11]. For clarity, we

define the contravariant functor Φ : DL→ PRIgiving rise to this dual equivalence. It maps an object L of DL toΦ(L) = (L∗, τ,⊆), where (L∗,⊆)is the poset of prime

filters of L, ordered by inclusion. As for the topology τ, it will be the one generated by the sub-basis

{ϕ(a): a∈ A} ∪ {L∗\ϕ(a): a∈ A},

where ϕ(a):= {F∈ L∗: a∈ F}. We call L∗ := Φ(L)the Priestley dual of L. We note

that by the definition of τ, the clopen upsets of L∗ are exactly the subsets of the form

ϕ(a), for a∈ A, while the clopens are of the form

n

[

i=1

ϕ(ai) \ϕ(bi),

for a1, . . . , an, b1, . . . , bn ∈ L and n ∈ ω. Given a morphism f : L → L0 ∈ DL, our

functor sends it to the Priestley morphism f∗: L0∗ →L∗ defined by f∗(F):= f−1F, for

(24)

2.6. Distributive lattices and (co-)Heyting algebras

Conversely, the inverse ofΦ maps a Priestley spaceX to X∗ := (CpU p(X ),∪,∩,∅, X),

which we call the distributive lattice dual of X, and sends a Priestley morphism f : X → X0 _{to f}∗_: _X0∗ _{→ X}∗_{defined by f}∗₍_U₎_:₌ _f−1_{U, for U}_∈_{CpU p}_(X0₎_.

We note that L and (L_∗)∗ are isomorphic as bounded distributive lattices, and thatX and(X∗)∗ are isomorphic as Priestley spaces (by which we mean there exists

an invertible Priestley morphism between them). For an example, see Figure 2.2.

L L∗

Figure 2.2: The lattice L and its dual Priestley space L∗

Definition 2.6.11. A Heyting algebra is a bounded distributive lattice A such that, for every a, b∈ A, there exists an element a→b∈ A satisfying, for every c∈ A:

c≤ a→b ⇐⇒ a∧c≤ b.

We call→the Heyting implication, and for every a∈ A, we define¬a :=a →0. Again, we can also give the previous definition axiomatically, as shown by the next proposition. For a proof, see [22].

Proposition 2.6.12. Let A= (A,∨,∧,→, 0, 1)be an algebra of typeF := (∨,∧,→,⊥,>), such that(A,∨,∧, 0, 1)is a bounded distributive lattice. Then A is a Heyting algebra if and only if→is a binary operation satisfying, for all a, b, c∈ A:

(i) a→a=1,

(ii) a∧ (a→b) =a∧b, (iii) b∧ (a→b) =b,

(iv) a→ (b∧c) = (a→b) ∧ (a→c).

We use HA to denote the category of Heyting algebras and their homomorphisms, while HA denotes the class of Heyting algebras. From Proposition2.6.2and

Proposi-tion2.6.12, it follows that this class is equationally definable. Thus, HA is in fact a

variety, by Theorem2.5.5. We note that in the setting of Heyting algebras, we use = to

interpret the symbol≈.

Definition 2.6.13. A Boolean algebra is a Heyting algebra A satisfying the equation p∨ ¬p≈ >, that is, we have

a∨ ¬a =1,

(25)

Next we list some properties of Heyting algebras that will be useful throughout. We will refer to this result as “the Properties of Heyting algebras”. For a proof, just dualize the proofs of the analogous properties in Proposition2.7.3.

Proposition 2.6.14. Let A be a Heyting algebra and a, b, c ∈ A. We have: 1. a→b=W{d∈ A : a∧d≤ b}, 2. a∧ ¬a=0, 3. a≤b ⇐⇒ a→b=1, 4. a≤b =⇒ ¬b≤ ¬a, 5. a=0 ⇐⇒ ¬a=1, 6. ¬(a∨b) = ¬a∧ ¬b, 7. a≤ ¬¬a, 8. ¬¬¬a= ¬a, 9. a∧ (b→c) =a∧(a∧b) → (a∧c), 10. a∧b=0 ⇐⇒ b≤ ¬a.

The next theorem gives us a characterization of the SI Heyting algebras. For a proof, see [17].

Theorem 2.6.15. A Heyting algebra A is subdirectly irreducible if and only if it has a second

largest element.

Definition 2.6.16. A Priestley spaceX is an Esakia space if it satisfies the condition ∀U ∈Cp(X ) ↓U∈Cp(X ).

IfX = (X, τ, R)andX0 = (X0, τ0, R0)are Esakia spaces and f : X → X0 is a Priestley morphism, we say that f is an Esakia morphism, or a p-morphism (between Esakia spaces), if it satisfies the forth condition, i.e.,

∀x∈ X,∀y0 ∈ X0 f(x)R0y0 =⇒ ∃y∈ ↑x(f(y) =y0).

We denote the category of Esakia spaces and Esakia morphisms by ESA.

The following theorem is attributed to Esakia [15] and is usually treated as folklore.

For a detailed proof, see [17].

Theorem 2.6.17. (Esakia) RestrictingΦ : DL → PRIto HA yields a duality of categories between HA and ESA.

(26)

2.7. Bi-Heyting algebras

Definition 2.6.18. A co-Heyting algebra is a bounded distributive lattice A such that for every a, b∈ A, there exists an element a←b∈ A satisfying, for every c∈ A:

a←b≤c ⇐⇒ a≤b∨c.

We call← the Heyting co-implication, and for every a ∈ A, we define∼a := 1 ← a. Moreover, we denote the category of co-Heyting algebras and their homomorphisms by coHA.

Example 2.6.19. Every Heyting algebra A gives rise to a co-Heyting algebra by dualizing its order (i.e., by defining a ≤op b iff b ≤a, for all a, b∈ A), and vice-versa.

Again, we can also give the previous definition axiomatically. For a proof, just dualize the proof of Propostion2.6.12. In fact, all the results in the remainder of this

section can be obtained by dualizing the proofs of the analogous results for Heyting algebras.

Proposition 2.6.20. Let A= (A,∨,∧,←, 0, 1)be an algebra of typeF := (∨,∧,←,⊥,>) such that(A,∨,∧, 0, 1)is a bounded distributive lattice. Then A is a co-Heyting algebra if and only if←is a binary operation satisfying, for every a, b, c∈ A:

(i) a←a=0, (ii) a∨ (a←b) =a, (iii) b∨ (a←b) =a∨b,

(iv) (a∨b) ←c= (a←c) ∨ (b←c).

Theorem 2.6.21. A co-Heyting algebra A is subdirectly irreducible if and only if it has a

second least element.

Definition 2.6.22. A Priestley spaceX is a co-Esakia space if it satisfies the condition ∀U ∈Cp(X ) ↑U∈Cp(X ).

If X = (X, τ, R) and X0 = (X0, τ0, R0) are co-Esakia spaces and f : X → X0 is a Priestley morphism, we say that f is a co-Esakia morphism if it satisfies the back condition, i.e.,

∀x∈ X,∀z0 ∈X0 z0R0f(x) =⇒ ∃z∈ ↓x (f(z) =z0).

We denote the category of co-Esakia spaces and co-Esakia morphisms by coESA.

Theorem 2.6.23. The categories coHA and coESA are dually equivalent.

2.7 Bi-Heyting algebras

In this section, we define bi-Heyting algebras and discuss many of their algebraic properties. We also characterize the normal filters of an arbitrary bi-Heyting algebra A, and make the connection between these filters and the congruences on A explicit.

(27)

Definition 2.7.1. A bounded distributive lattice A is a bi-Heyting algebra if it is both a Heyting and a co-Heyting algebra, i.e., for every a, b∈ A, there are elements a→b, a←b∈ A satisfying, for each c∈ A:

c≤ a→b ⇐⇒ a∧c≤b and

a←b≤c ⇐⇒ a≤b∨c.

Recall that we define ¬a := a→0 and∼a :=1←a, for every a∈ A.

As with the previous algebraic structures, bi-Heyting algebras can also be defined in a purely axiomatic way. This is shown by the following proposition.

Proposition 2.7.2. Let A = (A,∨,∧,→,←, 0, 1) be a universal algebra of type F := (∨,∧,→,←,⊥,>), such that(A,∨,∧, 0, 1)is a bounded distributive lattice. Then A is a bi-Heyting algebra if and only if→and←are binary operations satisfying, for all a, b, c∈ A:

(i) a→a=1, (ii) a∧ (a→b) =a∧b, (iii) b∧ (a→b) =b, (iv) a→ (b∧c) = (a→b) ∧ (a→c), (v) a←a=0, (vi) a∨ (a←b) =a, (vii) b∨ (a←b) =a∨b, (viii) (a∨b) ←c= (a←c) ∨ (b←c).

Proof. This follows directly from the definition of a bi-Heyting algebra, together with Proposition2.6.12and Proposition2.6.20.

We use bHA to denote the category of bi-Heyting algebras and their homomor-phisms, while the class of bi-Heyting algebras is denoted by bHA. By Theorem2.5.5,

it follows that this class is in fact a variety, as it is equationally definable, by the previous proposition and Proposition2.6.2.

Next we list some useful properties of bi-Heyting algebras. We will refer to this proposition as “the Properties of bi-Heyting algebras”.

Proposition 2.7.3. Let A be a bi-Heytig algebra and a, b, c ∈A. We have: 1. a←b=V{d∈ A : a≤d∨b},

2. a≤b ⇐⇒ a←b=0,

3. a←a=0,

4. a←1=0,

(28)

2.7. Bi-Heyting algebras 6. 0←a=0, 7. a∨ ∼a=1, 8. ∼a≤b ⇐⇒ a∨b=1, 9. a←b≤ a, 10. b≤c =⇒ a ←c≤ a←b, 11. b≤c =⇒ b←a≤c←a, 12. a≤b =⇒ ∼b≤ ∼a, 13. a← (b∧c) = (a←b) ∨ (a ←c), 14. a∨ (b←c) =a∨(a∨b) ← (a∨c), 15. a∨b← (b∨c)=a, 16. ∼a=0 ⇐⇒ a=1, 17. ¬ ∼a =1 ⇐⇒ a =1, 18. ∼¬a=0 ⇐⇒ a=0, 19. ∼(a∧b) = ∼a∨ ∼b, 20. ∼(a∨b) ≤ ∼a∧ ∼b, 21. ¬ ∼(a∧b) = ¬ ∼a∧ ¬ ∼b, 22. ∼¬(a∨b) = ∼¬a∨ ∼¬b; 23. ¬a≤ ∼a, 24. ∼∼a≤a, 25. ∼∼∼a = ∼a, 26. ¬ ∼a ≤ ∼∼a ≤a≤ ¬¬a ≤ ∼¬a.

Proof. 1. By the definition of a ←b, we have a≤ (a ←b) ∨b, hence a←b∈ {d∈ A : a≤ d∨b}.

Again by definition of a←b, it follows that for every c∈ {d∈ A : a≤d∨b}we have a←b≤ c. We conclude

a ←b=^{d∈ A : a≤d∨b}. 2. Suppose a≤b. Hence a ≤0∨b=b, and clearly we have

a←b=^{d∈ A : a≤d∨b} =0.

(29)

3. This follows directly from (2).

4. This follows directly from (2).

5. a←0=V{d∈ A : a≤d∨0= d}is clearly equal to a. 6. This follows directly from (2).

7. By Proposition2.7.2.(vi), we have

a∨ ∼a= a∨ (1←a) =a∨1=1.

8. Suppose∼a≤b. Hence 1= ∼a∨a≤b∨a≤1, and it now follows that a∨b=1. Conversely, if a∨b=1, then we have b∈ {d∈ A : 1≤d∨a}. Thus,

1←a = ∼a=^{d∈ A : 1≤d∨a} ≤b. 9. This follows from a≤ a∨b.

10. If b≤ c, then(a←b) ∨b≤ (a←b) ∨c. Since we also have

a≤ a∨b= (a←b) ∨b by Proposition2.7.2.(vii), it follows that

a←c= ^{d ∈ A : a≤d∨c} ≤a←b.

11. If b ≤ c, then b ≤ c ≤ c∨a = (c ← a) ∨a, where the equality follows from Proposition2.7.2.(vii). Hence we have

b← a=^{d∈ A : b≤d∨a} ≤c←a. 12. This follows directly from (10).

13. Firstly, since b∧c ≤ b, c, it follows from (10) that a ← b, a ← c ≤ a ← (b∧c). Consequently,(a ← b) ∨ (a ←c) ≤a ← (b∧c). On the other hand, let us note that a ≤ (a←b) ∨ (a←c) ∨b ∧ (a←b) ∨ (a←c) ∨c = = (a ←b) ∨ (a←c) ∨ (b∧c), since we have both

a≤a∨b= (a←b) ∨b≤ (a ←b) ∨b∨ (a←c) and

a ≤a∨c= (a←c) ∨c≤ (a ←c) ∨c∨ (a ←b), by Proposition2.7.2.(vii). We conclude that

a← (b∧c) =^{d∈ A : a≤d∨ (b∧c)} ≤ (a← b) ∨ (a←c), as desired.

(30)

14. Firstly, note that by conditions (vi) and (viii) of Proposition2.7.2, we have

a∨

(a∨b) ← (a∨c)

= a∨a← (a∨c)

∨b← (a∨c)

=a∨b← (a∨c). Since c≤ a∨c, it follows from (10) that b← (a∨c) ≤b←c, hence we have

a∨b← (a∨c)

≤ a∨ (b←c). To achieve the desired equality, it remains to prove

a∨ (b←c) ≤a∨b← (a∨c).

To this end, let x ∈ A and suppose a∨ [b ← (a∨c)] ≤ x, hence we have both a ≤x and b← (a∨c) ≤x. Note that by the definition of←, the latter inequality is equivalent to b≤x∨ (a∨c). Since a ≤x, i.e., x=x∨a, it follows that

b≤ x∨ (a∨c) =x∨c.

Again by the definition of←, we have b≤x∨c iff b←c≤ x. This, together with a ≤x, now entail a∨ (b←c) ≤x. If we take x =a∨b← (a∨c), this discussion yields

a∨ (b←c) ≤a∨b← (a∨c), as desired.

15. This follows from (2), since b≤b∨c.

16. If∼a =0, then a=a∨0=a∨ ∼a. By (7), this implies a=1. Conversely, if a=1, then

∼a=^{d∈ A : 1≤d∨a=d∨1=1} =0.

17. This follows directly from (16) and the equivalence(∼a) = 0 iff¬(∼a) = 1 (see the Properties of Heyting algebras2.6.14).

18. This follows directly from (16) and the equivalence a = 0 iff ¬a = 1 (see the Properties of Heyting algebras2.6.14).

19. This follows directly from (13) and the definition of∼.

20. Since a, b≤ a∨b, (12) implies∼(a∨b) ≤ ∼a,∼b. Consequently,

∼(a∨b) ≤ ∼a∧ ∼b.

21. This follows directly from (19) and the equality¬(∼a∨ ∼b) = ¬(∼a) ∧ ¬(∼b) (see the Properties of Heyting algebras2.6.14).

22. This follows directly from (19) and the equality ¬(a∨b) = ¬a∧ ¬b (see the Properties of Heyting algebras2.6.14).

23. Suppose that¬a _∼a. By the Prime Filter Theorem 2.6.7, there exists a prime

filter F of A containing ¬a but omitting ∼a. Note that, since F is ∧-closed and a∧ ¬a = 0 /∈ F, then a cannot belong to F. But F is prime, so a∨ ∼a = 1 ∈ F implies∼a∈ F, a contradiction. We conclude that¬a≤ ∼a, as desired.

(31)

24. This follows directly from (7) and (8).

25. By (24) we have∼∼a≤a, so (12) yields ∼a ≤ ∼∼∼a. Conversely, since we have ∼a∨ ∼∼a =1 by (7), it follows from (8) that∼∼∼a≤ ∼a, hence the equality. 26. The first inequality follows directly from (23), the second is just (24), the third

comes from Proposition2.6.14, and the fourth follows again directly from (23).

Example 2.7.4. Given a lattice L, if the infimum operator V

and the supremum operator W

are well-defined on arbitrary subsets, L is said to be complete. Clearly, every finite bounded distributive lattice is complete, and since we can characterize the Heyting implication and co-implication usingW

andV

, respectively, it follows that every finite bounded distributive lattice can be viewed as a bi-Heyting algebra. In particular, this holds for all finite Heyting algebras.

We could generalize this example in the following manner: a complete bounded lattice L is said to be infinite distributive if it satisfies the infinite distributivity laws, i.e., for a ∈L and S⊆ L, we have:

• a∧W S=W {a∧s : s∈S}, • a∨V S=V {a∨s : s∈S}.

Using these equalities, it is clear that L has well-defined Heyting implications, hence complete bounded infinite distributive lattices can be viewed as bi-Heyting algebras.

Example 2.7.5. Every Boolean algebra A can be viewed as a bi-Heyting algebra, where the Heyting co-implication is given by a ←b= a∧ ¬b, for all a, b∈ A.

Example 2.7.6. Given a Kripke frame F= (W, R), then(U p(F),∪,∩,→,←,∅, W)is a bi-Heyting algebra, where the implications are defined as follows, for all U, V ∈ U p(F):

• U →V :=W\ ↓(U\V) =x ∈W :∀y∈ ↑x(y∈U =⇒ y∈V) , • U ←V := ↑(U\V) =x∈W : ∃y ∈U\V (y≤x) .

Example 2.7.7. Recall Example2.7.4and let(X, τ)be a topological space. It is easy to

see that(τ,W,V,∅, X)is a complete bounded lattice, where

_

S=[S and ^S=int(\S):= the interior of \S, for S⊆τ. Notice that for S∪ {U} ⊆τ, we have

U∩_

S=U∩[

S =[{U∩V : V ∈S} =_

{U∩V : V ∈ S}.

Now, if in (X, τ)we have that arbitrary intersections of opens are open (i.e., τ is an Alexandrov topology), then we also haveV

S= T

S, hence

U∪^

S=^{U∪V : V ∈ S}.

If this is the case, (τ,W,V,∅, X) is then a complete bounded infinite distributive

(32)

Definition 2.7.8. Given a bi-Heyting algebra A, we call the unary operation¬ ∼on A the double negation operation. For n ∈ω, we define(¬ ∼)n as usual (see Section 2.1).

Having a good understanding of the congruences of an algebra is always useful, and the particular case of bi-Heyting algebras is no exception. It is well-known that the filters of a Heyting algebra are in a one-to-one correspondence with its congruences [17]. This is not the case for bi-Heyting algebras; there are filters that

do not give rise to congruences. To achieve a similar one-to-one correspondence, we need to restrict our domain to filters that are closed under double negation. These filters, called the normal filters, were first introduced by Katri ˇn´ak in [23], in the setting

of double p-algebras (i.e., bounded distributive lattices with well-defined negation and co-negation operators), while their connection with congruences of a bi-Heyting algebra was first established by K ¨ohler in [24]. Next we define these filters, describe

the normal filter generated by a non-empty subset of a bi-Heyting algebra, and state their correspondence with congruences explicitly.

Definition 2.7.9. Let A be a bi-Heyting algebra and F∈ Filt(A). We say that F is a normal filter if it is ¬ ∼-closed, i.e., for all a∈ F, we have¬ ∼a ∈ F. Since filters are upsets, and we have¬ ∼a≤a by (26) of the Properties of bi-Heyting algebras2.7.3,

we could rephrase this as:

F∈ Filt(A)is normal ⇐⇒ ∀n∈ω,∀a∈ A(a ∈F ⇐⇒ (¬ ∼)na∈ F).

We denote the set of normal filters of A by NFilt(A).

Proposition 2.7.10. Let A be a bi-Heyting algebra. If B a non-empty subset of A, then the

normal filter of A generated by B (i.e., the least normal filter of A containing B) is given by NF(B):= {a∈ A : ∃b1, . . . , bt∈ B,∃n∈ω (¬ ∼)n(b1∧ · · · ∧bt) ≤a}.

Proof. We first show NF(B) ∈ NFilt(A). Suppose we have b, c∈ NF(X)and b ≤ a, for some a ∈ A. Then there exist b1, . . . , bt, c1, . . . , cr∈ B and n, m∈ ωsuch that

(¬ ∼)n(b1∧ · · · ∧bt) ≤b and (¬ ∼)m(c1∧ · · · ∧cr) ≤c.

Thus, for n0 := Max{n, m}, we have

(¬ ∼)n0(b1∧ · · · ∧bt∧c1∧ · · · ∧cr) = (¬ ∼)n 0 (b1∧ · · · ∧bt) ∧dn 0 (c1∧ · · · ∧cr) ≤ ≤ (¬ ∼)n(b1∧ · · · ∧bt) ∧ (¬ ∼)m(c1∧ · · · ∧cr) ≤b∧c,

where the equality follows from a successive application of condition (21) of the Properties of bi-Heyting algebras 2.7.3, the first inequality comes from2.7.3.(26), and

the last inequality comes from our definition of the bi and the cj. Thus b∧c∈ NF(X),

and since we also have (¬ ∼)n(b1∧ · · · ∧bt) ≤b≤a, then a∈ NF(X). So NF(X)is

a non-empty∧-closed upset of A, i.e., a filter. Moreover,

(¬ ∼)n(b1∧ · · · ∧bt) ≤b implies(¬ ∼)n+1(b1∧ · · · ∧bt) ≤ ¬ ∼b,

again by 2.7.3.(26). Hence, ¬ ∼b ∈ NF(X), and we conclude that this is indeed a

normal filter.

Now, suppose B⊆F, for some F∈ NFilt(A). Note that, for all b1, . . . , bt ∈ B, we

have b1∧ · · · ∧bt ∈ F, since filters are∧-closed. So F being a normal filter entails

(¬ ∼)n₍_b

1∧ · · · ∧bt) ∈ F, for all n∈ω. Hence↑(¬ ∼)n(b1∧ · · · ∧bt) ⊆F since filters

(33)

Definition 2.7.11. Let A be a bi-Heyting algebra and F ∈ NFilt(A). We define the congruence of A associated with F by

aθFb ⇐⇒ ∃e∈ F(a∧e =b∧e),

for every a, b∈ A.

The following theorem provides us with the desired one-to-one correspondence between the normal filters of a bi-Heyting algebra and its congruences. For a proof, see [24]. As in the Heyting case, we define F

θ :=J1Kθ, for θ ∈Con(A).

Theorem 2.7.12. (K¨ohler) Let A be a bi-Heyting algebra, θ∈Con(A)and F ∈ NFilt(A). Then:

1. θ_F∈ Con(A);

2. F

θ ∈NFilt(A);

3. The maps G∈ NFilt(A) 7→_θ_G and δ∈Con(A) 7→F

δ are lattice isomorphisms, and are

inverses of each other.

2.8 Algebraic completeness of Bi-IPC and its extensions

In this section, we highlight the algebraic completeness of Bi-IPC and its extensions. This is a crucial result for what follows, as it allows us to freely alternate between an algebraic framework (e.g., the study of a variety V of bi-Heyting algebras) and a logical one (the study of the logic of V, i.e., the set of formulas which are true in all algebras of V).

In the setting of bi-Heyting algebras, the concept of truth in an universal algebra is interpreted as follows. The variables of type F = (∨,∧,→,←,⊥,>)are the elements of Prop, while the terms of type F are the formulas in Form(L), and we use = to interpret the symbol≈. Given a formula φ and a bi-Heyting algebra A, we say that φ is true, or valid, or holds in A, written as A|= φ, if A|= φ≈ >(i.e., for every valuation

v : Prop → A, we have v(φ) = v(>) = 1). For the case of Heyting algebras, we

restrict our domain to Form(L0)andF0 _{= (∨}_,_∧_,_→_,_⊥_,_>)_.

Using the well-known Lindenbaum-Tarski construction (see, e.g., [10]) we obtain

the algebraic completeness of IPC, Bi-IPC, and CPC.

Theorem 2.8.1. Let ϕ∈ Form(L0)and φ∈ Form(L)The following conditions hold: • IPC` ϕ if and only if ϕ is valid in every Heyting algebra;

• Bi-IPC `φ if and only if φ is valid in every bi-Heyting algebra;

• CPC` φ if and only if φ is valid in every Boolean algebra.

We note that everything that is said in the following discussion can be applied to intermediate logics and to varieties of Heyting algebras, by taking the appropriate restrictions.

Bi-Gödel algebras and co-trees

Master Thesis

Bi-G ¨odel algebras and co-trees

Miguel Martins

dr. Nick Bezhanishvili

March 15, 2021

dr. Tommaso Moraschini

(University of Barcelona)

Abstract

Contents

CHAPTER

1

Introduction

CHAPTER

2

Bi-Heyting algebras

2.1

Notation and conventions

2.2

Bi-intuitionistic propositional logic

2.3

Kripke semantics of Bi-IPC

2.4

Universal algebra

∏

2.5

Varieties

2.6

Distributive lattices and (co-)Heyting algebras

2.7

Bi-Heyting algebras

2.8

Algebraic completeness of Bi-IPC and its extensions