An algebraic study of S5-modal Gödel logic
Diego Castaño, Cecilia Cimadamore, José Patricio Díaz Varela, Laura Rueda
AAn algebraic study of S5-modal G¨odel logic
Diego Casta˜no, Cecilia Cimadamore, Jos´e Patricio D´ıaz Varela, Laura RuedaJune 19, 2020
Abstract
In this paper we continue the study of the variety MG of monadic G¨odel algebras. Thesealgebras are the equivalent algebraic semantics of the S5-modal expansion of G¨odel logic,which is equivalent to the one-variable monadic fragment of first-order G¨odel logic. Weshow three families of locally finite subvarieties of MG and give their equational bases. Wealso introduce a topological duality for monadic G¨odel algebras and, as an application ofthis representation theorem, we characterize congruences and give characterizations of thelocally finite subvarieties mentioned above by means of their dual spaces. Finally, we studysome further properties of the subvariety generated by monadic G¨odel chains: we present acharacteristic chain for this variety, we prove that a Glivenko-type theorem holds for thesealgebras and we characterize free algebras over n generators. In this article we assume a general knowledge of H´ajek’s Basic Logic and its equivalent algebraicsemantics, the variety of BL-algebras (see [13]).In [13] H´ajek introduced the S5-modal expansion S C ) of any axiomatic extension C of hisBasic Logic BL . This logic is defined on the language of basic logic augmented with the unaryconnectives (cid:3) and ♦ by interpreting formulas on structures based on C -chains. This expansionis particularly interesting because it is equivalent to the one-variable fragment of the first-orderextension of C (see [13]).One of the most important extensions of BL is G¨odel logic G , obtained by adding the axiom p → p . This logic has been studied in recent papers (e.g. [6, 7, 8]). In [8] we showed a strongcompleteness theorem for S G ) based on an algebraic semantics for S G ). This algebraicsemantics is the variety MG of monadic G¨odel algebras . We started the study of this variety in[8]; the main objective of this article is to expand our understanding of this variety, especiallyof some of its subvarieties, which naturally correspond to axiomatic extensions of S G ). MG is a subvariety of the variety MBL of monadic BL-algebras introduced in [7]. MonadicBL-algebras are BL-algebras endowed with two unary operations ∀ and ∃ that satisfy the fol-lowing identities:(M1) ∀ x → x ≈ ∀ ( x → ∀ y ) ≈ ∃ x → ∀ y .(M3) ∀ ( ∀ x → y ) ≈ ∀ x → ∀ y . (M4) ∀ ( ∃ x ∨ y ) ≈ ∃ x ∨ ∀ y .(M5) ∃ ( x ∗ x ) ≈ ∃ x ∗ ∃ x .The class MBL was defined as a candidate for the equivalent algebraic semantics of S BL );note that we use ∀ and ∃ instead of (cid:3) and ♦ , respectively. The variety MG is obtained by addingthe identity x ≈ x to those for MBL (see [7, 8]). Note also that the identity (M5) becomestrivial when x ≈ x holds.Monadic Heyting algebras were introduced by Monteiro and Varsavksy in [16] and laterstudied in depth by Bezhanishvili in [2]. In [7] we also showed that monadic G¨odel algebras1 a r X i v : . [ m a t h . L O ] J un oincide with monadic Heyting algebras that satisfy the prelinearity identity ( x → y ) ∨ ( y → x ) ≈ MG is not locally finite;however, all the subvarieties introduced are proved to be locally finite varieties.The aim of Section 3 is to give a topological representation of monadic G¨odel algebrasusing Priestley spaces. The duality established is based on the duality given by Cignoli in[9] for distributive lattices with an additive closure operator. As applications, we characterizecongruences on monadic G¨odel algebras by means of saturated closed increasing subsets ofthe dual space, and we describe the dual spaces of the algebras belonging to the subvarietiesintroduced in Section 2.We devote Section 4 to study the subvariety generated by monadic G¨odel chains in moredepth. First we produce a characteristic chain for this subvariety, that is, a totally orderedalgebra that generates the whole variety. Then we prove a Glivenko-type theorem for thisvariety. Recall that Glivenko showed in [12] that a propositional formula is provable in theclassical propositional logic if and only if its double negation is provable in the intuitionisticpropositional logic. This result has an algebraic formulation, that is, the double negation is ahomomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Weprove a Glivenko-type theorem (in an algebraic version) establishing a relation between algebrasin this variety and the class of monadic G¨odel algebras such that the image of ∃ is a Booleanalgebra. Finally, we close Section 4 with a full description of free algebras over a finite number ofgenerators in this variety; we give a procedure to calculate the dual spaces of the free algebras.We finish this section summarizing the basic properties of monadic G¨odel algebras; all theproofs can be found in [7] in the broader context of monadic BL-algebras. For brevity, if A isa G¨odel algebra and we enrich it with a monadic structure, we denote the resulting algebra by (cid:104) A , ∃ , ∀(cid:105) . The next lemma collects some of the basic properties that hold true in any monadicG¨odel algebra. We abbreviate “finitely subdirectly irreducible” as f.s.i. Lemma 1.1.
Let (cid:104) A , ∃ , ∀(cid:105) be a monadic G¨odel algebra. Then: (1) ∃ A = ∀ A ; (2) ∃ A is a subalgebra of A ; (3) ∃ a = min { c ∈ ∃ A : c ≥ a } and ∀ a = max { c ∈ ∃ A : c ≤ a } for every a ∈ A ; (4) the lattices of congruences of (cid:104) A , ∃ , ∀(cid:105) and ∃ A are isomorphic; (5) (cid:104) A , ∃ , ∀(cid:105) is f.s.i. if and only if ∃ A is totally ordered; (6) (cid:104) A , ∃ , ∀(cid:105) is subdirectly irreducible if and only if ∃ A is totally ordered and there exists u ∈∃ A \ { } such that a ≤ u for all a ∈ ∃ A \ { } . The next lemma includes several arithmetical properties that are used constantly throughoutthe paper.
Lemma 1.2.
Let (cid:104) A , ∃ , ∀(cid:105) be a monadic G¨odel algebra. Then, for any a, b ∈ A and c ∈ ∃ A : (1) ∀ ∃ and ∀ ∃ ; (2) ∀ c = ∃ c = c ; (3) ∀ a ≤ a ≤ ∃ a ; (4) if a ≤ b , then ∀ a ≤ ∀ b and ∃ a ≤ ∃ b ; (5) ∀ ( a ∨ c ) = ∀ a ∨ c ; (6) ∃ ( a ∨ b ) = ∃ a ∨ ∃ b ; ∀ ( a ∧ b ) = ∀ a ∧ ∀ b ; (8) ∃ ( a ∧ c ) = ∃ a ∧ c ; (9) ∀ ( a → c ) = ∃ a → c ; (10) ∃ ( a → c ) ≤ ∀ a → c ; (11) ∀ ( c → a ) = c → ∀ a ; (12) ∃ ( c → a ) ≤ c → ∃ a ; (13) ∀¬ a = ¬∃ a ; (14) ∃¬ a ≤ ¬∀ a . In [7] we give a characterization of those subalgebras of a given BL-algebra that may be therange of the quantifiers ∀ and ∃ . Given a BL-algebra A , we say that a subalgebra C ≤ A is m -relatively complete if the following conditions hold:(s1) For every a ∈ A , the subset { c ∈ C : c ≤ a } has a greatest element and { c ∈ C : c ≥ a } hasa least element.(s2) For every a ∈ A and c , c ∈ C such that c ≤ c ∨ a , there exists c ∈ C such that c ≤ c ∨ c and c ≤ a .(s3) For every a ∈ A and c ∈ C such that a ∗ a ≤ c , there exists c ∈ C such that a ≤ c and c ∗ c ≤ c .Under certain circumstances these conditions can be simplified. For example, if A is finite,condition (s1) is trivially satisfied. If C is totally ordered, condition (s2) may be replaced bythe following simpler equivalent form:(s2 (cid:96) ) If 1 = c ∨ a for some c ∈ C , a ∈ A , then c = 1 or a = 1.If A is a G¨odel algebra, condition (s3) is immediate since ∗ coincides with ∧ .Given a BL-algebra A and an m -relatively complete subalgebra C ≤ A , if we define on A the operations ∃ a := min { c ∈ C : c ≥ a } , ∀ a := max { c ∈ C : c ≤ a } , then (cid:104) A , ∃ , ∀(cid:105) is a monadic BL-algebra such that ∀ A = ∃ A = C . In this section we introduce some subvarieties of MG . We prove that there are equations thatconstrain the “height” or “width” of an algebra, and that the subvarieties determined by theseconditions are locally finite.We start by recalling that the variety MG is not locally finite. In fact, there are f.s.i. algebrasthat are not locally finite. For example, consider the monadic G¨odel algebra A := (cid:104) [0 , N G , ∃ , ∀(cid:105) ,where [0 , G is the standrad G¨odel algebra and ∃ A is the set of constant sequences in [0 , N G .Let a ∈ A be the sequence defined by a ( n ) := 1 − n , n ∈ N . We claim that the subuniverse of A generated by a is infinite. Indeed, consider the following sequences defined by recursion: a = a ; a k +1 := a k ∨ ( a k → ∀ a k ) for k ∈ N . It is straightforward to check that a k ( n ) = 1 for n < k , and a k ( n ) = 1 − n for n ≥ k . Since all these sequences are different, the subalgebra generated by a in A is infinite.We now introduce the notions of “height” and “width” that allow us to give examples oflocally finite subvarieties of MG .It is known from [14] that the variety of G¨odel algebras generated by the chain with n elements is characterized in G by means of the equation n (cid:95) i =1 ( x i → x i +1 ) ≈ .
3e denote by H n the subvariety of MG characterized by this identity. The f.s.i. algebras in H n are precisely the monadic G¨odel algebras (cid:104) A , ∃ , ∀(cid:105) where A is a subdirect product of G¨odelchains with at most n elements; observe that ∃ A is a G¨odel chain with at most n elements. Wesay that the “height” of these algebras is at most n . It is worth noting that H is the variety ofmonadic Boolean algebras.We can define larger subvarieties of MG if we only require the subalgebra ∃ A to be of finiteheight. Let H ∃ n be the subvariety of MG axiomatized by the equation n (cid:95) i =1 ( ∃ x i → ∃ x i +1 ) ≈ . A f.s.i. monadic G¨odel algebra (cid:104) A , ∃ , ∀(cid:105) belongs to H ∃ n if and only if ∃ A is a G¨odel chain withat most n elements.Observe that, for any monadic G¨odel algebra (cid:104) A , ∃ , ∀(cid:105) , we have that (cid:104) A , ∃ , ∀(cid:105) ∈ H ∃ if andonly if ∃ A is a Boolean algebra. Moreover, (cid:104) A , ∃ , ∀(cid:105) is a f.s.i. algebra in H ∃ if and only if ∃ A = { , } . Now, since the two-element chain is the only simple G¨odel algebra, by Lemma 1.1(4) (cid:104) A , ∃ , ∀(cid:105) is a simple algebra in MG if and only if ∃ A = { , } . Thus H ∃ is the subvarietyof MG generated by its simple members. It is easy to see that H ∃ is a discriminator variety (infact, the largest one contained in MG ) since the term t ( x, y, z ) := ( ∀ (( x → y ) ∧ ( y → x )) ∧ z ) ∨ ( ¬∀ (( x → y ) ∧ ( y → x )) ∧ x )gives the ternary discriminator function on each f.s.i. (simple) algebra in H ∃ .The varieties H n and H ∃ n are characterized by some bounded “height”. We can also definesubvarieties of MG based on a notion of “width”. Consider the identity (cid:94) ≤ i Let (cid:104) A , ∃ , ∀(cid:105) be a f.s.i. monadic G¨odel algebra. The following conditions areequivalent: (1) (cid:104) A , ∃ , ∀(cid:105) satisfies equation ( α k ) ; (2) any orthogonal set in A has at most k elements; (3) there are prime filters P , . . . , P r in A , r ≤ k , such that (cid:84) ri =1 P i = { } and P i ∩ ∃ A = { } for ≤ i ≤ r .Proof. We first prove that (1) implies (2). Assume (1) holds and suppose there is an orthogonalset S = { s , . . . , s k , s k +1 } with k + 1 elements. Then ∀ ( s i ∨ s j ) = ∀ i (cid:54) = j , but (cid:87) k +1 i =1 ∀ s i = ∀ s i ≤ s i < i , since ∃ A is totally ordered. This contradicts the validityof ( α k ).Now assume condition (2) is true. Let r ≤ k be the maximal cardinality of orthogonal setsin A and fix an r -element orthogonal set S = { s , . . . , s r } . Consider the sets P i = { x ∈ A : x ∨ s i = 1 } . We claim that P i are the desired prime filters. Indeed, it is clear that P i are filterson A . To prove that P i is prime, suppose x ∨ y ∈ P i but x, y / ∈ P i . Since ( x ∨ s i ) ∨ ( y ∨ s i ) = 1,the set S (cid:48) = { x ∨ s i , y ∨ s i } ∪ S \ { s i } is an orthogonal set. Hence S (cid:48) must contain at most r elements. There are three possibilities: if x ∨ s i = s j , i (cid:54) = j , then x ∨ s i = s j ∨ s i = 1, so x ∈ P i ,4 contradiction; if y ∨ s i = s j , i (cid:54) = j , then y ∨ s i = s j ∨ s i = 1, so y ∈ P i , a contradiction; if x ∨ s i = y ∨ s i , then x ∨ s i = x ∨ y ∨ s i = 1, so x ∈ P i , a contradiction. Thus P i is prime for1 ≤ i ≤ r . Moreover, if x ∈ (cid:84) ri =1 P i , it must be that x = 1, since otherwise S ∪ { x } would bean orthogonal set with r + 1 elements. Finally, note that if c ∈ P i ∩ ∃ A , then c ∨ s i = 1 and, bycondition ( s c = 1. This concludes the proof that (2) implies (3).Finally we prove that (3) implies (1). By the assumption we may consider A ≤ A × . . . × A r with r ≤ k , where each A i is totally ordered. Let ¯ a , . . . , ¯ a k +1 ∈ A . Let ¯ b = (cid:86) j The variety H ∃ n is locally finite for every n .Proof. Since the variety of G¨odel algebras is locally finite, for each natural number m the sizeof an m -generated G¨odel algebra has an upper bound N m . Now let (cid:104) A , ∃ , ∀(cid:105) be a f.s.i. monadicG¨odel algebra in H ∃ n generated by a set S of size m . We know that ∃ A has at most n elements.Thus A is a G¨odel algebra generated by the set S ∪ ∃ A , which has at most m + n elements.Therefore the size of A is bound by N m + n . This shows that the class of f.s.i. algebras in H ∃ n isuniformly locally finite. Thus, H ∃ n is a locally finite variety (see [3, Theorem 3.7]).Since H n ⊆ H ∃ n we also get the following result. Corollary 2.6. The variety H n is locally finite for every n . We now prove that W k is also a locally finite variety for every k . Theorem 2.7. The class of f.s.i. algebras in W k is uniformly locally finite. roof. Let (cid:104) A , ∃ , ∀(cid:105) be a f.s.i. monadic G¨odel algebra of width k . Let ϕ : A → A × . . . × A k be a subdirect representation of A as given by Corollary 2.2. Without loss of generality assumethat π j ◦ ϕ is the identity on ∃ A , thus ∃ A is a subalgebra of each A j .For each j define two partial operations ∃ j , ∀ j on A j by ∃ j x := min { c ∈ ∃ A : c ≥ x } , ∀ j x := max { c ∈ ∃ A : c ≤ x } , provided these elements exist.Fix a , . . . , a n ∈ A and let B be the subalgebra of (cid:104) A , ∃ , ∀(cid:105) generated by { a , . . . , a n } . Weshow that B is finite.Put ϕ ( a i ) = ( a i , . . . , a ik ), 1 ≤ i ≤ n . For each j , let S j := { a ij : 1 ≤ i ≤ n } ∪ { , } ,and put C j := ∀ j S j ∪ ∃ j S j . Let S ∗ j := S j ∪ C ∪ . . . ∪ C k . Observe that S ∗ j is finite for every j . We claim that ϕ ( B ) ⊆ S ∗ × . . . × S ∗ k which suffices to prove that B is finite. Indeed,we show that A (cid:48) := ϕ − ( S ∗ × . . . × S ∗ k ) is a subuniverse of (cid:104) A , ∃ , ∀(cid:105) . It is clear that A (cid:48) isclosed under G¨odel operations. Now fix x ∈ A (cid:48) and put ϕ ( x ) = ( x , . . . , x k ). Consider thedown-sets D j := { c ∈ ∃ A : c ≤ x j } , 1 ≤ j ≤ k . Then (cid:84) kj =1 D j = { c ∈ ∃ A : c ≤ x } andmax (cid:84) kj =1 D j exists (in fact, it is precisely ∀ x ). Since the family of down-sets { D j : 1 ≤ j ≤ k } is a chain (because ∃ A is totally ordered), there is j such that (cid:84) kj =1 D j = D j . Hence ∀ x =max (cid:84) kj =1 D j = max D j = ∀ j x j . Now, since x j ∈ S ∗ j , we get that ∀ j x j ∈ S ∗ j for 1 ≤ j ≤ k ,so ϕ ( ∀ x ) = ( ∀ x, . . . , ∀ x ) = ( ∀ j x j , . . . , ∀ j x j ) ∈ S ∗ × . . . × S ∗ k , which proves that ∀ x ∈ A (cid:48) .Analogously ∃ x ∈ A (cid:48) . This proves that A (cid:48) is a finite subuniverse of A containing B .Finally, note that | B | ≤ | S ∗ × . . . × S ∗ k | ≤ ( n + 2 + k (2 n + 4)) k . Thus the size of n -generatedsubalgebras of (cid:104) A , ∃ , ∀(cid:105) is uniformly bound.Applying [3, Theorem 3.7] we get the following corollary. Corollary 2.8. The variety W k is locally finite for every k . We close this section with an example that shows that there is much more to say aboutlocally finite algebras in MG . Example 2.9. Let A := (cid:104) [0 , G , ∃ , ∀(cid:105) where ∃ A = [0 , B := (cid:104) [0 , N G , ∃ , ∀(cid:105) where ∃ B = { , } . Observe that both A and B are f.s.i. algebras in MG ; in addition, note that A ∈ W but has infinite height; on the other hand, B ∈ H ∃ but has infinite width. Nowconsider the ordinal sum C := A ⊕ B where the top element in A is identified with the bottomelement in B . It is straightforward to check that C is locally finite, but has infinite width and ∃ C is also infinite. In this section we give a topological representation for monadic G¨odel algebras. As an applica-tion, we characterize congruences by means of saturated closed increasing subsets of the dualspace. We also describe the subvarieties introduced in the previous section by means of theirdual spaces. We will see in the next section that this duality is a most useful tool to characterizefree algebras in the variety W .We start by recalling the definitions needed to state the duality. For a poset (cid:104) X, ≤(cid:105) and Y ⊆ X , let ( Y ] = { x ∈ X : x ≤ y for some y ∈ Y } and [ Y ) = { x ∈ X : x ≥ y for some y ∈ Y } .We write [ x ), ( x ] instead of [ { x } ), ( { x } ], respectively. We say that Y is decreasing if Y = ( Y ]and increasing if Y = [ Y ).A triple (cid:104) X ; ≤ , τ (cid:105) is a totally order-disconnected topological space if (cid:104) X, ≤(cid:105) is a poset, τ is atopology on X , and for x , y ∈ X , if x (cid:54)≤ y , then there exists a clopen increasing set U ⊆ X suchthat x ∈ U and y / ∈ U . A compact totally order-disconnected space is called a Priestley space .6n [17] it is proved that the category of bounded distributive lattices and homomorphisms isdually equivalent to the category of Priestley spaces and order-preserving continuous functions.More precisely, Priestley defined contravariant functors I and X as follows. If X is a Priestleyspace, then I ( X ) is the lattice of clopen increasing subsets of X (we denote by I( X ) the universeof this lattice), and for each morphism f : X → X (cid:48) , I ( f ) is defined by I ( f )( U ) := f − ( U ) for each U ∈ I( X (cid:48) ). If L is a bounded distributive lattice, we denote by X( L ) the set of prime filters of L . Then X ( L ) is the Priestley space obtained by ordering X( L ) by set inclusion and consideringthe topology generated by the sets of the form σ ( a ) := { P ∈ X( L ) : a ∈ P } and X( L ) \ σ ( a ) foreach a ∈ L . If h : L → L (cid:48) is a homomorphism, then X ( h ) is defined by X ( h )( P ) = h − ( P ) foreach P ∈ X( L (cid:48) ). It follows that σ : L → I ( X ( L )) is a lattice isomorphism, and that the mapping ε : X → X ( I ( X )) defined by the formula ε ( x ) := { U ∈ I( X ) : x ∈ U } is a homeomorphism andan order isomorphism. We refer the reader to [10] for basic properties of Priestley spaces andPriestley duality.Next we define the kind of Priestley spaces that will prove to be the duals of monadic G¨odelalgebras. Definition 3.1. An MG-space X = (cid:104) X ; ≤ , τ, E (cid:105) is a Priestley space (cid:104) X ; ≤ , τ (cid:105) enriched with anequivalence relation E defined on X that satisfies the following conditions:(c1) ( Y ] is clopen for every clopen Y ⊆ X .(c2) [ x ) is a chain for every x ∈ X .(c3) The relation E satisfies:(c3a) ∃ U ∈ I( X ) for each U ∈ I( X ), where ∃ U is the union of all the equivalence classesthat contain an element of U ,(c3b) ∀ U ∈ I( X ) for each U ∈ I( X ), where ∀ U is the union of all the equivalence classesthat are contained in U ,(c3c) the equivalence classes determined by E are closed in X .We denote by ¯ x the equivalence class of an element x in a Priestley space enriched with anequivalence relation E .Let us recall that a Priestley space fulfilling condition (c1) is a Heyting space, that is, thePriestley space associated to a Heyting algebra [11, 18]. If, in addition, the space satisfiescondition (c2), then it is possible to prove that the category of these spaces and its morphismsis dually equivalent to the category of G¨odel algebras and homomorphisms [15].Observe also that if a Priestley space is enriched with an equivalence relation E which satisfiesconditions (c3a) and (c3c), then the space is a Q-space [9]. Cignoli proved that the category ofQ-spaces is dually equivalent to the category of bounded distributive lattices with a quantifier ∃ (denoted by ∇ in his paper) that satisfies the identities ∃ ≈ x ∧ ∃ x ≈ x , ∃ ( x ∧ ∃ y ) ≈ ∃ x ∧ ∃ y , ∃ ( x ∨ y ) ≈ ∃ x ∨ ∃ y , all of them valid in monadic G¨odel algebras.Let L be a bounded distributive lattice. A unary operation ∀ : L → L is called an interioroperator if it satisfies the following identities ∀ ≈ x ∧∀ x ≈ ∀ x , ∀∀ x ≈ ∀ x and ∀ ( x ∧ y ) = ∀ x ∧∀ y (see [5]). Theorem 3.2. Let L be a bounded distributive lattice with an interior operator ∀ and put E := { ( P, Q ) ∈ X( L ) : P ∩ ∀ L = Q ∩ ∀ L } , an equivalence relation over X( L ) . The followingare equivalent properties: (i) Given P, Q ∈ X( L ) such that P ∩ ∀ L ⊆ Q ∩ ∀ L , there exists R ∈ X( L ) such that ( P, R ) ∈ E and R ⊆ Q . (ii) If P ∈ X( L ) , a ∈ P and P ⊆ σ ( a ) , then ∀ a ∈ P . For each a ∈ L , ∀ σ ( a ) = σ ( ∀ a ) , where ∀ σ ( a ) is the union of the equivalence classescontained in σ ( a ) . (iv) For all a, b ∈ L , ∀ ( ∀ a ∨ b ) = ∀ a ∨ ∀ b .Proof. (i) implies (ii). Let P ∈ X( L ) such that a ∈ P . We will prove that, if ∀ a / ∈ P , thenthere exists R ∈ X( L ) such that ( P, R ) ∈ E and a / ∈ R . For that, let us consider the filter F of L generated by P ∩ ∀ L and the principal ideal J of L generated by a (note that P ∩ ∀ L is closed under ∧ ). Then F ∩ J = ∅ . Indeed, if c ∈ F ∩ J , there is k ∈ P ∩ ∀ L such that k ≤ c ≤ a . Then k = ∀ k ≤ ∀ a and this contradicts that ∀ a / ∈ P . By the Prime Filter Theorem,there exists a prime filter Q such that F ⊆ Q and Q ∩ J = ∅ . We have that Q ∈ X( L ) and P ∩ ∀ L ⊆ F ∩ ∀ L ⊆ Q ∩ ∀ L . Now by (i) we obtain R in X( L ) such that ( P, R ) ∈ E and R ⊆ Q .So, a / ∈ R .(ii) implies (iii). Let P ∈ ∀ σ ( a ). By the definition of ∀ , a ∈ P and P ⊆ σ ( a ). Considering(ii) we have that ∀ a ∈ P , and then P ∈ σ ( ∀ a ). Suppose now that P ∈ σ ( ∀ a ). Then P ∈ σ ( a ).Let Q ∈ P , that is, P ∩ ∀ L = Q ∩ ∀ L . Since ∀ a ∈ P ∩ ∀ L , we have that ∀ a ∈ Q and then a ∈ Q .Therefore P ⊆ σ ( a ), which implies that P ∈ ∀ σ ( a ).(iii) implies (iv). From (iii) and since σ is a lattice isomorphism, we have that σ ( ∀ ( ∀ a ∨ b )) = ∀ σ ( ∀ a ∨ b ) = ∀ ( ∀ σ ( a ) ∪ σ ( b )). Let us see that ∀ ( ∀ σ ( a ) ∪ σ ( b )) = ∀ σ ( a ) ∪ ∀ σ ( b ). Indeed, if P ∈ ∀ ( ∀ σ ( a ) ∪ σ ( b )), then P ⊆ ∀ σ ( a ) ∪ σ ( b ). If P ∩ ∀ σ ( a ) (cid:54) = ∅ , then P ⊆ ∀ σ ( a ). On the otherhand, if P ∩∀ σ ( a ) = ∅ then P ⊆ σ ( b ). In consequence, P ∈ ∀ σ ( a ) ∪∀ σ ( b ). Let P ∈ ∀ σ ( a ) ∪∀ σ ( b ).Then P ∈ ∀ σ ( a ) or P ∈ ∀ σ ( b ). So, P ⊆ σ ( a ) or P ⊆ σ ( b ). Clearly, if P ⊆ σ ( a ), then P ⊆ ∀ σ ( a ).Thus, P ⊆ ∀ σ ( a ) ∪ σ ( b ) and, from the definition of ∀ , we have that P ∈ ∀ ( ∀ σ ( a ) ∪ σ ( b )). Onceagain, from (iii) and since σ is a lattice isomorphism, we obtain ∀ σ ( a ) ∪ ∀ σ ( b ) = σ ( ∀ a ∨ ∀ b ). So, σ ( ∀ ( ∀ a ∨ b )) = σ ( ∀ a ∨ ∀ b ).(iv) implies (i). Let P, Q ∈ X( L ) such that P ∩ ∀ L ⊆ Q ∩ ∀ L . Consider the filter F generatedby P ∩ ∀ L and the ideal J generated by ( L \ Q ) ∪ ( ∀ L \ P ). Let us see that F ∩ J = ∅ . Indeed,if c ∈ F ∩ J , then there exist a ∈ L \ Q and ∀ b / ∈ P such that c ≤ a ∨ ∀ b (note that ∀ L \ P isclosed under ∨ by condition (iv)). Also, there exists d ∈ P ∩ ∀ L such that d ≤ c . By (iv), wehave that d = ∀ d ≤ ∀ c ≤ ∀ ( a ∨ ∀ b ) = ∀ a ∨ ∀ b . Since d ∈ P ∩ ∀ L , we have that ∀ a ∨ ∀ b ∈ P ∩ ∀ L .From a / ∈ Q , we obtain that ∀ a / ∈ Q . But P ∩ ∀ L ⊆ Q ∩ ∀ L , so ∀ a / ∈ P and then ∀ b ∈ P , whichis a contradiction. By the Prime Filter Theorem there exists a prime filter R such that F ⊆ R and R ∩ J = ∅ . Since L \ Q ⊆ J ⊆ L \ R , we have that R ⊆ Q . Let us prove that ( P, R ) ∈ E . If c ∈ P ∩ ∀ L , then c ∈ F and so c ∈ R . Thus, c ∈ R ∩ ∀ L . Conversely, let c ∈ R ∩ ∀ L . If c / ∈ P ,then c ∈ ∀ L \ P and c ∈ J ⊆ L \ R . So, c / ∈ R . Then, c ∈ P ∩ ∀ L . Remark . Cignoli proved in [9, Theorem 2.2] a theorem analogous to Theorem 3.2 for boundeddistributive lattices with an additive closure operator. We included the proof of Theorem 3.2because neither of the results follows directly from the other one, since these algebras are notsymmetric and the properties of ∀ are not mere consequences of the corresponding properties of ∃ . Let A be a monadic G¨odel algebra and let us consider the enriched Priestley space (cid:104) X( A ); ⊆ , τ, E (cid:105) where E = { ( P, Q ) ∈ X( A ) : P ∩ ∃ A = Q ∩ ∃ A } . We already know thatthis space satisfies (c1), (c2), (c3a) and (c3c) ([9], [15]). Moreover, from Theorem 3.2 and thefact that monadic G¨odel algebras satisfy the identity ∀ ( ∀ x ∨ y ) ≈ ∀ x ∨ ∀ y , we have that (c3b)is also satisfied. So, the next result follows. Proposition 3.4. Let A ∈ MG . Then (cid:104) X( A ); ⊆ , τ, E (cid:105) is an MG-space. Let X be an MG-space. Let us consider the lattice I ( X ), where we define U → V = X \ ( U \ V ],for U, V ∈ I( X ), and, ∃ and ∀ as in (c3a) and (c3b). Lemma 3.5. The algebra (cid:104) I( X ); ∩ , ∪ , → , ∅ , X, ∃ , ∀(cid:105) satisfies: (cid:104) I( X ); ∩ , ∪ , → , ∅ , X (cid:105) is a G¨odel algebra; and • the following identities: (1) ∀ ≈ , (2) ∃ ≈ , (3) ∀ x → x ≈ , (4) x → ∃ x ≈ , (5) ∀∃ x ≈ ∃ x , (6) ∃∀ x ≈ ∀ x , (7) ∀ ( x ∧ y ) ≈ ∀ x ∧ ∀ y , (8) ∃ ( x ∨ y ) = ∃ x ∨ ∃ y , (9) ∃ ( x ∧ ∃ y ) ≈ ∃ x ∧ ∃ y , (10) ∀ ( ∃ x ∨ y ) = ∃ x ∨ ∀ y .Proof. The fact that (cid:104) I( X ); ∩ , ∪ , → , ∅ , X (cid:105) is a G¨odel algebra follows immediately from the knownduality for G¨odel algebras (see [10, 15]). Clearly from the definitions of ∃ and ∀ we have (1),(3), (5) and (6). From [9], we have (2), (4), (8) and (9). It only remains to prove (7) and (10).Indeed, clearly ∀ ( U ∩ V ) ⊆ ∀ U ∩ ∀ V . If x ∈ ∀ U ∩ ∀ V , then x ⊆ U and x ⊆ V . So, x ⊆ U ∩ V and then x ∈ ∀ ( U ∩ V ). To see (10), let x ∈ ∃ U ∪ ∀ V . If x ∈ ∃ U then x ⊆ ∃ U . So, x ⊆ ∃ U ∪ V which means that x ∈ ∀ ( ∃ U ∪ V ). On the other hand, if x ∈ ∀ V then x ⊆ V . So, x ⊆ ∃ U ∪ V and then x ∈ ∀ ( ∃ U ∪ V ). For the other inclusion, let x ∈ ∀ ( ∃ U ∪ V ). Then x ⊆ ∃ U ∪ V . If x ⊆ ∃ U then x ⊆ ∃ U ∪ ∀ V . If x ∩ ∃ U = ∅ then x ⊆ V . So, x ∈ ∀ V .From Lemma 3.5 we have that I ( X ) is a monadic Heyting algebra (as defined by Bezhanishviliin [2]) that also satisfies the prelinearity identity and (M4). From [7, Theorem 5.9], we obtainthe following. Proposition 3.6. If (cid:104) X ; ≤ , τ, E (cid:105) is an MG-space then I ( X ) is a monadic G¨odel algebra. Having established the correspondence between objects, we turn now to morphisms. Definition 3.7. Let X and X (cid:48) be MG-spaces. An MG-morphism f from X to X (cid:48) is a con-tinuous order-preserving map f : X → X (cid:48) such that f ([ x )) = [ f ( x )), ∃ f − ( U (cid:48) ) = f − ( ∃ U (cid:48) ) and ∀ f − ( U (cid:48) ) = f − ( ∀ U (cid:48) ) for every U (cid:48) ∈ I( X (cid:48) ).From the known dualities for G¨odel algebras and Q-distributive lattices we have that if X , X (cid:48) are MG-spaces and f : X → X (cid:48) is an MG-morphism, then I ( f ) : I ( X (cid:48) ) → I ( X ) is a homomor-phism. Conversely, if A , A (cid:48) are monadic G¨odel algebras and h : A → A (cid:48) is a homomorphism,consider X ( h ) : X ( A (cid:48) ) → X ( A ). Again most of the conditions to check follow from the dualitiesfor G¨odel algebras and Q-distributive lattices; the only condition that remains to be proven isthat ∀ X ( h ) − ( σ ( a )) = X ( h ) − ( ∀ σ ( a )) for every a ∈ A . Indeed, using item (iii) in Theorem 3.2 ∀ X ( h ) − ( σ ( a )) = ∀ σ ( h ( a )) = σ ( ∀ h ( a )) = σ ( h ( ∀ a )) = X ( h ) − ( σ ( ∀ a )) = X ( h ) − ( ∀ σ ( a )) . Thus, X ( h ) is an MG-morphism.Clearly, σ : A → I ( X ( A )) is an isomorphism. The mapping ε : X → X ( I ( X )) is a homeo-morphism, an order isomorphism and satisfies the condition:( x, y ) ∈ E ⇔ ( ε ( x ) , ε ( y )) ∈ { ( P, Q ) ∈ X( I ( X )) × X( I ( X )) : P ∩ ∃ I( X ) = Q ∩ ∃ I( X ) } [9, Theorem 2.6]. Finally, the naturality of σ and ε follows immediately from the originalPriestley duality. So, we have proved the following theorem. Theorem 3.8. The categories of monadic G¨odel algebras and MG-spaces are dually equivalent. .1 Some applications of the duality Recall that if A is a Heyting algebra and Y is a closed increasing subset of X( A ), then θ ( Y ) := { ( a, b ) ∈ A × A : σ ( a ) ∩ Y = σ ( b ) ∩ Y } is a congruence on A (see [11]). Moreover, thecorrespondence Y (cid:55)→ θ ( Y ) establishes an anti-isomorphism from the lattice of closed increasingsets of X( A ) onto the congruence lattice of A . This properties are clearly inherited by G¨odelalgebras. Next we derive a similar result for monadic G¨odel algebras.Let X be an MG-space. A subset Y ⊆ X is called saturated if ∃ Y = Y . Clearly, if Y issaturated, then ∀ Y = Y . Theorem 3.9. Let A ∈ MG . Then, θ is a congruence of A if and only if θ = θ ( Y ) := { ( a, b ) ∈ A × A : σ ( a ) ∩ Y = σ ( b ) ∩ Y } for some saturated closed increasing subset Y of X( A ) .Consequently, Y (cid:55)→ θ ( Y ) is an anti-isomoprhism from the lattice of saturated closed increasingsubsets of X ( A ) onto the lattice of congruences of A .Proof. Let θ be a congruence of A . We know that there is a closed increasing subset Y ⊆ X( A )such that θ = θ ( Y ). It only remains to show that Y is saturated. Let us suppose that thereexists P ∈ ∃ Y \ Y . Then, P (cid:54)⊆ Y . Since Y is increasing, if Q ∈ Y , then we have that Q (cid:54)⊆ P .Thus, for each Q ∈ Y , there is a Q ∈ A such that Q ∈ σ ( a Q ) and P / ∈ σ ( a Q ). Then, Y ⊆ (cid:91) Q ∈ Y σ ( a Q ) . Since Y is compact, there exists a ∈ A such that Y ⊆ σ ( a ) and P / ∈ σ ( a ). From σ ( a ) ∩ Y = Y = σ (1) ∩ Y , we obtain that ( a, ∈ θ . Let us see that ( ∀ a, / ∈ θ , which is a contradiction.Indeed, P (cid:54)⊆ Y , P / ∈ σ ( a ) and P ∈ ∃ Y . Then, there exists R ∈ P ∩ Y , that is, R / ∈ σ ( ∀ a ) ∩ Y .But R ∈ σ (1) ∩ Y = Y .Conversely, let Y be a saturated closed increasing subset of X( A ) such that θ = θ ( Y ). Inparticular, we know that θ is a congruence of the G¨odel reduct of A . Moreover, by [9, Lemma3.1], we also know that θ preserves ∃ . We need to prove that θ preserves ∀ . Let a, b ∈ A suchthat σ ( a ) ∩ Y = σ ( b ) ∩ Y . By taking into account Theorem 3.2, and since Y is saturated, wehave that ∀ ( σ ( a ) ∩ Y ) = ∀ σ ( a ) ∩ ∀ Y = σ ( ∀ a ) ∩ Y . Analogously ∀ ( σ ( b ) ∩ Y ) = σ ( ∀ b ) ∩ Y , whichimplies what we wanted.In the following theorem we characterize the MG-spaces corresponding to the algebras in thesubvarieties introduced in Section 2. Given an MG-space (cid:104) X ; ≤ , τ, E (cid:105) , observe that, since anyclass x is closed, min x (cid:54) = ∅ , where min x is the set of minimal elements of x . Moreover, sinceprincipal decreasing subsets of X are also closed, x ⊆ [min x ) (see [10]). Theorem 3.10. Let A ∈ MG , X ( A ) be its associated MG-space. Then: (1) A ∈ W k if and only if the equivalence class P has at most k minimal elements for every P ∈ X( A ) . (2) A ∈ W if and only if the equivalence class P has exactly one minimal element for every P ∈ X( A ) . (3) A ∈ H n if and only if the chain [ P ) has at most n − elements for every P ∈ X( A ) . (4) A ∈ H ∃ n if and only if [ P ) /E has at most n − elements for every P ∈ X( A ) .Proof. (1) Let A be a monadic G¨odel algebra that satisfies ( α k ). Let us suppose that thereexists P ∈ X( A ) such that P has k + 1 minimal elements { N i : 1 ≤ i ≤ k + 1 } . For each i and j , i (cid:54) = j , take a ij ∈ A such that a ij ∈ N i \ N j , and let a i = (cid:94) j ∈{ , ··· ,k +1 }\{ i } a ij . Clearly10 i ∈ N i , but a i (cid:54)∈ N j for j (cid:54) = i . Take b i = a i → (cid:87) j (cid:54) = i a j , 1 ≤ i ≤ k + 1. Then, for i (cid:54) = t wehave b i ∨ b t = (cid:95) j (cid:54) = i ( a i → a j ) ∨ (cid:95) j (cid:54) = t ( a t → a j ) = 1by prelinearity. So, (cid:86) ≤ i 0) if b = 0( a + 1 , 0) if b (cid:54) = 0 , and ∀(cid:62) = (cid:62) , and, ∀ ( a, b ) = ( a, . It is clear that A = (cid:104) A ; ∨ , ∧ , → , ∃ , ∀ , (0 , , (cid:62)(cid:105) is a monadic G¨odel chain. Let us show thatevery finite chain C ( m,m ,...,m r ) is a homomorphic image of a subalgebra of A .Let S be the subalgebra of A whose subuniverse is given by S = { ( i, j ) : 0 ≤ i ≤ r, ≤ j ≤ m i } ∪ { ( r + 1 , } ∪ {(cid:62)} . Let us rename the elements of C ( m,m ,...,m r ) in the following way { a (0 , = 0 , a (0 , , . . . , a (0 ,m ) , b = a (1 , , . . . , a (1 ,m ) , . . . , a ( r, = b r , . . . , a ( r,m r ) , (cid:62) = b r +1 } . If we define h : S → C ( m,m ,...,m r ) as h (( i, j )) = a ( i,j ) , h (( r + 1 , h ( (cid:62) ) = (cid:62) , then it is straightforward to prove that h is an homomorphism and C ( m,m ,...,m r ) is a homomor-phic image of S . So, all finite chains of W are in the variety generated by A and consequently, A is a characteristic algebra for this variety. 12 .2 A Glivenko-type theorem for W We recall some definitions of special elements. Given a (monadic) G¨odel algebra A , an element a ∈ A is said to be: • dense if ¬ a = 0; • regular if ¬¬ a = a ; • boolean if a ∨ ¬ a = 1.We denote by D ( A ), Reg ( A ) and B ( A ) the set of dense, regular and boolean elements of A ,respectively.Let A be a G¨odel algebra. In this case B ( A ) = Reg ( A ) is a subuniverse of A and wedenote the corresponding subalgebra by Reg ( A ). Moreover, D ( A ) is an filter and, by Glivenko’stheorem, we have that Reg ( A ) ∼ = A /D ( A ) and the map r : A → Reg ( A ) defined by r ( a ) = ¬¬ a is a surjective homomorphism. Note that, since r ( a ) = r ( ¬¬ a ) for all a ∈ A , ¬¬ a → a ∈ D ( A ). Lemma 4.1. For any monadic G¨odel algebra A , B ( A ) = Reg ( A ) is a subuniverse of A .Proof. We already know that B ( A ) is a subuniverse of the G¨odel reduct of A . It remains toshow that it is also closed under ∃ and ∀ . Let b ∈ B ( A ). Then ∃ b ∨¬∃ b = ∃ b ∨∀¬ b = ∀ ( ∃ b ∨¬ b ) = ∀ ∃ b ∈ B ( A ). Analogously, ∀ b ∨ ¬∀ b ≥ ∀ b ∨ ∃¬ b = ∀ ( b ∨ ∃¬ b ) = ∀ ∀ b ∈ B ( A )too.We introduce now some definitions pertaining specifically to monadic algebras. Let A be amonadic G¨odel algebra. We denote by D ∀ ( A ) the set of elements a ∈ A such that ∀ a ∈ D ( A ),that is, D ∀ ( A ) = { a ∈ A : ¬∀ a = 0 } . It is straightforward to see that the set D ∀ ( A ) is amonadic filter of A . Moreover, A /D ∀ ( A ) ∈ H ∃ . Indeed, since ∃ a → ¬¬∃ a = 1 ∈ D ∀ ( A ) and ¬∀ ( ¬¬∃ a → ∃ a ) = ¬ ( ¬¬∃ a → ∃ a ) = 0, we have that ∃ a and ¬¬∃ a are identified in the quotient A /D ∀ ( A ); thus, ∃ ( A /D ∀ ( A )) is a Boolean algebra. Bezhanishvili in [4, Corollary 6] proved asimilar result for monadic Heyting algebras by means of Esakia spaces.We denote by Reg ∀ ( A ) the set of elements a ∈ A such that ∀ a ∈ Reg ( A ), that is, Reg ∀ ( A ) = { a ∈ A : ¬¬∀ a = ∀ a } . The following lemma shows that for algebras in the subvariety W the set Reg ∀ ( A ) may be endowed with a structure of monadic G¨odel algebra in a natural way. Observethat algebras in W satisfy the identity ¬¬ x ≈ ¬¬∃ x . Lemma 4.2. If A ∈ W , then Reg ∀ ( A ) := (cid:104) Reg ∀ ( A ); ∨ , ∧ , → , ∃ r , ∀ , , (cid:105) ∈ H ∃ ∩ W , where ∃ r a := ¬¬ a = ¬¬∃ a .Proof. Let a, b ∈ Reg ∀ ( A ). As ∀ ( a ∨ b ) = ∀ a ∨ ∀ b ∈ Reg ( A ), we have a ∨ b ∈ Reg ∀ ( A ), andsince ∀ ( a ∧ b ) = ∀ a ∧ ∀ b ∈ Reg ( A ), we also have that a ∧ b ∈ Reg ∀ ( A ). It is easy to see that ina monadic G¨odel chain, if ∀ b = ¬¬∀ b , then ∀ ( a → b ) = ¬¬∀ ( a → b ); so, a → b ∈ Reg ∀ ( A ).Thus, (cid:104) Reg ∀ ( A ); ∨ , ∧ , → , , (cid:105) is a subalgebra of (cid:104) A ; ∨ , ∧ , → , , (cid:105) . It remains to show that (cid:104) Reg ∀ ( A ); ∃ r , ∀(cid:105) satisfies axioms (M1)-(M4) and (M6). Clearly ∃ r a, ∀ a ∈ Reg ∀ ( A ), for all a ∈ Reg ∀ ( A ), and (M1), (M3), (M4) and (M6) hold. To see (M2), since ∀ ( a → ∀ b ) = ∃ a → ∀ b ,we need to prove ∃ a → ∀ b = ∃ r a → ∀ b . Indeed: ∃ a → ∀ b = ∃ a → ¬¬∀ b = ¬∀ b → ¬∃ a = ¬∀ b → ¬∃ r a = ∃ r a → ¬¬∀ b = ∃ r a → ∀ b. Thus, Reg ∀ ( A ) ∈ W . Finally, since ∃ r Reg ∀ ( A ) = Reg ( A ), then Reg ∀ ( A ) ∈ H ∃ .We can now prove a Glivenko theorem for algebras in W . Theorem 4.3. If A ∈ W , then Reg ∀ ( A ) ∼ = A /D ∀ ( A ) .Proof. We define the map g : A → Reg ∀ ( A ) by g ( a ) = a ∨ ¬¬∀ a . Let us see that g isa homomorphism of A onto Reg ∀ ( A ) such that g − ( { } ) = D ∀ ( A ). Indeed, g ( a ∨ b ) =( a ∨ b ) ∨¬¬∀ ( a ∨ b ) = ( a ∨ b ) ∨¬¬ ( ∀ a ∨∀ b ) = ( a ∨¬¬∀ a ) ∨ ( b ∨¬¬∀ b ) = g ( a ) ∨ g ( b ). It is easy to see13hat the following holds in monadic G¨odel chains: ( a ∧ b ) ∨ ¬¬∀ ( a ∧ b ) = ( a ∨ ¬¬∀ a ) ∧ ( b ∨ ¬¬∀ b )and ( a → b ) ∨ ¬¬∀ ( a → b ) = ( a ∨ ¬¬∀ a ) → ( b ∨ ¬¬∀ b ). Then, clearly, g ( a ∧ b ) = g ( a ) ∧ g ( b )and g ( a → b ) = g ( a ) → g ( b ). Also, g ( ∀ a ) = ∀ a ∨ ¬¬∀ a = ∀ ( a ∨ ¬¬∀ a ) = ∀ g ( a ), and, g ( ∃ a ) = ∃ a ∨ ¬¬∀∃ a = ∃ a ∨ ¬¬∃ a = ¬¬∃ a = ¬¬∃ a ∨ ¬¬∀ a = ¬¬∃ ( a ∨ ¬¬∀ a ) = ∃ r g ( a ).So g is a homomorphism. Clearly g is onto Reg ∀ ( A ), since g ( b ) = b ∨ ¬¬∀ b = b ∨ ∀ b = b for b ∈ Reg ∀ ( A ).Finally, if ¬∀ a = 0, then g ( a ) = 1. And, if 1 = a ∨¬¬∀ a , then 1 = ∀ ( a ∨¬¬∀ a ) = ∀ a ∨¬¬∀ a = ¬¬∀ a , so a ∈ D ∀ ( A ). Then, g − ( { } ) = D ∀ ( A ). Corollary 4.4. If A ∈ W , then A /D ∀ ( A ) is semisimple. In summary we have shown that for every A ∈ W the quotient algebra A /D ∀ ( A ) belongsto H ∃ ∩ W . Since H ∃ is a discriminator variety, we have that A /D ∀ ( A ) is a Boolean productof simple monadic G¨odel chains. W Having a clear description of free algebras in a variety greatly improves the understanding ofthe structures at hand. We devote the last part of this section to characterize the free algebra F ( n ) in W with n generators.First, we state some results for any finite monadic G¨odel algebra. Let A ∈ W be a finite algebra. We denote by Π( A ) the family of join-irreducible elements of A . We know that P ∈ X( A ) if and only if P = [ p ) is the filter generated by an element p ∈ Π( A ). Let us recallthat (cid:104) X( A ) , ⊆(cid:105) is dually isomorphic to the ordered set (cid:104) Π( A ) , ≤(cid:105) . The equivalence relation E defined on X( A ) (see § 3) naturally induces an equivalence relation on Π( A ) which we will alsodenote by E , by an abuse of notation. Note that if p, q ∈ Π( A ), ( p, q ) ∈ E iff [ p ) ∩ ∃ A = [ q ) ∩ ∃ A iff ∃ p = ∃ q . We will use the set (cid:104) Π( A ) , E (cid:105) instead of (cid:104) X( A ) , E (cid:105) . Recall that in any G¨odelalgebra the family of prime filters which contain a prime filter is a chain. Thus, if p ∈ Π( A )then ( p ] ∩ Π( A ) is a chain. In the sequel we write ( p ] Π instead of ( p ] ∩ Π( A ). Note also that( p ] Π = ( p ] − { } .We say that p ∈ Π( A ) ∩ ∃ A has coordinates ( m, m , . . . , m r ) if( p ] Π = { p , . . . , p m , p m +1 , p m +2 , . . . , p m + m +1 , p m + m +2 , . . . , p m +1 = p } has m + 1 elements and ( p ] Π ∩ ∃ A = { p m +1 , p m + m +2 , . . . , p m +1 = p } . That is, ( p ] Π is a chain with m + 1 elements like( p ] Π = { p , . . . , p m (cid:124) (cid:123)(cid:122) (cid:125) not in ∃ A , p m +1 (cid:124) (cid:123)(cid:122) (cid:125) in ∃ A , p m +2 , . . . , p m + m +1 (cid:124) (cid:123)(cid:122) (cid:125) m elements not in ∃ A , p m + m +2 (cid:124) (cid:123)(cid:122) (cid:125) in ∃ A , . . . , p m +1 = p (cid:124) (cid:123)(cid:122) (cid:125) in ∃ A } . Remark . If A ∈ W is finite and p ∈ Π( A ) ∩ ∃ A has coordinates ( m, m , . . . , m r ), then A / [ p ) ∼ = C ( m,m ,...,m r ) (recall the definition of C ( m,m ,...,m r ) from Section 4.1). Indeed, h : A → C ( m,m ,...,m r ) defined by h ( x ) = x ∈ [ p ) ,a i if x ∈ [ p i ) \ [ p i +1 ) , ≤ i ≤ m, x / ∈ [ p ) , is a surjective homomorphism such that [ p ) = h − ( { } ). Observe that the coordinates of a prime p ∈ Π( A ) ∩ ∃ A fully determine the number of prime elements in A below p as well as which ofthem belong to ∃ A . 14 emma 4.6. For every finite algebra A in W we have that: (i) If p ∈ Π( A ) , then ∃ p ∈ Π( A ) . (ii) Π( ∃ A ) = ∃ Π( A ) = Π( A ) ∩ ∃ A , (iii) max Π( A ) = max ∃ Π( A ) ,Proof. (i) Let p ∈ Π( A ) and let a, b ∈ A such that ∃ p = a ∨ b . Then ∃ p = ∀ ( a ∨ b ) = ∀ a ∨ ∀ b , so p = p ∧ ∃ p = ( p ∧ ∀ a ) ∨ ( p ∧ ∀ b ). Since p is an irreducible element, p = p ∧ ∀ a or p = p ∧ ∀ b .Suppose that p = p ∧ ∀ a . Then p ≤ ∀ a , and so ∃ p ≤ ∀ a ≤ a ≤ ∃ p . In consequence, ∃ p = a .The other case is similar.(ii) The inclusion ∃ Π( A ) ⊆ Π( A ) ∩∃ A follows from the previous item, and Π( A ) ∩∃ A ⊆ Π( ∃ A )is trivial. It remains to show that Π( ∃ A ) ⊆ ∃ Π( A ). Let a ∈ Π( ∃ A ). Since ∃ a = a , itsuffices to show that a ∈ Π( A ). Let b, c ∈ A such that a = b ∨ c . Then a = ∀ a = ∀ ( b ∨ c ) = ∀ b ∨ ∀ c . Thus, a = ∀ b or a = ∀ c . If a = ∀ b , then a ≤ b ≤ b ∨ c = a , and consequently a = b .Similarly, if a = ∀ c , then a = c .(iii) If m ∈ max Π( A ), then ∃ m ∈ Π( A ) by (i). Since m ≤ ∃ m , from the maximality of m , wehave that m = ∃ m . On the other hand, if m ∈ max ∃ Π( A ) and there is m (cid:48) ∈ Π( A ) suchthat m ≤ m (cid:48) , then m ≤ m (cid:48) ≤ ∃ m (cid:48) and from the maximality of m we have that m = ∃ m (cid:48) and so m = m (cid:48) . Remark . If we know the poset ∃ Π( A ) and the coordinates of its maximal elements, we canfully determine (cid:104) Π( A ) , E (cid:105) . Indeed, since max Π( A ) = max ∃ Π( A ), every p ∈ Π( A ) lies belowa maximal prime element m ∈ ∃ Π( A ). Moreover, the coordinates of m completely describe thechain of prime elements below it, including which of those elements belong to ∃ A (see Remark4.5). Thus the values of ∃ on Π( A ) are immediate and the equivalence relation E is then easilycalculated because ( p, q ) ∈ E if and only if ∃ p = ∃ q .For example, suppose ∃ Π( A ) is the poset shown in Figure 1 ( a ) and assume the coordinatesof the maximal elements m , m , m are (4 , , , , , 0) and (4 , , , A ) is the one given in Figure 1 ( b ), where the elements of ∃ Π( A ) are highlighted.The equivalence relation E is now evident and shown in Figure 1 ( c ). m m m ( a ) m ( b ) m m m ( c ) m m Figure 1: From ∃ Π( A ) to (cid:104) Π( A ) , E (cid:105) 15n what follows we characterize the free algebra F ( n ) in W with n generators using theprocedure described in the last remark. We build the ordered set (cid:104) Π( F ( n )) , E (cid:105) of its join-irreducible elements together with the equivalence relation E that determines the quantifiersfrom the ordered set ∃ Π( F ( n )) and the coordinates of its maximal elements. For the sake ofsimplicity we write Π( n ) instead of Π( F ( n )) and ∃ Π( n ) instead of ∃ Π( F ( n )).Let F ( m,m ,...,m r ) be the set of all functions f from the set G of free generators of F ( n ) into C m such that the subalgebra generated by f ( G ) is C ( m,m ,...,m r ) . If f ∈ F ( m,m ,...,m r ) , then f can be extended to a unique surjective homomorphism ¯ f : F ( n ) → C ( m,m ,...,m r ) and it isknown that ¯ f − ( { } ) = [ p f ) is a monadic prime filter of F ( n ) where p f ∈ ∃ Π( n ). Moreover, p f has coordinates ( m, m , . . . , m r ). On the other hand, if p ∈ ∃ Π( n ) is such that F ( n ) / [ p ) ∼ = C ( m,m ,...,m r ) and we consider the canonical map h : F ( n ) → F ( n ) / [ p ) and the restriction f = h (cid:22) G , then clearly f ∈ F ( m,m ,...,m r ) , ¯ f = h and p f = p . Therefore, there is a bijection betweenthe set F ( m,m ,...,m r ) and the elements of ∃ Π( n ) with coordinates ( m, m , . . . , m r ).Now we want to characterize functions f : G → C m whose image generates C ( m,m ,...,m r ) .Let f : G → C m be such that the subalgebra generated by f ( G ) is C ( m,m ,...,m r ) , that is, f ( G ) ∪ ∃ f ( G ) ∪ ∀ f ( G ) ∪ { , } = C m . Note that, if | G | = n , then m ≤ n .Let ∃ C m = { b = 0 , b , b , . . . , b r , b r +1 = 1 } and let M be the subset of ∃ C m of those elements whose predecessor and successor elements in C m are both in ∃ C m . That is, M = { b j ∈ ∃ C m : m j − = m j = 0 , ≤ j ≤ r } . Then, f ( G ) generates the chain C ( m,m ,...,m r ) if and only if M ∪ ( C m \ ∃ C m ) ⊆ f ( G ). So, | f ( G ) | ≥ | M | + r (cid:88) i =0 m i = | M | + m − r .For each m , 0 ≤ m ≤ n , let us consider the sets I m ( n ) = { ( m , . . . , m r ) : r (cid:88) i =0 m i = m − r and m − r + |{ j : m j − = m j = 0 }| ≤ n } . Observe that | I m ( n ) | is the number of nonisomorphic chains with m + 2 elements that canbe generated by a set of n generators.Let Λ( n ) = n (cid:91) m =0 (cid:91) ( m ,...,m r ) ∈ I m ( n ) F ( m,m ,...,m r ) . If f ∈ Λ( n ), then there is a unique( m , . . . , m r ) ∈ I m ( n ) such that f ∈ F ( m,m ,...,m r ) . Moreover, we have an injection from Λ( n )onto ∃ Π( n ) given by f (cid:55)→ p f . Then, each element of ∃ Π( n ) can be represented by an element ofΛ( n ). From the above results we have the following. Lemma 4.8. |∃ Π( n ) | = n (cid:88) m =0 (cid:88) ( m ,...,m r ) ∈ I m ( n ) | F ( m,m ,...,m r ) | . Example 4.9. For n = 1 we have I (1) = { (0) } , I (1) = { (1) , (0 , } , I (1) = { (1 , , (0 , } ,and I (1) = { (0 , , } . F (0 , is the set of functions from { g } (set of free generators) into C = { a , a } whose imagesgenerate the algebra C (0 , . There are two choices for the image of g in this case: a and a .We represent those functions by (0 , a ) and (0 , a ), writing the value of the functions on g after the semicolon. Thus F (0 , = { (0 , a ) , (0 , a ) } . In a similar way we can see that F (1 , = { (1 , a ) } , F (1 , , = { (1 , , a ) } , F (2 , , = { (2 , , a ) } , F (2 , , = { (2 , , a ) } , F (3 , , , = { (3 , , , a ) } . 16onsequently |∃ Π(1) | = (cid:88) m =0 (cid:88) ( m ,...,m r ) ∈ I m (1) | F ( m,m ,...,m r ) | = | F (0 , | + | F (1 , | + | F (1 , , | + | F (2 , , | + | F (2 , , | + | F (3 , , , | = 7.We say that q covers p if p < q and p ≤ r < q , implies p = r . Remark . In ∃ Π( n ), q covers p if and only if p < q , p has coordinates ( m, m , . . . , m r ),with ( m , . . . , m r ) ∈ I m ( n ), and q has coordinates ( m + m r +1 + 1 , m , . . . , m r , m r +1 ), with( m , . . . , m r , m r +1 ) ∈ I m + m r +1 +1 ( n ).In particular, from Remark 4.10, we have that p is minimal in the set ∃ Π( n ) if and only if p has coordinates ( m, m ), with 0 ≤ m ≤ n .Theorem 4.11 allows us to construct the ordered set ∃ Π( n ) and determine the coordinatesof all its elements. We follow a similar argument given in the proofs of [1, Theorem 3.14] and[19, Theorem 3.10]. Theorem 4.11. Let f , h ∈ Λ( n ) . Then p h covers p f in ∃ Π( n ) if and only if f ∈ F ( m,m ,...,m r ) ,where ≤ m ≤ n − and ( m , . . . , m r ) ∈ I m ( n ) , and h ∈ F ( m + m r +1 +1 ,m ,...,m r ,m r +1 ) , where m + m r +1 + 1 ≤ n and ( m , . . . , m r , m r +1 ) ∈ I m + m r +1 +1 ( n ) , and, for g ∈ G the followingconditions hold:(a) f ( g ) = a i if and only if h ( g ) = a i , ≤ i ≤ m .(b) f ( g ) = 1 = a m +1 if and only if h ( g ) = a i , m + 1 ≤ i ≤ m + m r +1 + 1 .Proof. If p h covers p f in ∃ Π( n ), then p < . . . < p m < p m +1 = p f < . . . < p m + m r +1 +2 = p h inΠ( n ) and the natural homomorphisms ¯ f , ¯ h are defined in the following way:¯ f ( x ) = x ∈ [ p m +1 ) ,a i if x ∈ [ p i ) \ [ p i +1 ) , ≤ i ≤ m, x / ∈ [ p ) , ¯ h ( x ) = x ∈ [ p h ) ,a i if x ∈ [ p i ) \ [ p i +1 ) , ≤ i ≤ m + m r +1 + 1 , x / ∈ [ p ) . In particular, we have that f ∈ F ( m,m ,...,m r ) , h ∈ F ( m + m r +1 +1 ,m ,...,m r ,m r +1 ) and conditions(a) and (b) hold.Conversely, let f ∈ F ( m,m ,...,m r ) , h ∈ F ( m + m r +1 +1 ,m ,...,m r +1 ) satisfying (a) and (b). Then, p f has coordinates ( m, m , . . . , m r ) and p h has coordinates ( m + m r +1 + 1 , m , . . . , m r , m r +1 ).From Remark 4.10, we need to prove that p f < p h .Consider in Π( n ) p < . . . < p m < p m +1 = p f and q < . . . < q m < q m +1 < . . . < q m + m r +1 +2 = p h the chains ( p f ] and ( p h ] respectively, and the following sets: S m + m r +1 +2 = [ p h ) ∩ [ p f ) ,S m + j = ([ q m + j ) \ [ q m + j +1 )) ∩ [ p m +1 ) , ≤ j ≤ m r +1 + 1 ,S i = ([ q i ) \ [ q i +1 )) ∩ ([ p i ) \ [ p i +1 )) , ≤ i ≤ m,S = F ( n ) \ ([ q ) ∪ [ p )) . Then 17 ∈ S m + m r +1 +2 if and only if ¯ h ( a ) = ¯ f ( a ) = 1, a ∈ S m + j if and only if ¯ h ( a ) = a m + j and ¯ f ( a ) = 1, 1 ≤ j ≤ m r +1 + 1, a ∈ S i if and only if ¯ h ( a ) = ¯ f ( a ) = a i , 0 ≤ i ≤ m .It is a routine matter to show that if S := (cid:83) m + m r +1 +2 k =0 S k , then S is a subalgebra of F ( n ) and G ⊆ S . Consequently S = F ( n ). Then we can write, [ p h ) = [ p h ) ∩ F ( n ) =[ p h ) ∩ ( (cid:83) m + m r +1 +2 i =0 S k ) = [ p h ) ∩ [ p f ). Since p h (cid:54) = p f , we have p f < p h . Theorem 4.11 induces an order in Λ( n ) isomorphic to that of ∃ Π( n ). Example 4.12. We already calculated the elements of Λ(1) in Example 4.9. Using Theorem 4.11we can build the corresponding poset, which is shown in Figure 2. p f p f p f p f p f p f p f f := (0 , a ) f := (1 , a ) f := (2 , , a ) f := (0 , a ) f := (1 , , a ) f := (2 , , a ) f := (3 , , , a )Figure 2: ∃ Π(1)Using Remark 4.7 and the coordinates of the maximal elements in ∃ Π(1) we can build (cid:104) Π(1) , E (cid:105) . Figure 3 shows this poset; we show the decreasing set corresponding to the generator g with a dash line as well as terms for each principal decreasing set. ¬ g ∀ gg ∧ ¬∀ g ∃ g ∧ ¬∀ g ¬ ( g → ∀ g ) ( g ∨ ¬ g ) → ∀ gg ∧ ¬¬∀ g ∃ g ∧ ¬¬∀ g (( g → ∀ g ) ∨ ¬∀ g ) → ∀ g Figure 3: (cid:104) Π(1) , E (cid:105) Remark . A few observations on the structure of ∃ Π( n ) can be derived from Theorem 4.11.1. f is maximal in Λ( n ) if and only if 1 / ∈ ∃ f ( G ). By Theorem 4.11, it is clear that if1 / ∈ ∃ f ( G ), then there is not h covering f . On the other hand, let f ∈ F ( m,m ,...,m r ) such that 1 = ∃ f ( a ), for some a ∈ G . Let us consider h ∈ F ( m +1 ,m ,...,m r , defined by h ( g ) = f ( g ), if g (cid:54) = a . If f ( a ) = 1, define h ( a ) = a m +1 , and if f ( a ) (cid:54) = 1, define h ( a ) = f ( a ).By Theorem 4.11, h covers f .2. f is minimal in Λ( n ) if and only if f ∈ F ( m,m ) , with 0 ≤ m ≤ n . For example, if n = 1, then | min Λ(1) | = | F (0 , | + | F (1 , | = 2 + 1 = 3, and, if n = 2, then | min Λ(2) | = | F (0 , | + | F (1 , | + | F (2 , | = 4 + 5 + 2 = 11. More generally, in the case with n freegenerators, for 0 ≤ m ≤ n we have that | F ( m,m ) | = S ( n, m ) + 2 S ( n, m + 1) + S ( n, m + 2),where S ( n, k ) is the number of surjective functions from an n -element set onto a k -elementset (recall that S ( n, k ) = k ! (cid:8) nk (cid:9) , where (cid:8) nk (cid:9) is a Stirling number of the second king).18. Let f ∈ min Λ( n ) such that | f − ( { } ) | = j . From Theorem 4.11 we know that h covers f ifand only if f ∈ F ( m,m ) and h ∈ F ( m (cid:48) ,m,m (cid:48) − m − , with 0 ≤ m ≤ n and m +1 ≤ m (cid:48) ≤ m + j +1,and where f and h also satisfy that(a) f ( g ) = a i if and only if h ( g ) = a i , for any i such that 0 ≤ i ≤ m ,(b) f ( g ) = 1 if and only if h ( g ) = a i , for any i such that m + 1 ≤ i ≤ m (cid:48) + 1.If f ∈ F ( m (cid:48) − m − ,m (cid:48) − m − is the function defined by f ( g ) = (cid:40) h ( g ) = a i , ≤ i ≤ m,a i − ( m +1) if h ( g ) = a i , m + 1 ≤ i ≤ m (cid:48) + 1 , then f is clearly a minimal element of Λ( n ). Let us see that [ h ) and [ f ) are isomorphic.Indeed, if we define α : [ h ) → [ f ) by means of α ( u ) = v , where v ( g ) = (cid:40) u ( g ) = a i , ≤ i ≤ m,a i − ( m +1) if u ( g ) = a i , m + 1 ≤ i ≤ m (cid:48) , then α is clearly an injection and onto mapping. By Theorem 4.11, it is straightforwardto see that α is an isomorphism.Finally, observe that F ( n ) ∼ = A × A , where A = ⊕ A . Example 4.14. With a little more effort we can calculate the elements of Λ(2) and use Theorem4.11 to build the ordered set ∃ Π(2), which turns out to have 71 elements. From this we canproduce the dual space (cid:104) Π(2) , E (cid:105) of the free algebra generated by two elements; in this case ithas 101 elements. The Hasse diagram is shown in Figure 4.Figure 4: (cid:104) Π(2) , E (cid:105) Remark . Observe that, from the previous work, we can obtain the structure of the n -generated free algebra F s ( n ) in H ∃ ∩ W . The subdirectly irreducible algebras in H ∃ ∩ W arethe simple algebras in W , that is, the algebras C ( m,m ) for m ≥ 0. Thus ∃ Π( F s ( n )) is isomorphicto min Λ( n ). Therefore, F ( n ) ∼ = n (cid:89) m =0 C | F ( m,m ) | ( m,m ) . See item 2 in Remark 4.13 for the values of | F ( m,m ) | .19 eferences [1] Abad, M. and Monteiro, L., On free L -algebras, Notas de L´ogica Matem´atica (1987),Univ. Nac. del Sur, Bah´ıa Blanca, 1–20.[2] Bezhanishvili, G., Varieties of monadic Heyting algebras I, Studia Logica (1998), no. 3,367–402.[3] Bezhanishvili, G., Locally finite varieties, Algebra Universalis (2001), n. 4, 531–548.[4] Bezhanishvili, G., Glivenko Type Theorems for Intuitionistic Logics, Studia Logica (2001), 89–109.[5] Blok, W. J., Varieties of interior algebras, Phd. Thesis, University of Amsterdam (1976).[6] Caicedo, X., Rodr´ıguez, R. O., Bi-modal G¨odel logic over [0 , J.Logic Comput. (2015), no. 1, 37–55.[7] Casta˜no, D. , Cimadamore, C. , D´ıaz Varela, J. P. , Rueda, L. , Monadic BL-algebras: Theequivalent algebraic semantics of H´ajek’s monadic fuzzy logic, Fuzzy Sets and Systems (2017), 40–59.[8] Casta˜no, D., Cimadamore, C., D´ıaz Varela, J. P., Rueda, L., Completeness for monadicfuzzy logics via functional algebras, Fuzzy Sets and Systems , in press (2020), DOI10.1016/j.fss.2020.02.002.[9] Cignoli, R., Quantifiers on distributive lattices, Discrete Mathematics (1991), n. 3, 183–197.[10] Davey, B. A., Priestley, H. A., Introduction to lattices and order , second edition, CambridgeUniversity Press, New York, 2002.[11] Esakia, L., Heyting algebras , translated from the Russian edition by Anton Evseev, Trendsin Logic—Studia Logica Library, 50, Springer, Cham, 2019.[12] Glivenko V., Sur quelques points de la logique de M. Brouwer, Bulletin de la Classe desSciences de l’Acad´emie Royale de Belgique (1929), 183–188.[13] H´ajek, P., Metamathematics of fuzzy logic , Trends in Logic—Studia Logica Library ,Kluwer Academic Publishers, Dordrecht, 1998.[14] Hecht, T. and Katriˇn´ak, T., Equational classes of relative Stone algebras, Notre Dame J.Formal Logic (1972), n. 2, 248–254.[15] Monteiro, A., Sur les Alg`ebres de Heyting Sym´etriques, Portugaliae Mathematica (1980),1–237.[16] Monteiro, A. and Varsavsky, O., ´Algebras de Heyting mon´adicas, Actas de las X Jornadasde la Uni´on Matem´atica Argentina , Bah´ıa Blanca, 1957, p. 52-62.[17] Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. (1970), 186–190.[18] Priestley, H. A., Ordered sets and duality for distributive lattices, Ann. Discrete Math. (1984), 39–60.[19] Rueda, L., The subvariety of Q -Heyting algebras generated by chains, Revista de la Uni´onMatem´atica Argentina50