Tense operators on MV -algebras and

Lukasiewicz-Moisil algebras

Denisa Diaconescu, George Georgescu∗

October 2006

ABSTRACT. We introduce the tense LMn-algebras and the tense MV -algebras as algebraic structures for some tense many-valued logics. A rep-resentation theorem for tense LMn-algebras is proved and the polynomiallyequivalence between tense LM3-algebras (resp. tense LM4-algebras) and tenseMV3-algebras (resp. tense MV4-algebras) is established. We study the pairsof dually-conjugated operations on MV -algebras and we use their propertiesin order to investigate how the axioms of tense operators are preserved bythe Dedekind-MacNeille completion of an Archimedean MV -algebra. A tensemany-valued propositional calculus is developed and a completeness theoremis proved.

1 Introduction

The n-valued Lukasiewicz-Moisil algebras (for short, LMn-algebras), introducedby Moisil in [22], were among the first structures of algebraic logic. The initialintension of Moisil was to find the algebras corresponding to Lukasiewicz logics.An example of A. Rose (1957) showed that, for n ≤ 5, the lukasiewiczian impli-cation cannot be defined in an LMn-algebra. In fact, Moisil discovered a newlogical system (named today Moisil logic [23], [8], [3]), whose algebraic modelsare LMn-algebras.

Moisil’s logic is based on the idea of nuance, expressed algebrically by thechrysippian endomorphisms. According to Moisil’s determination principle [23],any n-valued sentence is determined by its Boolean nuances. By [8], the n-valued Lukasiewicz logic and the n-valued Post logic (see [27]), can be developed in thecontext of the n-valued Moisil logic.

On the other hand, Chang defined in [5] the MV -algebras as algebraic struc-tures for the infinite valued Lukasiewicz propositional calculus (see also [9],[15]). He established a bijective correspondence between the totaly-orderedMV -algebras and the totally-ordered abelian l-groups with strong unit, thenused this result in order to obtain an algebraic proof for the completeness the-orem of Lukasiewicz logic [6]. In [24], Mundici extended Chang’s result byproving the categorial equivalence between MV -algebras and abelian l-groups

∗Faculty of Mathematics, University of Bucharest, Str. Academiei Nr. 14, Bucharest,Romania, ddenisuca@yahoo.com, georgescu@funinf.cs.unibuc.ro

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with strong unit. In [14], Grigolia introduced the MVn-algebras, a class ofstructures corresponding to the n-valued Lukasiewicz propositional logic.

According to [25] and [16], the LM3-algebras (resp. LM4-algebras) are poly-nomially equivalent with the MV3-algebras (resp. the MV4-algebras). Otherconnections between LMn-algebras and MVn-algebras were established in [8],[16], [20].

The classical tense logic is a logical system obtained from the bivalent logicby adding the tense operators G (”it is always going to be the case that”) andH (”it has always been the case that”). Starting with other logical systems(intuitionistic calculus, many-valued logics etc.) and adding appropiate tenseoperators we arrive to new tense logics.

This paper deals with the tense LMn-algebras and the tense MVn-algebras,structures obtained from LMn-algebras and MVn-algebras, respectively, byadding some tense operators. It is well-known that the tense Boolean-algebrasconstitute the algebraic basis for the bivalent tense logic [4], [18], [19]. Thusthe tense LMn-algebras and MVn-algebras can offer an algebraic framework inorder to develop some tense many-valued logics. These logical systems can bethough as many-valued logics for which the time is a parameter. If we considerKripke-type semantics, then, for any fixed time, the laws of the basic manyvalued logic hold, and, the temporal operators describe the transition from onetime to another.

This paper is organized as follows:Section 2 contains some background of LMn-algebras. In Section 3 we recall

some basic facts on the representation of tense Boolean algebras. In Section 4we define the tense LMn-algebras and we prove a representation theorem forthese structures. The proof of this result combines the representation theoremfor tense Boolean algebras and some technique based on Moisil’s determinationprinciple. The existence of a representation theorem creates the computationalfacilities in the tense LMn-algebras theory and assures ”good logical properties”for the corresponding logical system (see Section 7).

Section 5 deals with the tense MV -algebras. A source in defining the tenseMV -algebras is the canonical structure defined by a frame (X,R) on an MV -algebra of function LX (see Example 5.3). Another source is the axiomatizationof monadicMV -algebras [28]. We characterize the simple tenseMV -algebras, aswell as the subdirect irreducible tense MV -algebras. A representation theoremfor all tense MV -algebras seems to be difficult to prove; the section includessome results towards a representation theorem for some classes of tense MV -algebras.

In Section 6 we study the dually-conjugated operations on an MV -algebra,motivated by the last axiom in the definition of tense MV -algebras. We use theirproperties in analysing how the axioms of the tense MV -algebras are preservedby the Dedekind-MacNeille completion of an Archimedean MV -algebra.

Section 7 concerns with a tense logical system TMn based on the Moisillogic Mn. The Lindenbaum-Tarski algebra of TMn is a tense LMn-algebra,hence, by using the representation theorem given in Section 4, we obtain acompleteness theorem for TMn.

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The paper ends with a section containing comments and open questionsabout tense LMn-algebras, tense MVn-algebras and their logical systems.

2 Preliminaries on LMn-algebras

In this section we shall recall some definitions and basic facts on n-valued Lukasiewicz-algebras ([3], [8], [16], [22], [23]).

Definition 2.1 Let n ≥ 2. An n-valued Lukasiewicz-Moisil algebra (forshort, LMn-algebra) is a distributive lattice with a first and a last element,(L,∨,∧, 0, 1), such that:

LM-1) There is a map ¬ : L→ L with the properties:

• ¬(x ∨ y) = ¬x ∧ ¬ y,

• ¬(x ∧ y) = ¬x ∨ ¬ y,

• ¬(¬x) = x.

LM-2) There are (n − 1) maps ϕi : L → L, i = 1, . . . , n − 1 (known aschrysippian endomorphisms) which have the properties:

1. ϕi(0) = 0, ϕi(1) = 1, for any i = 1, . . . , n− 1,

2. ϕi(x ∧ y) = ϕi(x) ∧ ϕi(y) and ϕi(x ∨ y) = ϕi(x) ∨ ϕi(y), for anyx, y ∈ L and i = 1, . . . , n− 1,

3. ϕi(x) ∨ ¬ϕi(x) = 1 and ϕi(x) ∧ ¬ϕi(x) = 0, for any x ∈ L andi = 1, . . . , n− 1,

4. ϕh ◦ ϕk = ϕk, for any h, k = 1, . . . , n− 1,

5. ϕ1(x) ≤ ϕ2(x) ≤ . . . ≤ ϕn−1(x), for any x ∈ L,

6. ϕi(¬x) = ¬ϕj(x), where i+ j = n,

7. If ϕi(x) = ϕi(y), for any i = 1, . . . , n − 1, then x = y, for anyx, y ∈ L.

Axiom 7 is known as Moisil’s determination principle1.

Definition 2.2 Let L1, L2 be two LMn algebras. A morphism of LMn al-gebras f : L1 → L2 is a map who satisfies the conditions:

• f(0) = 0, f(1) = 1, f ◦ ¬ = ¬ ◦ f ,

• f(x ∨ y) = f(x) ∨ f(y), f(x ∧ y) = f(x) ∧ f(y), for any x, y ∈ L1,

1The determination principle does not appear as an axiom in the first definitions of LMn-algebras (for n = 3, 4) [22].

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• f ◦ ϕi = ϕi ◦ f , for any i = 1, . . . , n− 1.

Example 2.3 Let Ln = {0, 1n−1 ,

2n−1 , . . . ,

n−2n−1 , 1}. We consider the following

operations on Ln: x ∨ y = max(x, y), x ∧ y = min(x, y), ¬x = 1 − x for anyx, y ∈ Ln. We define the maps ϕ1, . . . , ϕn−1 : Ln → Ln by:

ϕi(j

n− 1) =

{0, if i+ j < n1, if i+ j ≥ n

,

for any 0 ≤ j ≤ n− 1 and 0 ≤ i ≤ n− 1.Ln with the operations ∨, ∧, ¬, ϕ1, . . . , ϕn−1 becomes an n-valued

Lukasiewicz-Moisil algebra.

Remark 2.4 Let L be an LMn-algebra. For any x, y ∈ L the following hold(i) x ≤ y iff ϕix ≤ ϕiy for all i = 1, . . . n− 1, and (ii) ϕ1x ≤ x ≤ ϕn−1x.

We fix an LMn-algebra L. Let us denote byB(L) the set of the complementedelements of L (an element x ∈ L is complemented if there exists y ∈ L such thatx ∨ y = 1 and x ∧ y = 0). It is easy to see that

B(L) = {x ∈ L | x ∨ ¬x = 1 and x ∧ ¬x = 0}

(B(L),∨,∧,¬, 0, 1) is a Boolean algebra. The following lemma characterizesthe elements of B(L):

Lemma 2.5 B(L) = {x ∈ L | ϕix = x, for any i = 1, . . . , n− 1}

If we interpret the elements of L as (equivalence classes of) propositionsin a many-valued logic, then the elements of B(L) will represent the booleanpropositions. The chrysippian endomorphisms ϕ1, . . . , ϕn−1 associate, with anyn-valued sentence x, the boolean sentences ϕ1x, . . . , ϕn−1x (the nuances of x).Thus, Moisil’s determination principle is a strong thesis expressing that everyn-valued proposition is determined by its nuances.

Let (B,∨,∧,¬, 0, 1) be a Boolean algebra. Let us consider the following set

D(B) = {(x1, . . . , xn−1) ∈ Bn−1 | x1 ≤ . . . ≤ xn−1}

We define on D(B) the following unary operations:

N : D(B) → D(B), N(x1, . . . , xn−1) = (¬xn−1, . . . ,¬x1),ϕi : D(B) → D(B), ϕi(x1, . . . , xn−1) = (xi, . . . , xi), i = 1 . . . , n− 1,

for all (x1, . . . , xn−1) ∈ D(B).

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Lemma 2.6 (D(B),∨,∧, N, ϕ1, . . . , ϕn−1, (0, . . . , 0), (1 . . . , 1)) is an LMn-algebra.

Remark 2.7 Let B be the category of Boolean algebras and Lukn the categoryof LMn-algebras. The assignments L 7−→ B(L) and B 7−→ D(B) define twoadjoint functors Lukn → B and B → Lukn. B is isomorphic to a reflectivesubcategory of Lukn. For any LMn-algebra L we denote by ψL : L→ D(B(L))the function defined by ψL(x) = (ϕ1x, . . . , ϕn−1x), for any x ∈ L. ψL is aninjective morphism of Lukn. By [3], p.169, L is an n-valued Post algebra iffψL is an isomorphism. Consequently, the category of n-valued Post algebrasis equivalent to the category of Boolean algebras [3]. Particularly, Ln (fromExample 2.3) is a Post algebra isomorphic to D(L2).

One of the main results in LMn-algebras theory is the following representa-tion theorem:

Theorem 2.8 (Moisil’s representation theorem) For any LMn-algebra L,there is a nonempty set I and an injective morphism of LMn-algebras d : L →LI

n.

The previous theorem reduces the calculus in an arbitrary LMn-algebra Lto the calculus in Ln.

In an LMn-algebra L let us define the following binary operation ⇒:

x⇒ y := y ∨n−1∨i=1

(¬(ϕix) ∨ ϕiy)

Proposition 2.9 (L,∨,∧,⇒, 0, 1) is a Heyting algebra.

Proposition 2.10 Any chrysippian endomorphism ϕi preserves the arbitrarysuprema and infima, whenever they exist.

3 Tense Boolean algebras

Tense Boolean algebras are algebraic structures corresponding to the proposi-tional tense logic [4], [11], [18]. In this section we shall remind some basic factson the representation of tense Boolean algebras [19].

Let (A,∨,∧,¬, 0, 1) be a Boolean algebra and two maps G,H : A→ A. Wedefine F, P : A→ A by F (x) = ¬G(¬x) and P (x) = ¬H(¬x), for any x ∈ A.

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Definition 3.1 (A,G,H) is a tense Boolean algebra if the following hold:

a) G(1) = 1, H(1) = 1,

b) G(x∧ y) = G(x)∧G(y) and H(x∧ y) = H(x)∧H(y), for any x, y ∈ A,

c) x ≤ GP (x) and x ≤ HF (x), for any x ∈ A.

The Lindenbaum-Tarski algebra of the tense propositional calculus is thefirst example of tense Boolean algebra.

Definition 3.2 A frame is a pair (X,R), where X is a nonempty set and Ris a binary relation on X.

The notion of frame allows us to construct the second example of tenseBoolean algebra. Given a frame (X,R), we define G∗,H∗ : LX

2 → LX2 by:

G∗(p)(x) =∧{p(y) | y ∈ X, xRy}, H∗(p)(x) =

∧{p(y) | y ∈ X, yRx}

for all p ∈ LX2 and x ∈ X.

Proposition 3.3 For any frame (X,R), (LX2 , G

∗,H∗) is a tense Boolean alge-bra.

It is easy to see that the derivate tense operators F ∗, P ∗ are given by:

F ∗(p)(x) =∨{p(y) | y ∈ X, xRy}, P ∗(p)(x) =

∨{p(y) | y ∈ X, yRx}

for all p ∈ LX2 and x ∈ X.

Definition 3.4 For any two tense Boolean algebras (A,G,H) and (A′, G′,H ′),a morphism of tense Boolean algebras f : (A,G,H) → (A′, G′,H ′)is a morphism of Boolean algebras which satisfies f(G(a)) = G′(f(a)) andf(H(a)) = H ′(f(a)), for any a ∈ A.

By this definition it follows that a morphism of tense Boolean algebras com-mutes to the tense operators F , P .

Theorem 3.5 (The representation theorem for tense Boolean algebras)For any tense Boolean algebra (A,G,H), there is a frame (X,R) and an injec-tive morphism of tense Boolean algebras d : (A,G,H) → (LX

2 , G∗,H∗).

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4 Tense operators on LMn-algebras

In this section we shall define the tense LMn-algebras and we shall prove arepresentation theorem for these structures. This result will provide a proof ofthe completeness theorem for the tense n-valued Moisil logic.

Definition 4.1 Let (A,∨,∧,¬, 0, 1, ϕ1, . . . , ϕn−1) be an LMn-algebra andG,H : A → A two unary operations on A. We say that (A,G,H) is a tenseLMn-algebra if the following axioms are satisfied:

(a) G(1) = 1, H(1) = 1,

(b) G(x ∧ y) = Gx ∧Gy, H(x ∧ y) = Hx ∧Hy,

(c) Gϕi = ϕiG, Hϕi = ϕiH, for any i = 1, . . . , n− 1,

(d) x ≤ GP (x), x ≤ HF (x), where F : A→ A and P : A→ A are definedby Fx = ¬G(¬x), Px = ¬H(¬x).

The previous definition contains the specific axiom (c), which says that thetense operators G,H commute with the chrysippian endomorphisms ϕi.

Remark 4.2 Let (A,G,H) be a tense LMn-algebra. If x ∈ B(A) then ϕiGx =Gϕix = Gx, for any i = 1, . . . , n−1, hence Gx ∈ B(A), and, similarly, x ∈ B(A)implies Hx ∈ B(A). Thus we can consider the unary operations G1 = G|B(A),H1 = H|B(A) of B(A) and (B(A), G1,H1) is a tense Boolean algebra.

According to this remark one gets the following result:

Proposition 4.3 The following properties hold in any tense LMn-algebra(A,G,H):

(1) If x ≤ y then Gx ≤ Gy, Hx ≤ Hy, Fx ≤ Fy, Px ≤ Py,

(2) GPx ∧ Fy ≤ F (Px ∧ y),

(3) Gx ∧ Fy ≤ F (x ∧ y),

(4) Gx ∧ Fy ≤ G(x ∧ y),

(5) G(x ∨ y) ≤ Gx ∨ Fy,

for any x, y ∈ A.

Proof. Conditions (1)-(5) hold in the tense Boolean algebra (B(A), G1,H1), so,by the determination principle, they are available in (A,G,H). a

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Proposition 4.4 Let G,H be two unary operations on the LMn-algebra L.If G,H satisfy G(1) = 1, H(1) = 1 then the axiom (b) of Definition 4.1 isequivalent to the following condition:

(b’) G(x⇒ y) ≤ Gx⇒ Gy, H(x⇒ y) ≤ Hx⇒ Hy, for all x, y ∈ L.

Proof. (b) ⇒ (b′). From (b) it follows that G and H are increasing. ByProposition 2.9 we have x∧y = x∧(x⇒ y), hence Gx∧G(x⇒ y) = G(x∧(x⇒y)) = G(x ∧ y) ≤ G(y), therefore G(x⇒ y) ≤ Gx⇒ Gy.

(b′) ⇒ (b). If x ≤ y then x ⇒ y = 1, so 1 = G(x ⇒ y) ≤ Gx ⇒ Gy, i.e.Gx ≤ Gy. Thus G is increasing. By Proposition 2.9, x ≤ y ⇒ (x ∧ y), henceGx ≤ G(y ⇒ (x ∧ y)) ≤ Gy ⇒ G(x ∧ y). Thus Gx ∧ Gy ≤ G(x ∧ y). Theconverse inequality G(x ∧ y) ≤ Gx ∧Gy follows from that G is increasing. a

Let (X,R) be a frame. We define G∗,H∗ : LXn → LX

n as follows:

(G∗p)(x) =∧{p(y) | y ∈ X,xRy}, (H∗p)(x) =

∧{p(y) | y ∈ X, yRx}

for all p ∈ LXn and x ∈ X.

Proposition 4.5 (LXn , G

∗,H∗) is a tense LMn-algebra.

Proof. Immediately from the definition. a

Definition 4.6 If (A,G,H) and (A′, G′,H ′) are two tense LMn-algebras, thena morphism of tense LMn-algebras f : (A,G,H) → (A′, G′,H ′) is a morphismof LMn-algebras such that f(G(a)) = G′(f(a)) and f(H(a)) = H ′(f(a)), forany a ∈ A.

Now we shall prove a representation theorem for tense LMn-algebras thatgeneralizes Theorem 3.5.

Proposition 4.7 Let (A,G,H) be a tense LMn-algebra. Then there exists aframe (X,R) and an injective morphism of tense LMn-algebras

Φ : (A,G,H) → (LXn , G

∗,H∗).

Proof. We consider the tense Boolean algebra (B(A), G1,H1). From the rep-resentation theorem for tense Boolean algebra (Theorem 3.5), there is a frame(X,R) and an injective morphism of tense Boolean algebras

d : (B(A), G1,H1) → (LX2 , G

∗1,H

∗1 ).

Let ϕ′1, . . . , ϕ′n−1 be the chryssipian endomorphisms of Ln. We recall that for

any u1, . . . , un−1 ∈ L2 such that u1 ≤ . . . ≤ un−1, there is an unique element

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u ∈ Ln such that ϕ′i(u) = ui, for any i = 1, . . . , n − 1 (since Ln is an n-valuedPost algebra).

We define Φ : A→ LXn as follows:

Φ(a)(x) = u iff d(ϕia)(x) = ϕ′i(u), i = 1, . . . , n− 1,

for any a ∈ A, x ∈ X and u ∈ Ln (because d(ϕ1a)(x) ≤ . . . ≤ d(ϕn−1a)(x)in L2, then there is an element u ∈ Ln with the property from the previousdefinition). It is easy to see that Φ commutes ∨, ∧, ¬, 0, 1. We check thatΦ commutes ϕi, for any i = 1, . . . , n − 1. This is equivalent to show thatϕ′i(Φ(a)(x)) = (Φ(ϕi(a)))(x), for all a ∈ A, x ∈ X and i = 1, . . . , n− 1.

Let Φ(a)(x) = u. We have d(ϕia)(x) = ϕ′i(u), for any i = 1, . . . , n − 1. Toshow that ϕ′i(u) = (Φ(ϕi(a)))(x) is the same with d(ϕj(ϕi(a)))(x) = ϕ′j(ϕ′i(u)),for any j = 1, . . . , n − 1 (by definition of Φ). The last equality is the samewith d(ϕi(a))(x) = ϕ′i(u), which we know. Let us show that Φ commutes withthe tense operators. Now we must prove that Φ(Ga)(x) = G∗(Φ(a))(x), for alla ∈ A and x ∈ X. We remark that

Φ(Ga)(x) = u iff d(ϕi(Ga))(x) = ϕ′i(u), i = 1, . . . , n− 1iff d(Gϕi(a))(x) = ϕ′i(u), i = 1, . . . , n− 1iff G∗1(d(ϕia))(x) = ϕ′i(u), i = 1, . . . , n− 1

iff∧xRy

d(ϕia)(y) = ϕ′i(u), i = 1, . . . , n− 1

and

G∗(Φ(a))(x) = u iff∧xRy

Φ(a)(y) = u

iff ϕ′i(∧xRy

Φ(a)(y)) = ϕ′i(u), i = 1, . . . , n− 1

iff∧xRy

ϕ′i(Φ(a)(y)) = ϕ′i(u), i = 1, . . . , n− 1

In accordance to these equivalences, it suffices to prove that d(ϕia)(y) =ϕ′i(Φ(a)(y)), for any i = 1, . . . , n − 1. Let j ∈ {1, . . . , n − 1}. We observe thatϕjϕia = ϕia and d(ϕia)(y) ∈ L2, so d(ϕjϕia)(y) = d(ϕia)(y) = ϕ′j(d(ϕia)(y)),for any j = 1, . . . , n − 1. Using the definition of Φ we get that d(ϕia)(y) =ϕ′i(Φ(a)(y)).

Now we must prove that Φ is injective. We show that if Φ(a) = 1, then a = 1.Assume Φ(a) = 1. Let x ∈ X. Then Φ(a)(x) = 1, hence d(ϕia)(x) = ϕ′i(1) = 1,for any i = 1, . . . , n. It follows that d(ϕia) = 1 for any i = 1, . . . , n− 1, hence,by the injectivity of d, we obtain ϕia = 1, for any i = 1, . . . , n − 1. Therefore,by the determination principle, a = 1. a

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Remark 4.8 Let (B,G,H) be a tense Boolean algebra and D(B) the LMn-algebra associated with B. If D(G), D(H) are the unary operations of D(B)defined by D(G)(x1, . . . , xn−1) = (Gx1, . . . , Gxn−1) and D(H)(x1, . . . , xn−1) =(Hx1, . . . ,Hxn−1), for any (x1, . . . , xn−1) ∈ D(B), then (D(B), D(G), D(H)) isa tense LMn-algebra. Thus, as in Remark 2.7, we obtain two adjoint functorsB 7→ D(B) and L 7→ B(L) between the category of tense Boolean algebras andthe category of tense LMn-algebras. The category of tense Boolean algebras isequivalent to the category of tense n-valued Post algebras.

Remark 4.9 (1Ln, 1Ln

) is the unique pair of tense operators on the LMn-algebra Ln. This follows by using that the LMn-algebras Ln and D(L2) areisomorphic.

5 Tense MV -algebras

MV -algebras are introduced by Chang in [5] as algebraic models for infinite-valued propositional Lukasiewicz logic. The basic texts on MV -algebras arein [8] and [15]. In this section we shall define the tense MV -algebras andwe shall establish some basic properties for these structures. We discuss therepresentation problem for tense MV -algebras and we characterize the simpleand subdirectly irreducible tense MV -algebras.

Definition 5.1 An MV -algebra is a structure (A,⊕,�,−, 0, 1), where ⊕ and� are binary operations, − is unary and 0,1 are constants, satisfying the fol-lowing axioms:

M1) (A,⊕, 0) and (A,�, 1) are commutative monoids,

M2) x� 0 = 0 and x⊕ 1 = 1, for any x ∈ A,

M3) x−− = x, for any x ∈ A,

M4) (x⊕ y)− = x− � y−, for any x, y ∈ A,

M5) (x� y−)⊕ y = (y � x−)⊕ x, for any x, y ∈ A.

In any MV -algebra A we can introduce the new operations ∨, ∧ and →:

x ∨ y = x⊕ (x− � y) = y ⊕ (y− � x), x ∧ y = x� (x− ⊕ y) = y � (y− ⊕ x), x→ y = x− ⊕ y.

The main examples of MV -algebra are the real interval [0, 1] and Ln, n ≥ 2,where: a⊕ b = min(a+ b, 1), a� b = max(a+ b− 1, 0), a− = 1− a.

Let A be an MV -algebra. If x ∈ A then we define nx and xn by induction:0x = 0, (n+ 1)x = nx⊕ x, x0 = 1, xn+1 = xn � x.

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Definition 5.2 Let (A,⊕,�,−, 0, 1) be an MV -algebra and G,H : A→ A twounary operations on A. The structure (A,G,H) is called tense MV -algebra ifthe following conditions are fulfiled:

A0: G(1) = 1, H(1) = 1,

A1: G(x → y) ≤ G(x) → G(y), H(x → y) ≤ H(x) → H(y), for anyx, y ∈ A,

A2: G(x)⊕G(y) ≤ G(x⊕ y), H(x)⊕H(y) ≤ H(x⊕ y), for any x, y ∈ A,

A3: G(x⊕ x) ≤ G(x)⊕G(x), H(x⊕ x) ≤ H(x)⊕H(x), for any x ∈ A,

A4: F (x)⊕ F (x) ≤ F (x⊕ x), P (x)⊕ P (x) ≤ P (x⊕ x), for any x ∈ A,

A5: x ≤ GP (x), x ≤ HF (x), where F and P are the unary operations ofA defined by Fx = (Gx−)−, Px = (Hx−)−, for all x ∈ A.

Example 5.3 Let L be a complete MV-chain, (X,R) be a frame and G∗, H∗

the unary operations on the MV -algebra LX defined by G∗(p)(x) =∧{p(y) | y ∈

X.xRy}, H∗(p)(x) =∧{p(y) | y ∈ X, yRx}, for all p ∈ LX and x ∈ X. Then

(LX , G∗,H∗) is a tense MV -algebra.

Proposition 5.4 The following hold in any tense MV -algebra (A,G,H):

(1) If x ≤ y then Gx ≤ Gy, Hx ≤ Hy, Fx ≤ Fy, Px ≤ Py;

(2) G(x→ y) ≤ Fx→ Fy, H(x→ y) ≤ Px→ Py;

(3) Gx�Gy ≤ G(x� y), Hx�Hy ≤ H(x� y);

(4) F (x⊕ y) ≤ Fx⊕ Fy, P (x⊕ y) ≤ Px⊕ Py;

(5) G(x ∨ y) ≤ Fx ∨Gy, H(x ∨ y) ≤ Px ∨Hy;

(6) G(x⊕ x) = Gx⊕Gx, G(x� x) = Gx�Gx;

(7) H(x⊕ x) = Hx⊕Hx, H(x� x) = Hx�Hx;

(8) F (x⊕ x) = Fx⊕ Fx, F (x� x) = Fx� Fx;

(9) P (x⊕ x) = Px⊕ Px, P (x� x) = Px� Px;

(10) x� Fy ≤ F (Px� y);

(11) PGx ≤ x, HFx ≤ x;

(12) PGP = P , GPG = G, HFH = H, FHF = F ;

(13) G and H preserve the arbitrary infima, whenever they exist;

11

(14) F and P preserve the arbitrary suprema, whenever they exist;

for any x, y ∈ A.

Proof. (1) If x ≤ y then x→ y = 1. From (A0) we obtain G(x→ y) = 1 andfrom (A1) Gx→ Gy = 1 hence Gx ≤ Gy. Similarly for H, F, P .

(2) G(x→ y) = G(y− → x−) ≤ Gy− → Gx− = (Gx−)− → (Gy−)− =Fx→ Fy.

(3) In any MV -algebra we have: a� (a→ b) = a� (a− ⊕ b) = a ∧ b. Thenwe have:

G(x− ⊕ y)�Gx = G(x→ y)�Gx ≤ (Gx→ Gy)�Gx = Gx ∧Gy ≤ Gy

By replacing y with x� y we obtain G(x− ⊕ (x� y))�Gx ≤ G(x� y), so

Gy �Gx ≤ G(x− ∨ y)�Gx = G(x− ⊕ (x� y))�Gx ≤ G(x� y).

(4) From (3) we have Gx− �Gy− ≤ G(x− � y−). Then

F (x⊕ y) = (G(x− � y−))− ≤ (Gx− �Gy−)− = Fx⊕ Fy.

(5) We know that x ∨ y = (x→ y) → y and by (A1) we obtain

G(x ∨ y) ≤ G(x→ y) → GyFx→ Gy = (Fx)− ⊕Gy = Gx− ⊕Gy ≤ G(x− ⊕ y) = G(x→ y).

We remark that

G(x→ y) → Gy = (G(x→ y))− ⊕Gy ≤ (Fx→ Gy)− ⊕Gy =(Fx� (Gy)−)⊕Gy = Fx ∨Gy.

So we have: G(x ∨ y) ≤ Fx ∨Gy.(6)-(9) are obvious from (A2), (A3), (A4), (3) and (4).(10) By (1) and (2), Gx ≤ G(y → x�y) ≤ Fy → F (x�y), hence Gx�Fy ≤

F (x�y). Thus, by using (A5) from Definition 5.2, we get x�Fy ≤ GPx�Fy ≤F (Px� y).

(11) By (A5), x− ≤ HFx− = H(Gx)−, so PGx = (H(Gx)−)− ≤ x.(12) By (A5) and (1), Px ≤ PGPx and, by (11), PGPx ≤ Px, so PGPx =

Px. The other equalities follows similarly.(13) We shall prove that G(

∧i ∈ I

xi) =∧i ∈ I

G(xi) whenever this meet exist.

By (1), G(∧i ∈ I

xi) ≤ G(xi), for all i ∈ I, so G(∧i ∈ I

xi) ≤∧i ∈ I

G(xi). Suppose

y ≤ G(xi), for all i ∈ I. Then, by (1) and (11), P (y) ≤ PG(xi) ≤ xi, for all i ∈I, therefore P (y) ≤

∧i ∈ I

xi. According to (A5) and (1), we get y ≤ GP (y) ≤ G(∧i ∈ I

xi). We have established the converse inequality∧i ∈ I

G(xi) ≤ G(∧i ∈ I

xi).

(14) Similar to (13). a

12

Remark 5.5 Assume that the unary operations G, H on the MV-algebra Averify the axiom (A0) of Definition 5.2. Then the axiom (A1) is equivalentwith the conjunction of the following conditions:

(A′

1) Gx�Gy ≤ G(x� y) and Hx�Hy ≤ H(x� y), for all x, y ∈ A.

(A′′

1 ) G and H are increasing, i.e. x ≤ y implies Gx ≤ Gy and Hx ≤ Hy,for all x, y ∈ A.

According to the proof of Proposition 5.4, it remains to show that (A′

1) and(A

′′

1 ) implies (A1). For all x, y ∈ A we have Gx � G(x → y) ≤ G(x � (x →y)) = G(x ∧ y) ≤ G(y), hence G(x→ y) ≤ Gx→ Gy.

Recall from [9] that the subset B(A) = {x ∈ A | x ⊕ x = x} of an MV -algebra A is a Boolean algebra. If (A,G,H) is a tense MV -algebra, then(B(A), G|B(A),H|B(A)) is a tense Boolean algebra.

Lemma 5.6 Let (A,G,H) be a tense MV -algebra, X a non-empty set, L acomplete MV -chain and f : A → LX a morphism of MV -algebras. For anyx, y ∈ X, let us denote:

αxy =∧a∈A

(f(a)(x) → f(Pa)(y)); βxy =∧b∈A

(f(b)(y) → f(Fb)(x));

γxy =∧c∈A

(f(Gc)(x) → f(c)(y)); δxy =∧d∈A

(f(Hd)(x) → f(d)(y)).

Then αxy = βxy = γxy = δxy.

Proof. We remark that

βxy =∧b∈A

(f(b)(y) → f(Fb)(x)) =∧c∈A

(f(c−)(y) → f(Fc−)(x))

=∧c∈A

(f((Fc−)−)(x) → (f(c−))−(y)) =∧c∈A

(f(Gc)(x) → f(c)(y))

= γxy

and similarly, αxy = δxy. Let c ∈ A. Since PGc ≤ c, it follows thatαxy ≤ f(Gc)(x) → f(PGc)(y) ≤ f(Gc)(x) → f(c)(y), therefore αxy ≤∧

c∈A(f(Gc)(x) → f(c)(y)) = γxy. In a similar way we can prove thatβxy ≤ δxy.a

Given a morphism of MV -algebras f : A→ LX we shall define the followingbinary relation Rf on X: Rf = {(x, y) ∈ X2 | αxy = 1}.

Let us consider the tense MV -algebra (LX , G∗f ,H∗f ) associated with the

frame (X,Rf ) (cf. Example 5.3).

13

Proposition 5.7 The following assertions are equivalent:

(a) f : (A,G,H) → (LX , G∗f ,H∗f ) is a morphism of tense MV -algebras;

(b) For all a ∈ A and x ∈ X, the inequalities∧{f(a)(y) | xRfy} ≤

f(Ga)(x) and∧{f(a)(y) | yRfx} ≤ f(Ha)(x) hold;

(c) For all a ∈ A and x ∈ X the inequalities f(Fa)(x) ≤∨{f(a)(y) | xRfy} and f(Pa)(x) ≤

∨{f(a)(y) | yRfx} hold.

Proof. Let a ∈ A and x ∈ X. According to the definition of Rf wehave f(Ga)(x) ≤ f(a)(y), for all y ∈ X such that xRfy. Thus the equal-ity f(Ga)(x) = G∗f (f(a))(x) is equivalent to (i). Similarly, f(Ha)(x) =H∗

f (f(a))(x) is equivalent to (ii), therefore (a) and (b) are equivalent. Therest of the proof follows similarly. a

Remark 5.8 Assume that (A,G,H) is a tense Boolean algebra, X is theprime spectrum SpecA of A and f : A → LX

2 is the Stone morphism. Thusf : (A,G,H) → (A,G∗f ,H

∗f ) is the morphism of tense Boolean algebras which

provides the proof of Theorem 3.5 (see [19]).

Let (A,G,H) a tense MV -algebra. We recall the following representationtheorems:

• If A is a semisimple MV -algebra then there exists a non-empty set X andan injective morphism of MV -algebras νA : A→ [0, 1]X (see [29]).

• If A is an arbitrary MV -algebra then there exists a non-empty set X andan injective morphism of MV -algebras µA : A → ([0, 1]∗)X , where [0, 1]∗

is an ultrapower of [0, 1] (see [26]).

In Section 4 we have proved a representation theorem for the tense LMn-algebras. The proof of a representation theorem for some classes of tense MV -algebras would invoke the existing representation theorems for MV -algebras.Thus one can formulate the following open problem:

Open problem 5.9 Characterize the tense MV -algebras (A,G,H) such thatνA : (A,G,H) → ([0, 1]X , G∗ν ,H

∗ν ) (resp. µA : (A,G,H) → (([0, 1]∗)X

, G∗µ,H∗µ))

is a morphism of tense MV -algebras.

Let (A,G,H) be a tense MV -algebra. We define the unary operations d andρ of A by dx = x∧Gx∧Hx and ρx = x�Gx�Hx, for any x ∈ A. We observethat ρx ≤ dx ≤ x, for any x ∈ A. If (A,G,H) is a tense Boolean algebra thend = ρ and we obtain the notion studied in [17], p.64. For n ∈ ω we define dnxand ρnx by induction: d0x = ρ0x = x, dn+1x = d(dnx) and ρn+1 = ρ(ρnx).

14

Lemma 5.10 For any x, y ∈ A and n ∈ ω, the following hold:

(1) dn0 = 0, dn1 = 1, dn+1x ≤ dnx;

(2) if x ≤ y then dnx ≤ dny;

(3) x ≤ dn(dnx−)−;

(4) if dx = x then dx− = x−;

(5) x = dx iff dnx = x.

Proof. (1) and (2) are obvious.(3) We remark that

x ≤ x ∧GPx ∧HFx ≤ (x ∨ Px ∨ Fx) ∧G(x ∨ Px ∨ Fx) ∧H(x ∨ Px ∨ Fx) =

= d(x∨Px∨Fx) = d(x∨(Hx−)−∨(Gx−)−) = d(x−∧Gx−∧Hx−)− = d(dx−)−.Then (dnx−)− ≤ d(d(dnx−)−−)− = d(dn+1x−)−, hence dn(dnx−)− ≤

dn+1(dn+1x−)−. Now (3) follows by induction.(4) Assume dx = x, hence, by (3), x− ≤ d(dx)− = dx−. The converse

inequality dx− ≤ x− is obvious, then dx− = x−.(5) Obvious. a

Proposition 5.11 The set Ad = {x ∈ A | dx = x} is closed under the MV -operations of A.

Proof. We remark that x ∈ Ad iff x ≤ Gx and x ≤ Hx. Assume x, y ∈ Ad,hence, by (A2) of Definition 5.2, x ⊕ y ≤ Gx ⊕ Gy ≤ G(x ⊕ y), and, similarly,x⊕y ≤ H(x⊕y). Then x⊕y ∈ Ad. According to Lemma 5.10, (4), Ad is closedunder the negation −, hence Ad is also closed under �. By Lemma 5.10, (1),the elements 0 and 1 belong to Ad. In conclusion, Ad is an MV -subalgebra ofA. a

Remark 5.12 Let f : (A1, G1,H1) → (A2, G2,H2) be a morphism of tenseMV -algebras. If x ∈ Ad

1 then f(x) ∈ Ad2, hence we can define a function

fd = f |Ad1

: Ad1 → Ad

2. fd is a morphism of MV -algebras. In fact, the assigmentA 7→ Ad and f 7→ fd defines a covariant functor (.)d from the category of tenseMV -algebras to the category of MV -algebras.

A nonempty subset Σ of an MV -algebra A is an MV -filter if it is closedunder � and x ∈ Σ, x ≤ y implies y ∈ Σ. Any MV -filter Σ of A determines anMV -congruence ∼Σ of A defined by: x ∼Σ y iff x → y ∈ Σ, y → x ∈ Σ. Theassignment Σ 7→∼Σ defines a bijective correspondence between the MV -filtersand the MV -congruences of A. A tense MV -filter of (A,G,H) is an MV -filterΣ of A which is closed under the operations G and H.

15

Remark 5.13 An MV -filter Σ of A is a tense MV -filter iff Σ is closed underd iff Σ is closed under ρ.

Proposition 5.14 There exists a bijective correspondence between the tenseMV -fliters of (A,G,H) and the congruences of (A,G,H).

Proof. Straightforward.

Proposition 5.15 For any x, y ∈ A and k, n ∈ ω the following hold:

(1) ρn0 = 0, ρn1 = 1, ρn+1x ≤ ρnx;

(2) if x ≤ y then ρnx ≤ ρny;

(3) ρk(x)� ρk(y) ≤ ρk(x� y);

(4) ρk(xn) = (ρkx)n.

Proof. (1) and (2) are obvious.(3) By Proposition 5.4, (3), we obtain ρ(x)� ρ(y) = x� y �G(x)�G(y)�

H(x) �H(y) ≤ x � y � G(x � y) �H(x � y) = ρ(x � y). Then (3) follows byinduction.

(4) According to Proposition 5.4, (6), we have ρ(an) = an�G(an)�H(an) =an� (Ga)n� (Ha)n = (a�Ga�Ha)n = (ρa)n. From this equality, (4) followsimmediately. a

Proposition 5.16 The tense MV -filter [ a)TMV of (A,G,H) generated by {a}has the following form:

[ a)TMV = {x ∈ A | ρn(an) ≤ x, for some n ∈ ω}.

Proof. Denote by Σ the right member of the above equality. Let x, y ∈ Σ,so ρn(an) ≤ x and ρn(an) ≤ y for some n ∈ ω. Thus, by Lemma 5.15, we getρ2n(a2n) ≤ ρn(a2n) = ρn(an)� ρn(an) ≤ x� y, hence x� y ∈ Σ. It follows thatΣ is an MV -filter. If ρn(an) ≤ x then ρn+1(an) ≤ Gρn(an) ≤ Gx, hence Σ isclosed under G. Similarly, Σ is closed under H, hence Σ is a tense MV -filter.The rest of the proof is obvious. a

The following two results are a straightforward consequence of Proposition5.16 (see [17], p.65, for the similar results in the case of tense Boolean algebras).

Proposition 5.17 For any tense MV -algebra (A,G,H) the following areequivalent:

(1) (A,G,H) is a simple tense MV -algebra;

16

(2) For any a ∈ A− {1} there exists n ∈ ω such that ρn(an) = 0.

Proposition 5.18 For any tense MV -algebra (A,G,H) the following areequivalent:

(1) (A,G,H) is a subdirectly irreducible tense MV -algebra;

(2) There exists b ∈ A − {1} such that for any a ∈ A − {1} there existsn ∈ ω such that ρn(an) ≤ b.

Example 5.19 Let X = ω = {0, 1, 2, . . .} and R the following binary rela-tion on X: R = {(n, n + 1) | n ∈ ω}. Consider the tense MV -algebra([0, 1]X , G∗,H∗) associated with the frame (X,R). In accordance to the defi-nition of G∗ and H∗ we obtain G∗(p)(n) = p(n+ 1) and H∗(p)(n+ 1) = p(n),for all p ∈ [0, 1]X and n ∈ ω. Therefore, for all p ∈ [0, 1]X and n ∈ ω, we have(dp)(n+1) = p(n)∧p(n+1)∧p(n+2) and (ρp)(n+1) = p(n)�p(n+1)�p(n+2).

It is clear that d and ρ are distinct. We observe that dp = p iff p(n+1) ≤ p(n)and p(n+ 1) ≤ p(n+ 2), for all n ∈ ω, hence ([0, 1]X)d consists of the constantfunctions p : X → [0, 1].

6 Connecting tense LMn-algebras and tenseMVn-algebras

In this section we shall prove that tense LM3-algebras (resp. tense LM4-algebras) are polynomially equivalent with tense MV3-algebras (resp. tenseMV4-algebras).

In [14], Grigolia defined the MVn-algebras as algebraic structures of then-valued Lukasiewicz propositional calculus. An MV -algebra A is an MVn-algebra if it satisfies the following conditions:

M6) (n− 1)x⊕ x = (n− 1)x and xn−1 � x = xn−1, for all x ∈ A,

M7) If n ≥ 4 the following axioms are added: [(jx)⊕(x−⊕((j−1)x)−]n−1 =0 and (n− 1)[xj ⊕ (x− ⊕ (xj−1))−] = 1,

for any x ∈ A and 0 < j < n− 1 such that j does not divide n-1.For example, an MV3-algebra is an MV -algebra with the property

x⊕ x⊕ x = x⊕ x.

We know that MV3-algebras and MV4-algebras are polynomially equivalentwith LM3-algebras and, respectively, LM4-algebras (see [25] and [16]).

We will extend these equivalences to the tense case. The definition of tenseMVn-algebra follows immediately from the definition of tense MV -algebra, bytaking A to be an MVn-algebra instead of an MV -algebra.

17

Proposition 6.1 Tense MV3-algebras are polynomially equivalent with tenseLM3-algebras.

Proof. Let(A,G,H) be a tense MV3-algebra. By [25], A has a canonical struc-ture of LM3-algebra. We recall from [25] that the chrysippian endomorphismsϕ1, ϕ2 are defined by ϕ1x = x� x and ϕ2x = x⊕ x, for any x ∈ A.

We must verify if the conditions (a)-(d) from Definition 4.1. Firstly, weremark that (a) and (d) are exactly (A0) and (A5), respectively.

(c) From Proposition 5.4, (6) we have ϕ2(Gx) = Gx⊕Gx = G(x⊕x) = Gϕ2xand ϕ1(Gx) = Gx�Gx = G(x� x) = Gϕ1x.

(b) We prove that G(x∧ y) = Gx∧Gy. In the tense Boolean-algebra B(A),a, b ∈ B(A) implies G(a) ∧ G(b) = G(a ∧ b). For any x, y ∈ A, ϕix, ϕiy ∈B(A), i = 1, 2. Then we get ϕiG(x ∧ y) = Gϕi(x ∧ y) = G(ϕix ∧ ϕiy) =Gϕix∧Gϕiy = ϕiGx∧ϕiGy = ϕi(Gx∧Gy), for i = 1, 2. From the determinationprinciple we have G(x ∧ y) = Gx ∧Gy.

Now, let (A,G,H) be a tense LM3-algebra. By [25], the LM3-algebra A canbe endowed to a structure of MV3-algebra. We recall from [25] that ⊕ and �are given by:

x⊕ y = (ϕ2x ∨ y) ∧ (ϕ2y ∨ x)x� y = (ϕ1x ∧ y) ∨ (ϕ1y ∧ x).

We will check axioms (A0)-(A5) from Definition 5.2.(A0) and (A5) are obvious.(A1) We show first that Gx�Gy ≤ G(x� y), for any x, y ∈ A. We have the

followings: Gx�Gy = (ϕ1Gx∧Gy)∨(ϕ1Gy∧Gx) = G(ϕ1x∧y)∨G(ϕ1y∧x) ≤G((ϕ1x ∧ y) ∨ (ϕ1y ∧ x)) = G(x� y).

Since (x� y−)− � x � y− = 0, we get (x� y−)− � x ≤ y, hence,G((x� y−)−)�Gx ≤ G((x� y−)−�x) ≤ Gy. Thus G(x→ y)�Gx�(Gy)− =0, so G(x → y) ≤ (Gx� (Gy)−)

−= Gx → Gy, because in an MV -algebra we

have (a� b−)− = a− ⊕ b = a→ b.(A2) G is increasing, hence the following implication hold: ϕ1x, y ≤ ϕ1x ∨

y ⇒ Gϕ1x,Gy ≤ G(ϕ1x ∨ y) ⇒ Gϕ1x ∨Gy ≤ G(ϕ1x ∨ y) ⇒ ϕ1Gx ∨Gy ≤G(ϕ1x ∨ y). In the same way we obtain ϕ1Gy ∨ Gx ≤ G(ϕ1y ∨ x), thereforeGx ⊕ Gy = (ϕ1Gx ∨ Gy) ∧ (ϕ1Gy ∨ Gx) ≤ G(ϕ1x ∨ y) ∧ G(ϕ1y ∨ x) =G((ϕ1x ∨ y) ∧ (ϕ1y ∨ x)) = G(x⊕ y).

Axioms (A3), (A4) are immediately. a

Proposition 6.2 Let (A,G,H) be a tense MV3-algebra. There is a frame(X,R) and an injectiv morphism of tense MV3-algebras

Φ : (A,G,H) → (LX3 , G

∗,H∗)

Proof. From Proposition 4.7 and Proposition 6.1. a

18

Proposition 6.3 Tense MV4-algebras are polynomially equivalent with tenseLM4-algebras.

Proof. Let (A,G,H) be a tense MV4-algebra. By [16], we can define on A acanonical structure of LM4-algebra. We recall from [16] that the chrysippianendomorphisms ϕ1, ϕ2, ϕ3 are defined by: ϕ1x = x � x � x = x3, ϕ2x =(2x)3 = 3x2 and ϕ3x = x⊕ x⊕ x = 3x.

We must prove that conditions (a)-(d) from Definition 4.1 are satisfied. Weshow only that (c) is satisfied, the other conditions being similar with the prooffor MV3-algebras.

For any x ∈ A we have x⊕x⊕x⊕x = x⊕x⊕x and x�x�x�x = x�x�x.Hence, by Proposition 5.4, (6) we get:

Gϕ3x = G(x ⊕ x ⊕ x) = G((x ⊕ x) ⊕ (x ⊕ x)) = G(x ⊕ x) ⊕ G(x ⊕ x) =Gx⊕Gx⊕Gx⊕Gx = Gx⊕Gx⊕Gx = ϕ3(Gx)

Gϕ1x = G(x � x � x) = G((x � x) � (x � x)) = G(x � x) � G(x � x) =Gx�Gx�Gx�Gx = Gx�Gx�Gx = ϕ1(Gx)

Gϕ2x = G(x2⊕x2⊕x2) = G(x2⊕x2⊕x2⊕x2) = G(x2⊕x2)⊕G(x2⊕x2) =G(x2)⊕G(x2)⊕G(x2)⊕G(x2) = (Gx)2 ⊕ (Gx)2 ⊕ (Gx)2 = ϕ2(Gx).

Now, let (A,G,H) be a tense LM4-algebra. Thus A becomes anMV4-algebra[16]. We recall from [16] that:

x⊕ y = (x ∨ ϕ3y) ∧ (y ∨ ϕ3x) ∧ (x− ∨ y− ∨ ϕ2x ∨ ϕ2y)x� y = (x ∧ ϕ1y) ∨ (y ∧ ϕ1x) ∨ (x− ∧ y− ∧ ϕ2x ∧ ϕ2y).

We will check axioms (A0)-(A5) from Definition 5.2. (A0) and (A5) are obvious.(A1) It is enough to show that Gx � Gy ≤ G(x � y). By Proposition 4.3 weknow Fa ∧Gb ≤ G(a ∧ b), for any a, b ∈ A. We remark that

Gx�Gy = (Gx ∧ ϕ1Gy) ∨ (Gy ∧ ϕ1Gx) ∨ ((Gx)− ∧ (Gy)− ∧ ϕ2Gx ∧ ϕ2Gy)= G(x ∧ ϕ1y) ∨G(y ∧ ϕ1x) ∨ (Fx− ∧ Fy− ∧G(ϕ2x ∧ ϕ2y)).

Using the previous observation we have

Fx− ∧ Fy− ∧G(ϕ2x ∧ ϕ2y) ≤ Fx− ∧G(y− ∧ ϕ2x ∧ ϕ2y)≤ G(x− ∧ y− ∧ ϕ2x ∧ ϕ2y)≤ G((x ∧ ϕ1y) ∨ (y ∧ ϕ1x) ∨ (x− ∧ y− ∧ ϕ2x ∧ ϕ2y)) = G(x� y).

Since G is increasing, we get G(x∧ϕ1y), G(y ∧ϕ1x) ≤ G(x� y). It results thatGx�Gy ≤ G(x� y).

(A2) Gx⊕Gy ≤ G(x⊕ y).We must show that

(Gx ∨ ϕ3Gx) ∧ (Gy ∨ ϕ3Gy) ∧ ((Gx)− ∨ (Gy)− ∨ ϕ2Gx ∨ ϕ2Gy)≤ G[x ∨ ϕ3y) ∧ (y ∨ ϕ3x) ∧ (x− ∨ y− ∨ ϕ2x ∨ ϕ2y)].

19

Let us denote by ε the right member of this inequality. Using the distributivityin the first member, it is enough to establish the inequalities:

Gx ∧Gy ∧ (Gx)− ≤ ε; Gx ∧Gy ∧ (Gy)− ≤ ε;Gx ∧Gy ∧ ϕ2Gx ≤ ε; Gx ∧Gy ∧ ϕ2Gy ≤ ε;Gx ∧ ϕ3Gx ∧ (Gx)− ≤ ε; Gx ∧ ϕ3Gx ∧ (Gy)− ≤ ε;Gx ∧ ϕ3Gx ∧ ϕ2Gx ≤ ε; Gx ∧ ϕ3Gx ∧ ϕ2Gy ≤ ε

and another 8 inequalities. Using Proposition 4.3 we have, for example, Gx ∧ϕ3Gx∧ (Gy)− = G(x∧ϕ3x)∧Fy− ≤ G(x∧ϕ3x∧y−) ≤ ε. By a simple calculuswe also get (A3) and (A4). a

7 Dedekind-MacNeille completion of anArchimedean tense MV -algebra

In this section we introduce the pairs (g, h) of dually-conjugated unary opera-tions on an MV -algebra A and we relate them to the tense operators on A. Weprove that any pair (g, h) of dually-conjugated operations on an ArchimedeanMV -algebra A can be extended to a pair (g∼, h∼) of dually-conjugated op-erations on the Dedekind-MacNeille completion A

∧

of A. We investigate howaxioms (A0)-(A5) of the tense MV -algebras (see Definition 5.2) are preservedby the Dedekind-MacNeille completion.

Let A be an MV -algebra. If f : A → A is an unary operation of A, thenthe dual fd : A → A of f is defined by fd(x) = (f(x−))−, for any x ∈ A. Forexample, in the case of a tense MV -algebra (A,G,H), P = Hd and F = Gd.

Lemma 7.1 If f is isotone then fd is isotone.

Two unary operations g : A → A, h : A → A of A are dually-conjugated if,for all x, y ∈ A, we have g(x)⊕ y = 1 iff x⊕ h(y) = 1.

Lemma 7.2 Let g, h be dually-conjugated. For any family {xi}i∈I ⊆ A, wehave

(i) g(∧i∈I

xi) =∧i∈I

g(xi); (ii) h(∧i∈I

xi) =∧i∈I

h(xi),

whenever these infimas exist.

Proof. For example, we shall prove (ii). Let {xi}i∈I be a family of elementsof A such that the infimum x =

∧i ∈ I

xi exists. Then, for any a ∈ A, the following

equivalence hold: a ⊕ h(x) = 1 iff g(a) ⊕ x = 1 iff g(a)⊕∧i ∈ I

xi = 1 iff

20

∧i ∈ I

(g(a)⊕ xi) = 1 iff g(a)⊕ xi = 1,for all i ∈ I, iff a⊕ h(xi) = 1, for all i ∈ I.

Thus a− ≤ h(x) iff a− ≤ h(xi), for all i ∈ I. Hence h(x) =∧i ∈ I

h(xi). a

Corollary 7.3 Any dually-conjugated operations g, h are isotone.

Proposition 7.4 Let g, h be two unary operations on A. If (A, g, h) satisfiesaxiom (A0) in Definition 5.2 then the following are equivalent:

(a) (A, g, h) satisfies the axiom (A5), i.e. x ≤ ghd(x) and x ≤ hgd(x), forany x ∈ A;

(b) g, h are dually-conjugated.

Proof. (a)⇒(b). Assume g(x)⊕y = 1, so y− ≤ g(x) = (gd(x−))−. It followsthat gd(x−) ≤ y, hence, by (A5), x− ≤ hgd(x−) ≤ h(y). Thus x⊕ h(y) = 1.

(b)⇒(a). Assume g, h are dually-conjugated. From g(x−)⊕gd(x) = g(x−)⊕(g(x−))− = 1 one gets x− ⊕ hgd(x) = 1, i.e. x ≤ hgd(x). Similarly, one obtainsx ≤ ghd(x). a

According to the previous proposition, in Definition 5.2, the axiom (A5) canbe replaced by that (G,H) is a pair of dually-conjugated operations of A.

For any unary operation g on A and a ∈ A, let us denote Kg(a) = {x | a⊕g(x) = 1}. If g, h are two dually-conjugated operations on A, then Kg(a) ={x | h(a)⊕ x = 1}.

Proposition 7.5 Let g, h be two dually-conjugated operations on A. Thenh(a) = (

∧Kg(a))− and g(a) = (

∧Kh(a))−, for all a ∈ A.

Proof. For any a ∈ A we have: h(a)� (h(a))− = 0 and h(a)⊕ (h(a))− = 1.hence a⊕ g((h(a))−) = 1.

Therefore h(a)⊕Kg(a) =∧{h(a)⊕x | h(a)⊕x = 1} = 1 and h(a)�Kg(a) =∧

{h(a)� x | a⊕ g(x) = 1} = 0. Thus h(a) = (∧Kg(a))−. a

Corollary 7.6 Let (A,G,H) be a tense MV -algebra. Then, for any a ∈ A, wehave G(a) = (

∧KH(a))− and H(a) = (

∧KG(a))−.

Recall that an MV -algebra A is Archimedean if na ≤ a−, for all n ∈ ω,implies that a = 0 (see [9]). By [2], [9], A is Archimedean iff it is semisimple.

Definition 7.7 Let A,A′ be two MV -algebras with the following properties:

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(a) A is an MV -subalgebra of A′;

(b) A′ is complete, i.e. every subset of A′ has a supremum (hence everysubset of A′ has an infimum);

(c) For all x ∈ A′ we have x =∨{a ∈ A | a ≤ x} =

∧{b ∈ A | x ≤ b}.

Then A′ is said to be a Dedekind-MacNeille completion of A.

The following result is well-known (see [1], p. 398-399):

Proposition 7.8 If A is an Archimedean MV -algebra then there exists anunique (up an isomorphism) Dedekind-MacNeille completion A

∧

of A.

Now, following the same ideas from [13], we shall extend any pair (g, h)of dually-conjugated operations on an Archimedean MV -algebra A to a pair(g∼, h∼) of dually-conjugated operations on A

∧

.Let A be an Archimedean MV -algebra and A

∧

its Dedekind-MacNeillecompletion. For any unary operation g : A → A on A let us define theunary operations g∼ :A

∧

→A∧

and g∼ :A∧

→A∧

by putting, for any x ∈A∧

,g∼(x) =

∨{g(a) ∈ A | a ≤ x} and g∼(x) =

∧{g(b) ∈ A | x ≤ b}, respec-

tively. In lattice theory, the two constructions g∼, g∼ appear in the contextof Dedekind-MacNeille completion of a lattice (see e.g. [13]). Now we shallprove some lemmas on g∼, g∼ which have an intrinsec interest. Particularly,for an Archimedean tense MV -algebra (A,G,H), we will obtain G∼ = G∼ andH∼ = H∼.

Lemma 7.9 For any x ∈A∧

, g∼(x) ≤ x ≤ g∼(x).

Lemma 7.10 If g is isotone then g∼ and g∼ are isotone.

Lemma 7.11 If g is isotone then (g∼)d(x) ≤ (gd)∼(x), for all x ∈ A.

Proof. We remark that

(g∼)d(x) = (g∼(x−))− = (∧{g(b) | b ∈ A, x− ≤ b})−

=∨{(g(b))− | b ∈ A, x− ≤ b},

(gd)∼(x) =∧{gd(a) | a ∈ A, x ≤ a}

=∧{(g(a−))− | a ∈ A, x ≤ a}.

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Let a, b ∈ A such that x ≤ a and x− ≤ b. Hence a− ≤ b. Since g isisotone we have g(a−) ≤ g(b), hence (g(b))− ≤ (g(a−))−. Thus the inequality(g∼)d(x) ≤ (gd)∼(x) follows. a

Lemma 7.12 If g, h are dually-conjugated then g∼, h∼ are dually-conjugated.

Proof. For any x, y ∈A∧

the following assertions are equivalent:

• x⊕ h∼(y) = 1

• x⊕∧{h(b) | b ∈ A, y ≤ b} = 1

•∧{x⊕ h(b) | b ∈ A, y ≤ b} = 1

• x⊕ h(b) = 1, for all b ∈ A, y ≤ b

• a⊕ h(b) = 1, for all a, b ∈ A, x ≤ a, y ≤ b

• g(a)⊕ b = 1, for all a, b ∈ A, x ≤ a, y ≤ b

• g(a)⊕ y = 1, for all a ∈ A, a ≤ x

•∧{g(a)⊕ y | a ∈ A, a ≤ x} = 1

• (∧{g(a) | a ∈ A, a ≤ x})⊕ y = 1

• g∼(y)⊕ y = 1.

Hence, g∼, h∼ are dually-conjugated. a

Proposition 7.13 If g, h are dually-conjugated then (g∼)d = (gd)∼ and(h∼)d = (hd)∼.

Proof. By Lemma 7.11, we must prove only the inequality (gd)∼(x) ≤(g∼)d(x), for all x ∈A

∧

. In order to obtain this inequality it suffices to showthat for all a ∈ A, (g∼)d(x) ≤ a implies (gd)∼(x) ≤ a.

Assume a ∈ A and (g∼)d(x) ≤ a, so (g∼(x−))− ≤ a, hence g∼(x−)⊕ a = 1.By Lemma 7.12, x− ⊕ h∼(a) = 1, hence x− ⊕ h(a) = 1 (because a ∈ A), i.e.x ≤ h(a). According to Lemma 7.1 and Corollary 7.3, (gd)∼ is isotone, hence(gd)∼(x) ≤ (gd)∼(h(a)) = gd(h(a)) = (g((h(a))−))−.

But h(a) ⊕ h(a)− = 1, hence a ⊕ g(h(a)−) = 1, i.e. (g(h(a)−))− ≤ a. Itfollows that (gd)∼(x) ≤ a. Thus (g∼)d(x) = (gd)∼(x). The equality (h∼)d(x) =(hd)∼(x) follows similarly. a

Corollary 7.14 If g, h are dually-conjugated then g∼ = g∼ and h∼ = h∼.

Proof. For any x ∈A∧

we have (g∼)d(x) = (∨{g(a) | a ∈ A, a ≤ x−})− =∧

{g(a)− | a ∈ A, a ≤ x−} =∧{(g(b−))− | b ∈ A, b− ≤ x−} =

∧{gd(b) | x ≤

b} = (gd)∼(x) = (g∼)d(x). Then (g∼)d = (g∼)d, hence g∼ = g∼. Similarly,h∼ = h∼. a

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Corollary 7.15 Let (A,G,H) be an Archimedean tense MV -algebra. ThenG∼ = G∼ and H∼ = H∼.

Proposition 7.16 Let (A,G,H) be an Archimedean tense MV -algebra. Thenthe structure (A

∧

, G∼,H∼) satisfies the axioms (A0), (A1), (A2) and (A5) inDefinition 5.2.

Proof. (A0). Obviously.(A1). Let x, y ∈A

∧

. Then, by Corollary 7.15, we have: G∼(x) =∨{G(a) | a ∈ A, a ≤ x}, G∼(y) =

∧{G(b) | b ∈ A, y ≤ b} and G∼(x →

y) =∨{G(c) | c ∈ A, c ≤ x→ y}.

Therefore G∼(x)�G∼(x→ y) =∨{G(a)�G(c) | a, c ∈ A, a ≤ x, c ≤ x→

y}.Let a, b, c ∈ A such that a ≤ x, y ≤ b and c ≤ x → y. Thus a � c ≤

x � (x → y) ≤ y ≤ b. Hence G(a) � G(c) ≤ G(a � c) ≤ G(b). We getG∼(x)�G∼(x→ y) → G∼(y), so G∼(x→ y) ≤ G∼(x) → G∼(y).

(A2). For any x, y ∈A∧

we have: G∼(x) =∨{G(a) | a ∈ A, a ≤ x}, G∼(y) =∨

{G(b) | b ∈ A, b ≤ y}.Therefore G∼(x) ⊕ G∼(y) =

∨{G(a) ⊕ G(b) | a, b ∈ A, a ≤ x, b ≤ y} ≤∨

{G(a⊕ b) | a, b ∈ A, a ≤ x, b ≤ y} ≤∨{G(c) | c ∈ A, c ≤ x⊕ y} = G∼(x⊕ y).

(A5). By Lemma 7.12 and Proposition 7.4. a

Corollary 7.17 If (A,G,H) is a tense Boolean algebra then (A∧

, G∼,H∼) isa tense Boolean algebra.

Open problem 7.18 Find an example of Archimedean tense MV -algebra(A,G,H) such that (A

∧

, G∼,H∼) is not a tense MV -algebra, i.e. it does notverify the axioms (A3) and (A4).

8 Tense Moisil logic

This section is concerning with the tense n-valued Moisil logic TMn, a logicalsystem obtained from the n-valued Moisil logic Mn (see [3], [8]) by adding twotemporal operators G and H. The main result of the section is a completenesstheorem for TMn, whose proof invokes the representation theorem for tenseLMn-algebras. The construction of TMn as a logical system will be based onthe intuitionistic implication ⇒.

The alphabet of TMn has the following primitive symbols:

(1) propositional variables p, q, r, . . . (the set V of propositional variables willbe assumed infinite);

24

(2) logical connectives ∨,∧,⇒,¬, ϕ1, . . . , ϕn−1;

(3) tense operators G,H;

(4) parantheses (, ).

The set E of sentences of TMn is defined by the canonical induction. Weshall use the following abbreviations: α ⇔ β for (α ⇒ β) ∧ (α ⇐ β), Fα for¬G¬α and Pα for ¬H¬α. Now we shall provide an axiomatization of the tensen-valued Moisil logic. We shall start from the axiomatization of Mn given byCignoli in [8] (see also [3], p. 478-479).

TMn has the following axioms:

(M0) the axioms of the Moisil logic Mn (the axioms (A1)-(A16) in [3], p.478-479);

(M1) G(α⇒ β) ⇒ (Gα⇒ Gβ); H(α⇒ β) ⇒ (Hα⇒ Hβ);

(M2) Gϕi(α) ⇔ ϕiG(α) and Hϕi(α) ⇔ ϕiH(α), for any i = 1, . . . , n− 1;

(M3) α⇒ GPα; α⇒ HFα.

The axioms (M1) and (M3) preserves, in terms of the Heyting implication⇒, the form of the two axioms of the classical tense logic. The axiom (M2)expresses the compatibility of the chrysippian endomorphisms with the modaloperators, hence the determination principle holds in TMn.

The notion of formal proof in TMn is defined in terms of the above axiomsand the following inference rules:

α, α⇒ β

β(modus ponens);

α

ϕ1α;α

Gα;

α

Hα.

We shall denote by `TMnα if α is provable in TMn. Let us consider on

E the following equivalence relation: α ∼ β iff `TMnα⇔ β. For any sentence

α ∈ E denote by [α] the equivalence class of α. We can define on the quotientset E/∼ the following operations: [α]∨ [β] = [α∨ β]; [α]∧ [β] = [α∧ β]; ¬[α] =[¬α]; ϕi([α]) = [ϕiα], for i = 1, . . . , n− 1; G([α]) = [Gα]; H([α]) = [Hα].

Lemma 8.1 < E/∼,∨,∧,¬, ϕ1, . . . , ϕn−1, G,H >, the Lindenbaum-Tarski al-gebra of TMn, is a tense LMn-algebra.

Proof. By [3], p.482 (Proposition 2.21), < E/∼,∨,∧,¬, ϕ1, . . . , ϕn−1 > is anLMn-algebra. The rest of the proof is straightforward. a

Remark 8.2 For any α ∈ E, `TMnα iff [α] = 1.

Let (X,R) be a frame. An evaluation of TMn in (X,R) is a functionI : E ×X → Ln such that, for all α, β ∈ E and x ∈ X, the following equalitieshold:

25

• I(α⇒ β, x) = I(α, x) ⇒ I(β, x),

• I(α ∨ β, x) = I(α, x) ∨ I(β, x),

• I(α ∧ β, x) = I(α, x) ∧ I(β, x),

• I(¬α, x) = ¬I(α, x),

• I(ϕiα, x) = ϕiI(α, x), for any i = 1, . . . , n− 1,

• I(Gα, x) =∧xRy

I(α, y), I(Pα, x) =∨yRx

I(α, y).

A sentence α ∈ E is universally valid in TMn ( |=TMn α ) if for everyframe (X,R) and for any evaluation I : E ×X → Ln we have I(α, x) = 1, forall x ∈ X.

Remark 8.3 If (X,R) is a frame then the elements of X can be interpreted asinstants of time and R the earlier-later relation (as in [4], p.73). The meaningof the first four conditions in the previous definition is obvious. The fifth axiomestablishes the action of the determination principle, while the last axiom relatesthe truth-value of a sentence to the truth-value of the same sentence in the futureor past tense. Thus, for any fixed time x ∈ X, the laws of the n-valued Moisillogic hold, and, the tense operators G and H describe the transition from onetime to another.

Theorem 8.4 (Completeness Theorem) A sentence α is provable in TMn

if and only if it is universally valid ( `TMnα iff |=TMn

α ).

Proof. (⇒). We proceed by induction on the complexity of sentence α. Forexample, we shall prove that the axiom (M1) is universally valid. Let (X,R)be a frame and I : E × X → Ln an evaluation. Using that ⇒ is the Heytingimplication we can write:

I(Gα, x) ∧ I(G(a⇒ β, x)) =∧xRy

(I(α, y) ∧ I(α⇒ β, y)) =∧xRy

(I(α, y) ∧ (I(α, y) ⇒ I(β, y))) ≤∧xRy

I(α, y) = I(Gα, x).

Thus, in accordance to the definition of evaluation we get:

I(G(α⇒ β) ⇒ (Gα⇒ β), x) = (I(Gα, x)∧I(G(α⇒ β), x)) ⇒ I(Gβ, x) = 1

(⇐). Assume α is not provable, i.e. [α] 6= 1. We apply Proposition 4.7 tothe tense LMn-algebra E/∼, hence there exists a frame (X,R) and an injectivemorphism of tense LMn-algebras Φ : (E/∼, G,H) → (LX

n , G∗,H∗).

Let us consider the function I : E×X → Ln defined by I(α, x) = Φ([α])(x),for all α ∈ E and x ∈ X. It is straightforward to prove that I is an evaluation.Since Φ is injective, Φ([α]) 6= 1, i.e. I(α, x0) = Φ([α])(x0) 6= 1, for some x0 ∈ X.Thus α is not universally valid. a

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Remark 8.5 In [7], a simple axiomatization of TM3 was obtained by usingthe Lukasiewiczian implication.

9 Concluding remarks

The tense Boolean algebras are obtained from Boolean algebras by adding thetense operators G and H. The representation theorem for tense Boolean al-gebras constitutes the algebraic counterpart of the completeness theorem ofclassical tense logic.

If we add some appropiate tense operators to another structure of logic thenwe obtain an algebraic framework for a new kind of tense logic. In this paper weapplied this thesis for two types of algebras of many-valued logics: MV -algebrasand LMn-algebras. We defined the tense operators G and H on MV -algebrasand on LMn-algebras. The definition preserves some of the old axioms of thetense Boolean operators and introduces new axioms, specific for each of the twocases.

We studied together algebraic properties of the tense operators on MV -algebras and of the tense operators on LMn-algebras. One of the main dis-tinctions between the two situations is the way of solving the representationproblem. In the meantime for the tense LMn-algebras we proved a represen-tation theorem (with a statement similar to the Boolean case), for the tenseMV -algebras the representation problem seems to be more difficult. Perhaps arepresentation theorem for an arbitrary tense MV -algebra does not hold. Twopossible ways for obtaining representation theorems are proposed. The repre-sentation theorem for tense LMn-algebras leads to a completeness theorem forthe tense Moisil logic. It remains an open problem to develope a complete tense Lukasiewicz propositional calculus.

We proved that the tense LM3-algebras (resp. tense LM4-algebras) are poly-nomially equivalent with the tense MV3-algebras (resp. tense MV4-algebras). In[8], it was introduced the proper LMn-algebras, structures polynomially equiv-alent with the MVn-algebras (cf. [8], [16], [20]). We conjecture that the tenseproper LMn-algebras are polynomially equivalent to the tense MVn-algebras.A representation theorem for the tense MVn-algebras would be a main step inproving this equivalence.

The pairs (g, h) of dually-conjugated operations on an MV -algebra A arerelated to the tense operators on A. Recall that A

∧

denotes the Dedekind-MacNeille extension of an Archimedean MV -algebra A. If (A,G,H) is anArchimedean tense MV -algebra then we get a pair (G∼,H∼) of dually-conjugated operations on A

∧

, and (A∧

, G∼,H∼) verifies some of the axiomsof the tense MV -algebras. To characterize the class of Archimedean tense MV -algebras (A,G,H) such that (A

∧

, G∼,H∼) is a tense MV -algebra is an openquestion. In the absence of the answer, it would be interesting to find remarkableclasses of Archimedean tense MV -algebras fulfiling this property (perhaps thetense n-valued Post algebras verify this condition). The paper does not contain

27

the analysis of a similar problem for tense LMn-algebras. An open problem is toinvestigate whenever the free Post extension ([3], p.411) of a tense LMn-algebrais again a tense LMn-algebra.

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