Some new axiomatic extensions of residuated logics

1 Introduction

The main domains of research and of application of the fuzzy logic are: Automation (production of the iron, purification of the water, chains of manufacture,...), Instrumentation (captors, instruments of measure, recognition of voice and of symbol,...), Judgment/conception (consultation, decision making, timetables of train,...), Intelligential processing (data bases, information looking-for, modeling of devices,...).

The origin of residuated lattices lies in Mathematical Logic. They have been investigated by Krull [16], Dilworth [6], Ward and Dilworth [21], Ward [22], Balbes and Dwinger [1] and Pavelka [17].

During relatively recent years, many algebras have been proposed as the semantical systems of logical systems. The connection with fuzzy logic was made and today we use it to speak about the algebra of fuzzy logic. In 1958, Chang introduced in [4] the notion of MV-algebras and in 1959 he proved in [3] the completeness theorem which stated the real unit interval [0 ;1] as a standard model of this logic. In 1998 BL-algebras (as commutative generalization of MV-algebra) was introduced in [10] by Hàjek in order to investigate many-valued logic by algebraic means. Haveshki’s and Eslami’s [7] introduced the notions of n-fold implicative basic logic and n-fold positive implicative basic logic. Later, Esko Turunen and al. in [18, 19] proved that n-fold implicative basic logic is Gödel logic and n-fold positive implicative basic logic is a fragment of Lukasiewicz logic. In [13] we introduce the notions of n-fold (Boolean, implicative, fantastic, involutive) pseudo-residuated lattices as new subvarieties of pseudo-residuated lattices. In this paper, we introduce the notions of n-fold (Boolean, implicative, fantastic, involutive, strong, strongly integral) logic and show that n-fold Boolean (resp., implicative, fantastic)logic are not Boolean (resp., Heyting, MV) logic. In [14], M. Kondo introduced the notion of extended filters in commutative residuated lattices. In this paper, we extend this notion to pseudo-residuated lattices and use it to give new characterizations of n-fold Boolean (resp., implicative, fantastic, involutive, strong, strongly integral) filters. The paper is organizes as follow: In Section 2, some basics concepts and properties are recalled. In Section 3, we study the notions of n-fold strongly integral filter, extended filter and give the characterizations of n-fold filters by extended filter. We also show that the classes of n-fold (strongly integral, boolean, implicative, fantastic, involutive, strong) pseudo-residuated lattices are new subvarieties of the variety of all pseudo-residuated lattices. In Section 4, we construct a new logical system of a pseudo-residuated logic and show that n-fold (strongly integral, boolean, implicative, fantastic, involutive, strong) pseudo-residuated lattices are the models of some axiomatic extensions of pseudo-residuated logic.

2 Preliminaries

In this section, we give some fundamental definitions and results. For more details, refer to the references.

A non-commutative residuated lattice or pseudo- residuated lattice is a non-empty set L with five binary operations ∧, ∨ , ⊗ , → , ⇝, and two constants 0, 1 satisfying: L-1: 𝕃 ( L ) : = ( L , ∧ , ∨ , 0 , 1 ) is a bounded lattice; L-2: (L, ⊗ , 1) is a monoid; L-3: x ⊗ y ≤ z iff x ≤ y → z iff y ≤ x ⇝ z (pseudo-Residuation). Recall that a non-empty subset F of A is called a filter if it satisfies: (F1): For every x, y ∈ F, x ⊗ y ∈ F; (F2): For every x, y ∈ L, if x ≤ y and x ∈ F, then y ∈ F.

It is clear that {1} and A are filters.

A filter H of L is called normal if for every x, y ∈ L, x → y ∈ F  {ifandonlyif  x ⇝ y ∈ F. We denote by F _n  (L) the set of all normal filters of L.

Proposition 2.1. [5] If L is a pseudo-residuated lattice, the following hold for all x, y, z ∈ L.

x ≤ y iff x → y = 1 = x ⇝ y;

Proposition 2.2. [5] Let ∅ ≠ F ⊆ L. The following conditions are equivalent:

(Ds1) F is a filter;

Lemma 2.3. [5] For all x, y ∈ L, we have x∨(y ⊗ z) ≥ (x ∨ y) ⊗ (x ∨ z). From this it follows that: (i) (x ∨ y)  ^mn  ≤ x ⁿ  ∨ y ^m for all m, n ≥ 1; (ii) (x ∨ a₁) ⊗ (x ∨ a₂) . . . ⊗ (x ∨ a _n ) ≤ x ∨ (a₁ ⊗ a₂ . . . ⊗ a _n ) for all n ≥ 1 and a₁, . . . , a _n  ∈ L. (iii) If F is a filter then, x, y ∈ F if and only if x ⊗ y ∈ F if and only if x ∧ y ∈ F if and only if x ⊗ (x ⇝ y) ∈ F if and only if (x ⇝ y) ⊗ x ∈ F.

Lemma 2.4. [5] Let L be a pseudo-residuated lattice. (1) If F is a filter, then 〈F〉 = F. (2) If A ⊆ L, then 〈A〉 = {x ∈ L :  a₁ ⊗ a₂ ⊗ . . . ⊗ a _n   ≤ x for some n ≥ 1 and a₁, . . . , a _n  ∈ A}. (3) 〈a〉 = {x ∈ L : a ⁿ   ≤ x for some n ≥ 1}. (4) If F is a filter and x ∈ L, then 〈F ∪ {x} 〉= {u∈ L :   (a₁ ⊗ x ^{n ₁} ) ⊗ (a₂ ⊗ x ^{n ₂} ) ⊗ (a₃ ⊗ x ^{n ₃} ) ⊗. . . ⊗ (a _m  ⊗ x ^{n _m} )   ≤ u  for some m ≥ 1,  n₁, n₂, n₃, . . . n _m  ≥ 0 and a₁, a₂, . . . , a _n  ∈ F}. (5) If F₁ and F₂ are are nonempty sets of L such that 1 ∈ F₁ ∩ F₂, then 〈 F 1 ∪ F 2 〉 = { u ∈ L : ( a 1 ⊗ b 1 n 1 ) ⊗ ( a 2 ⊗ b 2 n 2 ) ⊗ ( a 3 ⊗ b 3 n 3 ) ⊗ . . . ⊗ ( a m ⊗ b m n m ) ≤ u for some m ≥ 1 , n 1 , n 2 , n 3 , . . . n m ≥ 0 b 1 , b 2 , . . . , b n ∈ F 2 , a 1 , a 2 , . . . , a n ∈ F 1 }. (6) If F is a normal filter of L, then for all x ∈ L, we have: 〈F ∪ {x} 〉 = {y ∈ L : f ⊗ x ⁿ  ≤ y,  for some n ≥ 1 and f ∈ F} = {y ∈ L : x ⁿ  ⊗ f ≤ y,  for  some n ≥ 1 and f ∈ F} = {y ∈ L : x ⁿ  ⇝ y ∈ F  for some  n ≥ 1} = {y ∈ L : x ⁿ  → y ∈ F for some n ≥ 1}.

Lemma 2.5. [5] For all x, y ∈ L, we have (i) F ∨ 〈 {x} 〉 = 〈F ∪ {x} 〉. (ii) If x ≤ y, then 〈y〉 ⊆ 〈x〉. (iii) 〈x〉 ∨ 〈y〉 = 〈x ∧ y〉 = 〈x ⊗ y〉 = 〈y ⊗ x〉. (iv) 〈x〉 ∩ 〈y〉 = 〈x ∨ y〉. (v) 〈x → y〉 ∨ 〈x〉 = 〈x ⇝ y〉 ∨ 〈x〉.

Lemma 2.6. [5] A filter F of L is prime iff for all filters A, B of L, A ∩ B ⊆ F implies A ⊆ F or B ⊆ F.

Definition 2.7. [13] A filter F of M is said to be • n-fold integral if for all x, y ∈ M, ( x ⊗ y ) n ¯ ∈ F implies x n ˜ ∈ F or y n ˜ ∈ F and ( x ⊗ y ) n ˜ ∈ F implies x n ¯ ∈ F or y n ¯ ∈ F•n-fold strong if for all x ∈ M, [ ( x n ˜ ¯ ⇝ x ) ¯ ˜ ] ∧ [ ( x n ¯ ˜ → x ) ˜ ¯ ] ∈ F• n-fold fantastic if for all x, y ∈ M, ([(x ⁿ  → y) ⇝ y] → (x ∨ y)) ∧ ([(x ⁿ  ⇝ y) → y] ⇝ (x ∨ y)) ∈ F. • n-fold involutive (or n-fold IpRL filter) if [ x n ¯ ˜ → x ] ∧ [ x n ˜ ¯ ⇝ x ] ∈ F, for each x ∈ M. • n-fold Boolean if for all x, y ∈ M, ( x ∨ x n ¯ ) ∧ ( x ∨ x n ˜ ) ∈ F • n-fold implicative if for all x, y ∈ M, (x ⁿ  → x²ⁿ) ∧ (x ⁿ  ⇝ x²ⁿ) ∈ F. •prime if for every x, y ∈ L, x ∨ y ∈ F implies x ∈ F or y ∈ F.

Definition 2.8. [13] A residuated lattice M is said to be • n-fold integral if for all x, y ∈ M, (x ⊗ y)  ⁿ  = 0 implies x ⁿ  = 0 or y ⁿ  = 0. • n-fold strong if for all x ∈ M, [ ( x n ˜ ¯ ⇝ x ) ¯ ˜ ] ∧ [ ( x n ¯ ˜ → x ) ˜ ¯ ] = 1. • n-fold fantastic if for all x, y ∈ M, ([(x ⁿ  → y) ⇝ y] → (x ∨ y)) ∧ ([(x ⁿ  ⇝ y) → y] ⇝ (x ∨ y)) =1. • n-fold involutive (or n-fold IpRL filter) if [ x n ¯ ˜ → x ] ∧ [ x n ˜ ¯ ⇝ x ] = 1, for each x ∈ M. • n-fold Boolean if for all x, y ∈ M, ( x ∨ x n ¯ ) ∧ ( x ∨ x n ˜ ) = 1 • n-fold implicative if for all x, y ∈ M, x ⁿ  → xⁿ⁺¹= 1.

Definition 2.9. Let L₁ and L₂ be two pseudo-residuated lattices. Then a map f : L₁ → L₂ is called a pseudo-residuated lattice homomorphism if it satisfies the following conditions:

f (0) =0;

If f is bijective, the homomorphism f is called pseudo-residuated lattice isomorphism. In this case we write L₁ ≅ L₂.

Definition 2.10. Let F be a normal filter of L. The well known binary relation ≡ _F defined on L by x ≡  _F y if and only if x → y and y → x ∈ F, is a congruence on L. The quotient structure L/F is also a pseudo-residuated lattice where: x/F ∧ y/F = (x ∧ y)/F; x/F ∨/F = (x ∨ y)/F; x/F ⊗ y/F = (x ⊗ y)/F; x/F → y/F = (x → y)/F. x/F ⇝ y/F = (x ⇝ y)/F.

Note that L/{1} ≅ L.

We assume that the notion of propositional language L is defined as usual. We denote by T _ermL the free term algebra with signature L over a denumerable set of generators (in this context called the propositional variables). The universe of this free term algebra will be denoted by T _ermL and its elements are called formulas. In the following, K is an algebra and K is its universe. We recall the following definitions.

Definition 2.11. Let T _ermL be a set of formulas. (i) The endomorphisms of the algebra T _ermL are called L-substitutions; (ii) An L-consecution is a pair 〈Ω, φ〉 usually written as Ω ▹ φ, where Ω ∪ {φ} ⊆ T _ermL ; (iii) An L-consecution Ω ▹ φ is finitary if Ω is finite. We often identify a L-consecution of the form ∅ ▹ φ with the formula φ itself; (iv) A set of consecutions L can be viewed as a relation between sets of formulas and single formulas, we write Ω ⊢  _L φ instead of Ω ▹ φ ∈ L.

Definition 2.12. A logic L in the language L is a set of L-consecutions such that: 1. Ω, φ ⊢  _L φ, 2. if Ω ⊢  _L φ and for every χ ∈ Ω, ▵ ⊢  _L χ, then ▵ ⊢  _L φ, 3. Ω ⊢  _L φ implies σ [Ω] ⊢  _L σ (φ) for everyL-substitutions σ, 4. if Ω ⊢  _L φ, then there is a finite set ▵ ⊆ Ω such that ▵ ⊢  _L φ.

Definition 2.13. Let L be a logic. (i) The set Aχ of finitary L-consecutions is called an axiomatic system (or a presentation) of L if the relation ⊢ _L coincides with the provability relation given by Aχ as a Hilbert style calculi: i.e., Γ ⊢  _L φ iff there is a sequence of formulas 〈ψ₁, ψ₂, . . . ψ _n 〉 with ψ _n  = φ, where each ψ _i (1 ≤ i ≤ n) is an axiom or an element of Γ or is the result of the application of modus ponens on ψ _j , ψ _k (j, k ≤ i). (ii) Elements of Aχ of the form ∅ ▹ φ are called axioms, the remaining ones are called deduction rules.

Definition 2.14. Let L be a logic and R a set of consecutions. (i) The weakest logic containing L and R is denoted by L + R, so L + R is axiomatized by any of the presentations of L plus consecutions from R. (ii) A logic L′ is an (axiomatic) extension of L if there is a set R of consecutions (formulas) such that L′ = L + R.

3 Some examples of residuated lattices and filters

The following examples of residuated lattices and their filters will be used to illustrate the concepts treated in this paper.

Example 3.1. Let L = {0, a, b, c, d, 1} be a lattice such that 0 < a < c, 0 < b < c < d < 1, a and b are incomparable. Define the operations ⊗, → and ⇝ by the three tables below. Then L is a pseudo-residuated lattice which is not a pseudo-MTL algebra, since (a ⇝ b) ∨ (b ⇝ a) = c ≠ 1.

F = {1}; F₁ = {1, d} are the only proper filters of L. L is 2-fold implicative, but not a Heyting algebra.

Example 3.2. Let L = {0, a, b, c, d, e, 1} be a lattice such that 0 < a < c < d, 0 < a < b < d < e < 1, b and c are incomparable. Define the operations ⊗ and → by the three tables below. L is a pseudo-residuated lattice which is not a pseudo-MTL algebra, since (b ⇝ c) ∨ (c ⇝ b) = e ≠ 1.

F₁ = {1}; F₂ = {1, e}; F₃ = {1, e, d, c}; F₄ = {1, e, d, c, b, a} are the proper filters of L. L is 3-fold implicative, but not a Heyting algebra.

Example 3.3. Let L = {0, a, b, c, 1} be a lattice such that 0 < a < b < c < 1. Define the operations ⊗, → and ⇝ by the three tables below. Then L is a pseudo-residuated lattice which is not a pseudo-BL algebra, since b ⊗ (b ⇝ a) = b ⊗ b = 0 ≠ a = b ∧ a.

F₁ = {1}; F₂ = {1, c} are the only proper filters of L. L is 4-fold implicative, but not a Heyting algebra.

Example 3.4. Let L = {0, a, b, c, d, 1} be a lattice such that 0 < a < b < d < 1,0 < a < c < d < 1, b and c are incomparable. Define the operations ⊗, → and ⇝ by the three tables below. Then L is a pseudo-residuated lattice which is not a pseudo-MTL algebra, since (b → c) ∨ (c → b) = d ≠ 1.

F₁ = {1}; F₂ = {1, d}; F₃ = {1, d, c}; F₄ = {1, d, b} are the proper filters of L.

Example 3.5. Let L = {0, a, b, c, 1} be a lattice such that 0 < a < b < 1, 0 < a < c < 1, b and c are incomparable. Define the operations ⊗, → and ⇝ by the three tables below. Then L is a pseudo-residuated lattice which is not a pseudo-BL algebra, since (b → c) ⊗ b = c ⊗ b = 0 ≠ a = b ∧ c.

F₁ = {1}; F₂ = {1, c}; F₃ = {1, b} are the proper filters of L.

Example 3.6. Let L = {0, a, b, c, 1} be a lattice such that 0 < a < b < c < 1. Define the operations ⊗, → and ⇝ by the three tables below. Then L is a pseudo-residuated lattice which is not a pseudo-BL algebra, since (c → b) ⊗ c = c ⊗ c = a ≠ b = b ∧ c.

F₁ = {1} is the only proper filter of L. L is 2-fold Boolean, but not a Boolean algebra.

4 Some classes of filters

A class F of filters of L will be said to be closed under extension if for any filters F₁ and F₂ of L, (F 1 ∈ F and F₁ ⊆ F₂) implies F 2 ∈ F. It may be required that the filters in question satisfy a particular condition(for example, the condition of proper filters).

4.1 n-fold strongly integral filters

Definition 4.1. Let F be a filter of L, and n ≥ 1. (i) F is said to be n-fold strongly integral if for all x, y ∈ L, ( ( ( x ⊗ y ) n ¯ ) → ( x n ˜ ∨ y n ˜ ) ) ∧ ( ( ( x ⊗ y ) n ˜ ) → ( x n ¯ ∨ y n ¯ ) ) ∈ F. In particular, 1-fold strongly integral filters are strongly integral filters. (ii) L is said to be n-fold strongly integral if for all for all x, y ∈ L, ( ( ( x ⊗ y ) n ¯ ) → ( x n ˜ ∨ y n ˜ ) ) ∧ ( ( ( x ⊗ y ) n ˜ ) → ( x n ¯ ∨ y n ¯ ) ) = 1. In particular, a 1-fold strongly integral pseudo-residuated lattice is a strongly integral pseudo-residuated lattice.

It is easy to show the following result.

Lemma 4.2. Let F be a filter of the pseudo-residuated lattice L, and n ≥ 1 If F is prime and n-fold strongly integral, then it is n-fold integral.

Example 4.3. Let n ≥ 1. (i) The pseudo-residuated lattice of Example 3.2 is n-fold strongly integral and all its filters are n-fold strongly integral. (ii) The pseudo-residuated lattice of Example 3.5 is not 1-fold strongly integral, because ( ( a ⊗ b ) ¯ ) ⇝ ( a ˜ ∨ b ˜ ) = b ≠ 1.

4.2 t-filters

Let k ∈ ℕ, t be an k-airy term and M be a residuated lattice. By the symbol x →, we denote the abbreviation of (x₁, x₂, . . . , x _k ) ∈ M ^k , i.e. x → is a formal listing of variables x₁, x₂, . . . , x _k used in a given context. By the term t we always meant a term in the language of residuated lattices.

Definition 4.4. (i) A filter G of M is said to be a t-filter if for all x → ∈ M k, t ( x → ) ∈ G. (ii) A residuated lattice M is said to be a t-algebra if for all x → ∈ M k, t ( x → ) = 1. The following result are the direct consequences of the definition.

Lemma 4.5. The following properties are equivalent. (i) M is a t-algebra. (ii) Any filter of M is a t-filter. (iii) {1} is a t-filter.

Lemma 4.6. The class of t-filters is close under extension.

Lemma 4.7. A normal filter G of M is a t-filter iff M/G is a t-algebra.

4.3 Extended filters

The notion of extended filters of RL-monoid was firstly introduced in (2012) and their properties was considered by Haveshki and Mohamadhasani [8]. According to it, Michiro Kondo [14] defined an extended filter of a residuated lattice L. We here give a simple characterization theorem of extended filters in pseudo-residuated lattices.

We denote by: Fil (L) the set of all filters of L, Spec (L) the set of all prime filters of L, Max (L) the set of all maximal filters of L.

A lattice (L, ∧ , ∨) is said to be complete lattice if for all a ∈ L and {b _i  : i ∈ I} ⊆ L, ∨b _i and ∧b _i exist. A residuated lattice (L, ∧ , ∨ , ⊗ , → , ⇝ , 0, 1) is said to be complete if (L, ∧ , ∨) is complete.

Definition 4.8. [5] A complete lattice (L, ∧ , ∨) is called Browerian if it satisfies the identity a ∧ ⋁ b _i  = ⋁ (a ∧ b _i ).

Lemma 4.9. [5] (Fil (L) , ∧ , ∨) is a Browerian algebraic lattice, where ∨F _i  = 〈 ∪ F _i 〉, ⋀F _i  = ⋂ F _i and the compact elements being exactly the principal filter of L.

For F₁, F₂ ∈ Fil (L), we define F₁⊸F₂ : = ∨ {G : G ∩ F₁ ⊆ F₂}. Like the case of commutative residuated lattice [[14], Proposition 3], we also have F₁⊸F₂ = {x ∈ L : 〈x〉 ∩ F₁ ⊆ F₂}. From this and Lemma ref prirmai, we have the following result.

Theorem 4.10. (Fil (L) , ∧ , ∨ , ⊸, {1} , L) is a complete Heyting algebra(as a residuated lattice).

Definition 4.11. For a subset B ⊆ L and a filter F ∈ Fil (L), E _F  (B) = {x ∈ L : x ∨ b ∈ F for all b ∈ B} is called an extended filter associated with B.

Lemma 4.12. Let F ∈ Fil (L) and ∅ ≠ B ⊆ L Then, (1) E _F  (B) ∈ Fil (L). (2) F ⊆ E _F  (B).

Proof. Using Lemma 2.3(i), the proof is as in the case of commutative residuated lattice, [[14], Proposition 5].■

Example 4.13. Let L be the residuated lattice of Example 3.4 and B : = {d, c, b} ⊆ L. E _{F 2}  (B) = F₂.

At first we give a simple characterization theorem of extended filters.

Theorem 4.14. Let F ∈ Fil (L) and ∅ ≠ C, B ⊆ L. Then, (1) E _F  (B) = 〈B〉⊸F. (2) E_{E F (B)} (C) = E_{E F (C)} (B) = E _F  (〈B〉 ∩ 〈C〉). (3) For all (F _i ) _i∈Λ ⊆ Fil (L), we have ⋂_i∈ΛE _{F i}  (B) = E_{⋂i∈ΛF i}  (B).

Proof. Using Lemma 2.3(ii) and Theorem 4.10, the proof is as in the case of commutative residuated lattice, [[14], Theorem 1, Corollary 2, Corollary 3].

Lemma 4.15. Let F ∈ Fil (L) and ∅ ≠ B ⊆ L. Then, If F is a prime filter, then so is E _F  (B).

Proof. Using Lemmas 2.6 and 2.5, the proof is as in the case of commutative residuated lattice, [[14], Proposition 6].

Definition 4.16. Let F ∈ Fil (L) and ∅ ≠ B ⊆ L. F is called a stable filter relative to B if E _F  (B) = F.

We denote by S (B) the set of all stable filters relative to B.

Lemma 4.17. Let F ∈ Fil (L) and ∅ ≠ B ⊆ L. Then, (i) E _F  (B) is a stable filter relative to B. (ii) S (B) = {E _F  (B) : F ∈ Fil (L)}.

Proof. (i) By Theorem 4.14, we have E_{E F (B)} (B)= (〈B〉 ∧ 〈B〉) ⊸F = 〈B〉⊸F = E _F  (B).■ (ii) Follows from (i) and the definition of stable filter relative to B.

For F, G ∈ Fil (L), we define on S (B) two operations ⊓ and ⊔ as follows, E _F  (B) ⊓ E _G  (B) = E_F∧G (B) and E _F  (B) ⊔ E _G  (B) = E_F∨G (B). Let 〈B〉^∗ = E_{1} (B).

As in the case of commutative residuated lattice [[14], Theorem 4], using Theorems 4.14 and 4.10, we have the following result.

Theorem 4.18. Let F ∈ Fil (L) and ∅ ≠ B ⊆ L. Then, (S (B) , ⊓ , ⊔ , ⊸, 〈B〉^∗, L) is a complete Heyting algebra which is not a subalgebra of (Fil (L) , ∧ , ∨ , ⊸, {1} , L).

We now here give the characterization of t-filter by extended filters.

Theorem 4.19. Let M be a residuated lattice and F ∈ Fil (M). Then, F is a t-filter if and only if for all x → ∈ M k, E F ( 〈 t ( x → ) 〉 ) = M.

Proof. Let F ∈ Fil (M). F is a t-filter if and only if for all x → ∈ M k, t ( x → ) ∈ F if and only if for all x → ∈ M k, 〈 t ( x → ) 〉 ⊆ F if and only if for all x → ∈ M k, 〈 t ( x → ) 〉 ∩ 〈 0 〉 ⊆ F if and only if for all x → ∈ M k, 0 ∈ 〈 t ( x → ) 〉 ⊸ F if and only if for all x → ∈ M k, 0 ∈ E F ( 〈 t ( x → ) 〉 ) if and only if for all x → ∈ M k, E F ( 〈 t ( x → ) 〉 ) = M.

Remark 4.20. Let us taking the following term in the language of residuated lattice t 1 ( x , y ) = [ ( ( x ⊗ y ) n ¯ ) → ( x n ˜ ∨ y n ˜ ) ] ∧ [ ( ( x ⊗ y ) n ˜ )→ ( x n ¯ ∨ y n ¯ ) ]. t 2 ( x , y ) = ( ( x n ˜ ¯ ⇝ x ) ¯ ˜ ) ∧ ( ( x n ¯ ˜ → x ) ˜ ¯ ). t₃ (x, y) = ([(x ⁿ  → y) ⇝ y] → (x ∨ y)) ∧ ([(x ⁿ  ⇝ y) → y] ⇝ (x ∨ y)). t 4 ( x , y ) = [ x n ¯ ˜ → x ] ∧ [ x n ˜ ¯ ⇝ x ]. t 5 ( x , y ) = ( x ∨ x n ¯ ) ∧ ( x ∨ x n ˜ ). t₆ (x, y) = (x ⁿ  → x²ⁿ) ∧ (x ⁿ  ⇝ x²ⁿ).

Then, the class of n-fold strongly integral(resp., strong, fantastic, involutive, Boolean, implica-tive)residuated lattices are t₁, t₂, t₃, t₄, t₅, t₆-algebras, respectively. So using Theorem 4.19, we get the characterizations of an n-fold strongly integral(resp., strong, fantastic, involutive, boolean, implicative)filters by an extended filter.

Specially, we get the following characterization of a n-fold boolean filter.

Theorem 4.21. Let F ∈ Fil (L). Then, F is an n-fold boolean filter if and only if x ∈ E F ( 〈 x n ¯ 〉 ) ∩ E F ( 〈 x n ˜ 〉 ).

Proof. Let F ∈ Fil (L). F is an n-fold boolean filter if and only if x ∨ x n ¯ ∈ F and x ∨ x n ˜ ∈ F if and only if 〈 x ∨ x n ¯ 〉 ⊆ F and 〈 x ∨ x n ˜ 〉 ⊆ F if and only if 〈 x 〉 ∩ 〈 x n ¯ 〉 ⊆ F and 〈 x 〉 ∩ 〈 x n ˜ 〉 ⊆ F if and only if x ∈ 〈 x n ¯ 〉 ⊸ F and x ∈ 〈 x n ˜ 〉 ⊸ F if and only if x ∈ E F ( 〈 x n ¯ 〉 ) and x ∈ E F ( 〈 x n ˜ 〉 ) if and only if x ∈ E F ( 〈 x n ¯ 〉 ) ∩ E F ( 〈 x n ˜ 〉 ).

5 pseudo-Residuated logic

Let us start this section by the following result.

Theorem 5.1.Let us consider the following equations. (i) (x ⊗ y) → z = x → (y → z); (i’) (y ⊗ x) ⇝ z = x ⇝ (y ⇝ z); (ii) (x ⊗ (x ⇝ y)) ∧ y = x ⊗ (x ⇝ y); (ii’) ((x → y) ⊗ x) ∧ y = (x → y) ⊗ x; (iii) (x ∧ y) ⇝ y = 1; (iii’) (x ∧ y) → y = 1.

If L-1 and L-2 hold, then: L-3 ⇔ [(i) , (i′) , (ii) , (ii′) , (iii) , (iii′)]

Proof. Suppose that L-1 and L-2 hold ⇒ Assume that L-3 holds. (i) u ≤ (x ⊗ y) → z ⇔ u ⊗ (x ⊗ y) ≤ z ⇔ (u ⊗ x) ⊗ y ≤ z ⇔ u ⊗ x ≤ y → z ⇔ u ≤ x→ (y → z) (i’) u ≤ (y ⊗ x) ⇝ z ⇔ (y ⊗ x) ⊗ u ≤ z ⇔ y ⊗ (x ⊗ u) ≤ z ⇔ x ⊗ u ≤ y ⇝ z ⇔ u ≤ x⇝ (y ⇝ z). (ii) x ⊗ (x ⇝ y) ≤ y ⇔ x ⇝ y ≤ x ⇝ y. Hence (ii) holds. (ii’) (x → y) ⊗ x ≤ y ⇔ x → y ≤ x → y. Hence (ii’) holds. (iii) (x ∧ y) ⊗1 ≤ y ⇔ 1 ≤ (x ∧ y) ⇝ y ⇔ (x ∧ y)⇝y = 1. (iii’) 1 ⊗ (x ∧ y) ≤ y ⇔ 1 ≤ (x ∧ y) → y ⇔ (x ∧ y) → y = 1. ← Conversely, assume that (i), (i’), (ii), (ii’), (iii) and (iii’) hold. We have to show that L-3 holds. x ≤ y ⇔ x ∧ y = x; hence x → y = 1, by (iii’). x → y = 1 ⇒ (x → y) ⊗ x = x. Hence x ≤ y, by (ii’). So, x ≤ y ⇔ x → y = 1 (∗).

From this, we have: x ⊗ y ≤ z iff (x ⊗ y) → z = 1 iff x → (y → z) =1 (by (i)) iff x ≤ y → z, by (∗).

In addition, x ≤ y iff x ∧ y = x. Hence, x ⇝ y = 1, by (iii). x ⇝ y = 1 ⇒ x ⊗ (x ⇝ y) = x. Hence x ≤ y, by (ii). So, x ≤ y ⇔ x ⇝ y = 1 (∗∗).

From this, we have: x ⊗ y ≤ z iff (x ⊗ y) ⇝ z = 1 iff y ⇝ (x ⇝ z) =1 (by (i’)) iff y ≤ x ⇝ z, by (∗∗).

Corollary 5.2. The class RL of pseudo-residuated lattices is a variety.

Proof. It is known that 𝕃 ( L ) : = ( L , ∧ , ∨ , 0 , 1 ) is a bounded lattice iff it satisfies the following conditions: (a₁) (x ∨ y) ∨ z = x ∨ (y ∨ z) and (x ∧ y) ∧ z = x ∧ (y ∧ z); (a₂) x ∧ (x ∨ y) = x = x ∨ (x ∧ y); (a₃) x ∧ y = y ∧ x and x ∨ y = y ∨ x; (a₄) x ∨ x = x = x ∧ x; (a₅) x ∨ 0 = x = x ∧ 1.

It is also known that (L, ⊗ , 1) is a monoid iff it satisfies the following conditions: (a₆) x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z; (a₇) x ⊗ 1 = x = 1 ⊗ x.

The result follows from this and Theorem 5.1.■

Since the class epsfbox G:/Tex/IOSPRESS/IFS/0-2070/IF-01.eps of all t-algebras is equationally define, we have the following result.

Theorem 5.3. The class epsfbox G:/Tex/IOSPRESS/IFS/0-2070/IF-01.eps of all t-algebras is a sub-variety of RL.

So, the classes of n-fold strongly integral (resp., boolean, implicative, fantastic, involutive, strong) pseudo-residuated lattices are sub-varieties of RL.

5.1 pseudo-Residuated logic

Definition 5.4. Now we propose the logic 𝕌, which is later proved to be determined by the class of all pseudo-residuated lattices. The logic has the following language: Propositional variables: p₁, p₂, p₃ . . .. Constant: a, e. Logical symbols: ∧, ∨ , ∘ , → , ⇝. A formula of 𝕌 is defined as follows: (1) Every propositional variable is a formula; (2) Constants a, e are formulas; (3) If A and B are formulas, then so are A ∧ B, A ∨ B, A ∘ B, A → B, A ⇝ B.

We note that this logic is a negation-free. Let Φ₀ : = {p₀, p₁, p₂, . . . .} ∪ {a, e} and Φ be the set of all formulas of 𝕌. As a logical system, the logic 𝕌 has the following axioms and rules of inference: Axioms: (1) A → (A ∨ B). (2) (A ∨ B) → (B ∨ A). (3) (A ∨ A) → A. (3a) (A ∨ e) → e. (4) (A ∧ B) → A. (5) (A ∧ B) → (B ∧ A). (6) A → (A ∧ A). (7) (A ∧ (B ∧ C)) → ((A ∧ B) ∧ C)). (8) ((A ∧ B) ∧ C)) → (A ∧ (B ∧ C)). (9) (A ∨ (B ∨ C)) → ((A ∨ B) ∨ C)). (10) ((A ∨ B) ∨ C)) → (A ∨ (B ∨ C)). (11) (A ∘ e) → A. (12) (e ∘ A) → A. (13) (A ∘ (B ∘ C)) → ((A ∘ B) ∘ C)). (14) ((A ∘ B) ∘ C)) → (A ∘ (B ∘ C)). (15) e. (16) a → A. (17) A → (B → A); (18) A → (B ⇝ A); (19) (A → (B → C)) → ((A ∘ B) → C). (20) ((A ∘ B) → C) → (A → (B → C)). (21) (A ⇝ (B ⇝ C)) → ((B ∘ A) ⇝ C)). (22) ((B ∘ A) ⇝ C)) → ((A ⇝ (B ⇝ C)). (23) ((A ∘ (A ⇝ B)) ∧ B) → (A ∘ (A ⇝ B)). (24) (A ∘ (A ⇝ B)) → ((A ∘ (A ⇝ B)) ∧ B). (25) (((A → B) ∘ A) ∧ B) → ((A → B) ∘ A). (26) ((A → B) ∘ A) → (((A → B) ∘ A) ∧ B). (27) (A → B) → ((B → C) ⇝ (A → C)). (28) (A → B) → ((C → A) → (C → B)). (29) (A ⇝ B) → ((C ⇝ A) → (C ⇝ B)). (30) (A ⇝ B) → ((B ⇝ C) → (A ⇝ C)). (31) (A → B) → ((A ∘ C) → (B ∘ C)). (32) (A → B) → ((A ∧ C) → (B ∧ C)). (33) (A → B) → ((A ∨ C) → (B ∨ C)).

Rules of inference:

• implications:

A → B A ⇝ B; A ⇝ B A → B.

R_∨: A → C B → C ( A ∨ B ) → C, A ⇝ C B ⇝ C ( A ∨ B ) → C.

R_∧: A → B A → C A → ( B ∧ C ), A ⇝ B A ⇝ C A → ( B ∧ C ).

• Modus ponens: A A → B B. A A ⇝ B B.

Note A ⁿ  : = A^n-1 ∘ A for all n ≥ 1 and A⁰ : = e. A → a, A ⇝ a are denote by A ¯ and A ˜, respectively.

Proposition 5.5. For all A, B, C ∈ Φ, we have (1) A → B B → C A → C; A → B ( B → C ) ⇝ ( A → C ). (2a) A → B C → A C → B; A → B ( C → A ) → ( C → B ). (2b) A ⇝ B C ⇝ A C → B; A ⇝ B ( C ⇝ A ) → ( C ⇝ B ). (2c) A ⇝ B ( B ⇝ C ) → ( A ⇝ C ). (3) A → B ( A ∘ C ) → ( B ∘ C ); A ⇝ B ( C ∘ A ) ⇝ ( C ∘ B ). (4) A → B ( C ∧ A ) → ( C ∧ B ); A → B ( C ∨ A ) → ( C ∨ B ). (5) ⊢ 𝕌 ( A ∘ B ) → A. ⊢ 𝕌 ( A ∘ B ) → B. (6) ⊢ 𝕌 A n + 1 → A n.

Proof. (1) Follows from Axiom (27) and modus ponens. (2a) Follows from Axiom (28) and modus ponens. (2b) Follows from Axiom (29) and modus ponens. (2c) Follows from Axiom (30) and modus ponens. (3) Follows from Axiom (31) and modus ponens. (4) Follows from Axioms (32), (33) and modus ponens. (5) Follows from Axioms (32), (33) and modus ponens. (5) Follows from Axioms (17), (18), (19) and modus ponens. (6) Follows from (5).

Remark 5.6. Since the logic 𝕌 has axioms {A → (B → A)}, by using modus ponens, we can show the deduction theorem of the logic Γ ∪ { A } ⊢ 𝕌 B if and only if Γ ⊢ 𝕌 A → B.

Definition 5.7. A map v : Φ → L is called a valuation on L if it satisfies the following conditions: (v₁) v (A ∧ B) = v (A) ∧ v (B). (v₂) v (A ∨ B) = v (A) ∨ v (B). (v₃) v (A → B) = v (A) → v (B). (v₃) v (A ⇝ B) = v (A) ⇝ v (B). (v₄) v (A ∘ B) = v (A) ∘ v (B). (v₅) If A is an axiom of 𝕌, then v (A) =1

It is easy to see that v (e) =1 ; v (a) =0.

The semantical equational consequence relation ⊨ is define as follows. For Γ ∪ {A} ⊆ Φ, Γ ⊨ A if and only if v (A) =1 whenever v (γ) =1 for all γ ∈ Γ, for any valuation v on any pseudo-residuated lattice L.

Lemma 5.8.(Soundness Theorem): LetΓ ∪ {A}⊆Φ.

If Γ ⊢ 𝕌 A, then Γ ⊨ A.

Proof. Assume that Γ ⊢ 𝕌 A. Then, there is a sequence of formulas 〈ψ₁, ψ₂, . . . ψ _n 〉 with ψ _n  = A, where each ψ _i (1 ≤ i ≤ n) is an axiom or an element of Γ or is the result of the application of modus ponens on ψ _j , ψ _k (j, k ≤ i). Let v be a valuation on the pseudo-residuated lattice L such that for all γ ∈ Γ, v (γ) =1. We have to show that v (A) =1.

If each ψ _i (1 ≤ i ≤ n) is an axiom, then so is A = ψ _n and then v (A) =1.

If A ∈ Γ, then v (A) =1.

If each ψ _i (1 ≤ i ≤ n) is the result of the application of modus ponens on ψ _j , ψ _k (j, k ≤ i), then so is A = ψ _n and since v is a valuation, we obtain v (A) =1.

Using the well-known method called Lindenbaum-Tarski algebra, one can prove the converse direction (Completeness Theorem) of the above. Let Γ ⊆ Φ. At first define the relation ≡ _Γ on the set Φ of formulas of 𝕌: For A, B ∈ Φ, A ≡  _Γ B if and if Γ ⊢ 𝕌 A → B and Γ ⊢ 𝕌 B → A.

From Proposition 5.5 we have the following result:

Lemma 5.9. For all Γ ⊆ Φ, ≡ _Γ is a congruence on Φ.

Define operations on Φ/≡  _Γ :

[A] ⊓ [B] = [A ∧ B]; [A] ⊔ [B] = [A ∨ B]; [A]○ [B] = [A ∘ B]; [A] ↪ [B] = [A → B]; [A]  ↩ [B] = [A ⇝ B].

Define a relation ⊑ _Γ on Φ/≡  _Γ by [A] ⊑  _Γ  [B] if and only if Γ ⊢ 𝕌 A → B.

It is easy to show the following result.

Lemma 5.10. For all Γ ⊆ Φ, (Φ/≡  _Γ , ⊓, ⊔ , ○, ↩, ↩ [a] , [e]) is a pseudo-residuated lattice.

We have the following key lemma to prove the completeness theorem.

Lemma 5.11. For all Γ ⊆ Φ and any formula A ∈ Φ, we have: Γ ⊢ 𝕌 A ⇔ [ e ] ⊑ Γ [ A ] in Φ/≡  _Γ

Proof. Let Γ ⊆ Φ and A ∈ Φ. Assume that Γ ⊢ 𝕌 A, since A → (e → A) is an axiom, then by modus ponens, we deduce that Γ ⊢ 𝕌 e → A, that is [e] ⊑  _Γ  [A].

Conversely, assume that [e] ⊑  _Γ  [A], then Γ ⊢ 𝕌 e → A. Since e is an axiom, by modus ponens, we have Γ ⊢ 𝕌 A.

Theorem 5.12. (Strong Completeness theorem):

For all Γ ∪ {A} ⊆ Φ, we have: Γ ⊢ 𝕌 A ⇔ Γ ⊨ A.

Proof. Let Γ ∪ {A} ⊆ Φ. Assume that Γ ⊢ 𝕌 A, then, by Lemma 5.8, we have Γ ⊨ A.

Conversely, assume that Γ ⊨ A. Taking a canonical map v _Γ  : Φ → Φ/≡  _Γ defined by v _Γ  (A) = [A] as a particular valuation, we have the result.

Regular (or involutive) residuated lattices, Heyting algebras, Boolean algebras, MTL algebras, BL algebras, MV algebras are equivalent semantics of regular residuated logic, Heyting logic, Boolean logic, MTL, BL and MVL, respectively. The above logics are (axiomatic) extension of 𝕌; and are described informally as follows

We recall an algebraizability of a logic in the sense of Block and Pigozzi [2]. A class K of algebras is called an algebraic semantics if the consequence relation ⊢ can be interpreted in the semantical equational consequence relation ⊨ _K of K in a natural way. K is called an equivalent algebraic semantics for a logic L if there is an inverse interpretations of ⊨ _K in ⊢ _L . If L has an equivalent algebraic semantics, then it is called algebraizable. Concerning to algebraizability, we have a fundamental result:

Theorem 5.13.[15] A logic L is algebraizable if and only if there exist a system Δ of formulas in two variables and a system δ ≈ ɛ in a single variable such that the following conditions hold for all A, B, C ∈ Φ : (i) ⊢AΔA (ii) AΔB ⊢ BΔA (iii) AΔB, BΔC ⊢ AΔC

For every primitive logical symbol ϖ and all A₀, . . . A_n-1, . . , B₀, . . . B_n-1 ∈ Φ when n is the arity of ϖ (iv) A₀ΔB₀, . . , A_n-1ΔB_n-1ϖ (A₀, . . . A_n-1) Δϖ (B₀, . . . B_n-1). Finally, all A ∈ Φ (v) A ⊢ δ (A) Δɛ (A) and δ (A) Δɛ (A) ⊢ A.

In the case of the logic 𝕌, we have Δ = {p → q, q → p} , δ = p ∧ 1 and ɛ = 1.

Definition 5.14. (i) We said the logic L₁ generalize the logic L₂ if the model of L₁ generalize the model of L₂. (ii) We said the logics L₁ and L₂ coincide and we write L₁ = L₂ if the models of L₁ and L₂ coincide.

Example 5.15. Boolean logic = MVL + Heyting logic.

So, Heyting logic and MVL generalize Boolean logic.

We have the following table.

5.2 n-fold residuated logic

We recall that Φ is the set of all formulas of 𝕌(residuated logic). For A ∈ Φ and n ≥ 1, we denote by A ⁿ  : = A ∘ A^n-1, and A⁰ : = e.

Let us define informally the following logic as follow. nip 1 : 𝕌 + {φ ⁿ  → φⁿ⁺¹}

nip 2 : 𝕌 + { ( φ ∨ φ n ¯ ) ∧ ( φ ∨ φ n ˜ ) }

nip 3 : 𝕌 + {(((φ ⁿ  → χ) ⇝ χ) → (φ ∨ χ)) ∧ (((φ ⁿ  ⇝ χ) → χ) → (φ ∨ χ))}.

nip 4 : 𝕌 + {( ( φ n ˜ ¯ ⇝ φ ) ¯ ˜ ) ∧ ( ( φ n ¯ ˜ → φ ) ˜ ¯ ) }

nip 5 : 𝕌 + { ( ( φ ∘ χ ) n ¯ → ( φ n ˜ ∨ χ n ˜ ) ) ∧ ( ( φ ∘ χ ) n ˜ → ( φ n ¯ ∨ χ n ¯ ) ) }

nip 6 : 𝕌 + { ( φ n ¯ ˜ → φ ) ∧ ( φ n ˜ ¯ → φ ) }

It is clear that all the above logics are algebraizable since 𝕌 is algebraizable.

Now we show the Soundness and Completeness theorems of the above logics.

For example we show the Soundness and Completeness theorems of n-fold implicative pseudo-residuated logic or nIPL, in short.

The logic nIPL has the same language as 𝕌. We note that this logic is also a negation-free. Let Ψ be the set of all formulas of nIPL. As a logical system, nIPL is defined as an axiomatic extension of 𝕌 with the axiom A ⁿ  → Aⁿ⁺¹.

Definition 5.16. If L is an n-fold implicative pseudo-residuated lattice, then a valuation on L is an homomorphism from Ψ to L.

Define a semantical equational consequence relation ⊨ as follows. For Γ ∪ {A} ⊆ Ψ, Γ ⊨ A if and only if v (A) =1 whenever v (γ) =1 for all γ ∈ Γ and any valuation v on any n-fold implicative residuated lattice L. So, ∅ ⊨ A (or ⊨A) iff v (A) =1, that is A is a tautology iff v (A) =1.

A derivation of ρ from Γ (⊆ Ψ) is a finite sequence of formulas 〈ψ₁, ψ₂, . . . ψ _n 〉 with ψ _n  = φ, where each ψ _i (1 ≤ i ≤ n) is an axiom or an element of Γ or is the result of the application of modus ponens on ψ _j , ψ _k (j, k ≤ i). Let ⊢ be the derivation operator defined in this way, i.e., we write Γ ⊢  _nIPL ρ iff there is an derivation of ρ from Γ.

Now define cons (Γ) = {ρ ∈ Ψ : Γ ⊢  _nIPL ρ} and Cons (Γ) = {ρ ∈ Ψ : Γ ⊨ ρ}.

Remark 5.17. (i) Since n-fold implicative pseudo-residuated logic is an axiomatic extension of 𝕌 and 𝕌 has axioms A → (B → A), we get the deduction theorem of the n-fold implicative pseudo-residuated logic: Γ ∪ {A} ⊢  _nIPL B if and only if Γ ⊢  _nIPL A → B. (ii) For any valuation v on any n-fold implicative pseudo-residuated lattice L and any axioms φ of nIPL, v (φ) =1. (iii) For any ρ, σ ∈ Ψ, (v (ρ) =1 and v (ρ → σ) =1) implies v (σ) =1.

From the above remark, the following is easy to prove.

Lemma 5.18. (Soundness Theorem): cons (Γ) ⊆ Cons (Γ). That is Γ ⊢  _nIPL A implies Γ ⊨ A.

We use the well-known method called Lindenbaum-Tarski algebra to prove the converse direction (Completeness Theorem) of the above. Let Γ ⊆ Ψ. At first, we define a relation ≡ _Γ on the set Ψ of formulas of nIPL: For A, B ∈ Ψ, A ≡  _Γ B if and if Γ ⊢  _nIPL A → B and Γ ⊢  _nIPL B → A.

As a relation, we see from axioms 1 - 26 of 𝕌 together with Proposition 5.1 and the modus ponens that:

Lemma 5.19. ≡ _Γ is a congruence on Ψ.

So, we can define operations on Ψ/≡  _Γ : [A] ⊓ [B] = [A ∧ B]; [A] ⊔ [B] = [A ∨ B]; [A]○ [B] = [A ∘ B]; [A] ↪ [B] = [A → B], [A] epsfboxG :/Tex/IOSPRESS/IFS/0 -2070/IF - 02 . eps[B] = [A ⇝ B]. We define a relation ⊑ _Γ on Ψ/≡  _Γ by [A] ⊑  _Γ if and only if Γ ⊢  _nIPL A → B.

Theorem 5.20. (i) (Ψ/≡  _Γ , ⊓ , ⊔ , ○ , ↪ , ↩[a] , [e]) is an n-fold implicative pseudo-residuated lattice. (ii) There exists a homomorphic mapping v from Ψ to Ψ/≡  _Γ with v (φ) = [e] iff Γ ⊢  _nIPL φ.

Proof. (i) Since [A]  ⁿ  ↪ [A] ⁿ⁺¹ = [A ⁿ  → Aⁿ⁺¹] = [e], it follows that (Ψ/≡  _Γ , ⊓ , ⊔ , ○ , ↪ , ↩, [a] , [e]) is an n-fold implicative pseudo-residuated lattice. (ii) The result follows by taking a canonical map v _Γ  : Ψ → Ψ/≡  _Γ defined by v _Γ  (A) = [A].

From the above, we have the next theorem.

Theorem 5.21. (Strong Completeness Theorem): For Γ ⊆ Ψ, we have cons (Γ) = Cons (Γ). That is Γ ⊢  _nIPL A ⇔ Γ ⊨ A, for all A ∈ Ψ.

The Soundness and Completeness theorems of the rest of above logics are obtained similarly. From this, we have the following results:

Theorem 5.22. (i) The model of the n-fold involutive pseudo-residuated logic is an n-fold involutive pseudo-residuated lattice. (ii) The model of the n-fold implicative pseudo-residuated logic is an n-fold implicative pseudo-residuated lattice. (iii) The model of the n-fold boolean pseudo-residuated logic is an n-fold boolean pseudo-residuated lattice. (iv) The model of the n-fold fantastic pseudo-residuated logic is an n-fold fantastic pseudo-residuated lattice. (v) The model of the n-fold strong pseudo-residuated logic is an n-fold strong pseudo-residuated lattice. (vi) The model of the n-fold strongly integral pseudo-residuated logic is an n-fold strongly integral pseudo-residuated lattice.

Theorem 5.23 [13] Letn ≥ 1.

(i) A residuated lattice M is n-fold boolean iff it is n-fold fantastic and n-fold implicative. (ii) Any boolean algebra is an n-fold boolean residuated lattice, but the converse is not true. (ii) Any Heyting algebra is an n-fold implicative residuated lattice, but the converse is not true. (ii) Any MV algebra is an n-fold fantastic residuated lattice, but the converse is not true.

From the above theorem, it follows the following results.

Corollary 5.24. (i) n-fold boolean pseudo-residuated logic = n-fold fantastic pseudo-residuated logic+ n-fold implicative pseudo-residuated logic. (ii) n-fold involutive pseudo-residuated logic generalizes n-fold fantastic pseudo-residuated logic and n-fold strong pseudo-residuated logic generalizes n-fold involutive pseudo-residuated logic. (iii) n-fold boolean (resp., implicative, fantastic) pseudo-residuated logics generalize Propositional logic, Gödel logic and MV logic, respectively.

6 Conclusion

The results of this paper will be devoted to study some n-fold residuated logics and relationship among them. We introduce new logics which generalize Propositional logic, Gödel logic and MV logic, respectively.