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formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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Very true operators on MTL-algebras

Jun Tao Wang / Xiao Long Xin / Arsham Borumand Saeid
  • Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
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Published Online: 2016-12-10 | DOI: https://doi.org/10.1515/math-2016-0086

Abstract

The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.

Keywords: Very true MTL-algebra; Subdirectly irreducible; Representation; Stabilizer topology; Very true MTL-logic

MSC 2010: 03F50; 06F99

1 Introduction

Basic fuzzy logic (BL for short) is the many-valued residuated logic introduced by Hájek [1] to handle continuous t-norms and their residua. The fuzzy logics such as Łukasiewicz, Gödel and Product logic can be regarded as schematic extensions of BL. It is a well-known result that a t-norm has a residuum if and only if it is left-continuous; so this shows that BL is not the most general t-norm based logic. In fact, a logic weaker than BL, called monoidal t-norm-based logic (MTL for short), was introduced by Esteva and Godo in [2] and Jenei and Montagna [3] proved that MTL is indeed the logic of all left-continuous t-norms and their residua. In connection with the MTL logic, a new class of algebras is defined, called MTL-algebras [2]. In the last few years, the theory of MTL-algebras has been enriched with structure theorems [4, 5]. Many of these results have a strong impact with its algebraic structure. For example, Vetterlein [4] proved that most of MTL-algebras can be embeddable into the positive cone of a partially ordered group. He also proved that an MTL-algebra is a bounded, commutative, integral, prelinear residuated lattice [5]. As a more general residuated structure based on left-continuous t-norm logic, an MTL-algebra is a BL-algebra without the identity xy = x ⊙ (xy). Thus, MTL-algebras are the most fundamental residuated structures containing all algebras induced by (left) continuous t-norms and their residua. Therefore, MTL-algebra play an important role in studying fuzzy logics and their related structures. The filter theory of the MTL-algebras plays an important role in studying these algebras and the completeness of the MTL. From a logic point of view, various filters have natural interpretation as various sets of provable formulas. Recently, the filters on MTL-algebras have been widely studied and some important results have been obtained [2,6-8]. In particular, Esteva introduced the idea of filters and prime filters in MTL-algebras to prove the completeness and chain completeness of MTL [2]. After then, the concepts of implicative, positive and fantastic filters were defined in MTL-algebras in [6]. In [7], Borzooei was the first to systematically study filter theory in MTL-algebras, in which the relations between kinds of filters were obtained and some of their characterizations were presented. It was also proved that there exists at most one proper associative filter in any MTL-algebra, which is composed of all non-zero elements in this MTL-algebra in [8].

The concept of “very true” was introduced by Hájek [9] as an answer for the question “whether any natural axiomatization is possible and how far can even this sort of fuzzy logic be captured by standard methods of mathematical logic?”. In other words, very true operator as a tool for reducing the number of possible logical values in many-valued fuzzy logic. In fact, it is the same as the concept of hedge introduced by Zadeh [10], who gives some examples of handling these fuzzy truth values that seems uninterested in any sort of axiomatization. Apart from their important application in many valued fuzzy logic, very true operators were successfully used to formal concept analysis (FCA, in brief) (see [11]), which is another important branch of mathematics and becoming an popular method for analysis of object-attribute data. The main aim in FCA is to extract interesting clusters (called formal concepts) from tabular data, formal concepts correspond to maximal rectangles in a data table, hence the number of formal concepts in data can be extremely large. In order to reduce the number of formal concepts, Bělohlávek and Vyhodil [12] used the so called hedges, which are special cases of very true operators used in reducing the number of formal concepts in concept lattice. Since very true operator was successful in several distinct tasks in various branches of mathematics [10,12-14], it has been extended to other logical algebras such as MV-algebras [15], Rℓ-monoids [16], commutative basic algebras [17], equality algebras [18], effect algebras [19] and so on.

As we have mentioned in the above paragraph, very true operators have been studied on MV-algebras, BL-algebras, Rℓ-monoids and commutative basic algebras, etc. All the above-mentioned algebraic structures satisfy the divisibility condition xy = x ⊙ (xy). In this case, the conjunction ⊙ on the unit interval corresponds to a continuous t-norm. However, there are few research about the very true operators on residuated structures without the divisibility condition so far [19]. In fact, MTL-algebras are the more general residuated structure without the divisibility condition since it is an algebra induced by a left continuous t-norm and its corresponding residuum. Therefore, it is meaningful to study very true operators on MTL-algebras for treating a variant of the concept of very true operators within the framework of universal algebras and providing a solid algebraic foundation for reasoning about very true MTL logic. This is the motivation for us to investigate very true operators on MTL-algebras.

Based on the above considerations, we enrich the language of MTL by adding a very true operator to get algebras named very true MTL-algebras, which are the algebraic counterpart of very true MTL logic. This paper is structured in five sections. In order to make the paper as self-contained as possible, we recapitulate in Section 2 the definition of MTL-algebras, and review their basic properties that will be used in the remainder of the paper. In Section 3, we introduce very true operators on MTL-algebras and study some of their properties. Also, we give some characterizations of MV-algebra and Gödel algebra via such operator. In Section 4, we investigate very true filters of very true MTL-algebras and focus on an analogous of representation theorem for very true MTL-algebras and characterize subdirectly irreducible very true MTL-algebras by very true filters. Then, we introduce the left and right stabilizers of very true MTL-algebras and construct stabilizer topology via them. In Section 5, the corresponding very true MTL-logic is constructed, the soundness and completeness of this logic are proved based on the variety of very true MTL-algebras.

2 Preliminaries

In this section, we summarize some definitions and results about MTL-algebras, which will be used in the following and we shall not cite them every time they are used.

([2]).

An algebraic structure (L, ∧, ∨, ⊙, →, 0, 1) of type (2, 2, 2, 2, 0, 0) is called an MTL-algebra if it satisfies the following conditions:

  1. (L, ∧, ∨, 0,1) is a bounded lattice,

  2. (L, ⊙, 1) is a commutative monoid,

  3. xyz if and only if xyz,

  4. (xy) ∨ (yx) = 1,

    for any x, y, zL.

In what follows, by L we denote the universe of an MTL-algebra (L, ∧, ∨, ⊙, →, 0,1). For any xL, we define ¬x = x → 0.

([5]). In any MTL-algebra L, the following properties hold: for all x, y, zL,

  1. xy if and only if xy = 1,

  2. xyxy,

  3. 1 → x = x,

  4. x → (yz) = (xy) ∧ (xz),

  5. (xy) → z = (xz) ∧ (yz),

  6. xy implies xzyz,

  7. xy = x → (xy),

  8. xy = (xy) → y,

  9. xyz = (xz) ∨ (yz),

  10. iI(xiy) = ∨iI xiy, provided that both infimum as well as supremum exist,

  11. xy = ((xy) → y) ∧ ((yx) → x),

  12. (xy) → z = x → (yz).

([5]). Let L be an MTL-algebra. Then L is called:

  1. a BL-algebra if xy = x ⊙ (xy) for any x, yL.

  2. an MV-algebra if (xy) → y = (yx) → x for any x, yL.

  3. a Gödel algebra if xx = x for any xL.

A nonempty subset F of L is called a filter of L if it satisfies: (1) x, yF implies xyF;(2)xF, yL and xy implies yF. We denote by F[L] the set of all filers of L. A filter F of L is called a proper filter if FL. A proper filter F of L is called a prime filter if for each x, yF and xyF, implies xF or yF. For any filter F of L we can associate a congruence on L defined by x ~ F y if and only if (xy) ∧ (yx) ∈ F. We denote by L/F the set of congruence classes and L/F becomes an MTL-algebra with the natural operations induced by those of L. Note that a filter F of L is prime iff L/F is a linearly ordered MTL-algebra ([2, 6, 7]).

([20]). A Heyting algebra is an algebra (H, ∨, ∧, →, 0,1) of type (2, 2, 2, 0,0) for which (H, ∨, ∧, 0,1) is a bounded lattice and for a, bH, ab is the relative pseudocomplement of a with respect to b, i.e., acb if and only if cab.

([20]). Let L be a complete lattice and aL. Then element a is said to be compact if for every subset S of L, a ≤ ∨ S implies that a ≤ ∨ F for some finite subset F of S.

([21]). A topological space (X, T) is called a Baire space if for each countable collection of open dense sets, their intersection is dense.

At the end of this section, we review the known main results about representation theory of MTL-algebras, which is helpful for studying very true analogous representation theorem of MTL-algebras.

([22]). An element b of a lattice L is meet irreducible ifX = b implies bX, for any finite subset X of L.

A filter F of an MTL-algebra L is called prime if F is a finitely meet-irreducible element in the lattice F[L]. A prime filter F is called minimal if F is a minimal element in the set of prime filters of L ordered by inclusion. By Zorn’s lemma, every prime filter contains a minimal prime filter ([24]).

Let L be a MTL-algebra, and XL. The set X={aL|ax=1,for eachxX}

is called the co-annihilator of X in L[23]. For any aL, we write a instead of {a}.

([24]). For PF[L], the following conditions are equivalent:

  1. P is a minimal prime,

  2. P = ∪{a|aP}.

([24]). For an MTL-algebra L, the following conditions are equivalent:

  1. L is representable,

  2. There exists a set S of prime filters such thatS = {1}.

3 Very true operators on MTL-algebras

In this section, inspired by Hájek [9], we enlarge the language of MTL-algebra by introducing a very true operator, and investigate some related properties. As applications of very true operator, we discuss the structures of the fixed point set of a very true operator and give conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra.

Let L be an MTL-algebra. The mapping τ : LL is called a very true operator if it satisfies the following conditions:

(∨l) τ(1)=1,

(∨2) τ(x) ≤ x,

(∨3) τ(xy) ≤ τ (x) → τ(y),

(∨4) τ(x) ≤ τ τ(x),

(∨5) τ(xy) ∨ τ (yx)=1.

The pair (L, τ) is said to be a very true MTL-algebra.

Such a proliferation of conditions deserves some explanation. Then “1” seen in (∨l) is considered as the logical value absolutely true. First note that (∨l) means that absolutely true is very true, which is sound for each natural interpretation in many valued logic system. (∨2) means that if φ is very true then it is true. (∨3) means that if both φ and φψ are very true then so is ψ, that means the connective τ preserve modus ponens. (∨4) says that if φ is very true then τ(φ) is very true, which is a kind of necessitation. To obtain very true MTL-algebras that are representable as subdirect products of very true MTL-chains, we using (∨5).

  1. Let L be an MTL-algebra. One can easily check that idL is a very true operator on L, that is to say, every MTL-algebra can be seen as a very true MTL-algebra.

  2. Any linearly ordered MTL-algebra L can admit a very true operator; i.e., τ(1) = 1 and τ(x) = 0 for any x < 1.

  3. Let L = {0, a, b, 1} with 0 ≤ ab ≤ 1. Consider the operationandgiven by the following tables:

    Then (L, ∧, ∨, ⊙, →, 0,1) is an MTL-algebra. Now, we define τ as follows: τ(0) = 0, τ(a)=τ(b) = a, τ(1) = 1. One can easily check that τ is a very true operator on L. However, τ is not a homomorphisms on L since τ(bb) = a ≠ 0= τ(b) ⊙ τ (b) and τ (ba) = a ≠ 1 = τ(b) → τ(a).

Let τ be a very true operator on L. Then for any x, yL we have,

  1. τ(0) = 0,

  2. τ(x) = 1 if and only if x = 1,

  3. xy implies τ (x) ≤ τ (y),

  4. τ (¬ x) ≤ ¬ (τ x),

  5. τ(x) ⊙ τ (y) ≤ τ (xy),

  6. τ (xy) = τ(x) ∧ τ (y),

  7. τ (xy) = τ(x) ∨ τ (y),

  8. τ (xy) ∨ (yx ) = 1,

  9. τ2 (x) = τ (x),

  10. τ (x) ≤ y if and only if τ(x) ≤ τ(y),

  11. τ (L) = Fixτ (L), where Fixτ (L) = {xL | τ (x) = x},

  12. Fixτ (L) is closed under ⊙, ∧, ∨,

  13. If τ (L) = L, then τ = idL,

  14. Ker(τ) = {1}, where Ker(τ) = {xL|τ(x)=1},

  15. Ker(τ) is a filter of L.

  1. Applying (V2), we have τ(0) ≤ 0 and hence τ(0) = 0.

  2. If τ(x) = 1 for some xL then by (V2), 1 = τ(x) ≤ x giving x = 1. The converse follows by (Vl).

  3. If xy, then xy = 1. It follows from (Vl) and (V3) that τ(x) → τ (y) = 1. Thus, τ(x) ≤ τ (y).

  4. It follows from (1) and (V3) that τ (¬ x) = τ (x→ 0) ≤ τ(x) → 0 = ¬ τ(x).

  5. From xyxy, we get yx →(xy). By (V3) and (3), we have τ(y) ≤ τ (x →(xy)) ≤ τ(x) → τ(xy) and hence τ(x) ⊙ τ (y) ≤ τ (xy).

  6. On one hand, it is easy to see that τ(xy) ≤ τ(x) ∧ τ (y). On the other hand, by (V2),(V4) and Proposition 2.2 (4), we obtain, for all x, yL, 1 = τ (xy) ∨ τ (yx) = τ(x→ (xy)) ∨ τ (y→(xy))≤ (τ(x) → τ (xy))∨(τ(y) → τ (xy) and hence by Proposition 2.2 (9), we get 1 = (τ(x) ∧ τ (y)) → τ (xy). Therefore, τ(xy) = τ(x) ∧ τ (y).

  7. By(6) and Proposition 2.2 (11), we obtain, for all x, yL, τ (xy) = τ((xy) → y) ∧ τ ((yx) → x). By (3) and (V3), we get τ(xy) ≤ τ((xy) →(τ(x) ∨ τ (y))) ∧ (τ(yx) → (τ(x) ∨ τ(y))). Hence by Proposition 2.2 (10), we obtain τ(xy) ≤ (τ(xy)) ∨ (τ(yx)) →(τ(x)∨ τ(y)) and hence τ(xy) ≤ τ (x) ∨ τ(y). The other inequality follows easily from (3).

  8. Applying (V2) and (V5), we get τ (xy)∨(yx)≥ τ(xy) ∨ τ (yx) = 1.

  9. By (V2) and (V4), we have τ2(x) = τ(x).

  10. For all x, yL, assume that τ(x) ≤ y, we have τ τ(x) ≤ τ (y). By the (9), we get τ2(x) = τ(x). Thus τ (x) ≤ τ(y). Conversely, suppose that τ(x) ≤ τ (y), we have τ(x) ≤ τ(y) ≤y.

  11. Let y ∈ τ(L), so there exists xL such that y = τ(x). Hence τ(y) = τ τ (x) = τ(x) = y. This follows that yFixτ(L). Conversely, if y ∈ Fixτ(L), we have y ∈ τ(L). Therefore, τ(L) = Fixτ(L).

  12. It follows from (5)-(7).

  13. For any xL, we have x = τ(x0) for some x0L. By (9), we have τ(x) = τ (τ (x0)) = τ(x0) = x. Therefore, τ = idL.

  14. Assume that xL but x ≠ 1 such that τ(x) = 1. Applying (V2), we have 1 = τ(x) ≤ x and hence x = 1, which is a contradiction. Therefore, Ker(τ) ={1}.

  15. This is easy to check. Hence we omit the proof.

The assertion (7) of above proposition gives us an idea of introducing a very true operator on an MTL-algebra in a different way. Namely, we can consider a mapping τ : LL satisfying (1) – (4) and the following axiom (5′) which replaces the axiom (5): (5′) τ(xy) = τ(x) ∨ τ (y). From this point, one can check that the very true MTL-algebra essentially generalize very true BL-algebra, which was introduced by Hájek in 2001. A very true operator τ on an MV-algebra L was introduced in Leuştean (2006) as a mapping τ : LL satisfying conditions (∨1)-(∨3) and (8) in Proposition 3.3. From this point of view, the notion of very true MTL-algebra also generalizes that of very true MV-algebra.

Although the Fixτ(L) is not necessary a subalgebra of an MTL-algebra in general (in Example 3.2 (c), one can check that Fixτ(L) is not a subalgebra of L since it is not closed under →), while it forms an MTL-algebra after redefined its fuzzy implication.

Let τ be a very true operator on L. Then (Fixτ(L), ∧, ∨, ⊙, ⇝, 0,1) is an MTL-algebra, where xy = τ(xy) for all x, yFixτ(L).

First, we show that (Fixτ(L), ∧, ∨, 0,1) is a bounded lattice with 0 as the smallest element and 1 as the greatest element. From Proposition 3.3 (6),(7), we have that Fixτ(L) is closed under ∨ and ∧. Thus (Fixτ (L), ∧, ∨) is a lattice. For all x ∈ τ(L), one can easily check that x∨ 1 = 1 and x ∧ 0 = 0. Thus, 0 is the smallest element and 1 is the greatest element in Fixτ(L), respectively. Therefore (Fixτ(L), ∧, ∨, 0,1) is a bounded lattice.

Next, we prove that (Fixτ(L), ⊙, 1) is a commutative monoid with 1 as neutral element. By Proposition 3.3 (5), we have Fixτ(L) is closed under ⊙. It follows that (Fixτ(L), ⊙) is a commutative semigroup. For all xFixτ(L), we obtain that x ⊙ 1 = x, that is, 1 is a unital element.

Then, we prove that ⇝ and ⊙ form an adjoint pair. For all x, yFixτ(L), we define Xy = τ(xy). Now, we will show that xyz if and only if yXz for all x, y, zFixτ(L). From Proposition 3.3 (10), we have τ(x) ≤ y if and only if τ(x) ≤ τ(y). Hence we have xyz if and only if yxz if and only if τ(y) ≤ xz if and only if τ(y) ≤ τ (xz) if and only if τ(y) ≤ xy if and only if yxz for all x, y, zFixτ(L).

Finally, we prove that the prelinearity condition holds. For all x, yFixτ(L), by (∨5), we have (xy) ∨ (Yx) = τ (xy) ∨ τ(yx) = 1.

Therefore, we obtain that (Fixτ(L), ∧, ∨, ⊙,⇝, 0,1) is an MTL-algebra.

The result of Theorem 3.4 shows that the fixed point set Fixτ(L) of very true operator in an MTL-algebra L has the same structure as L, which reveals the essence of the fixed point set.

In the following, using the properties of very true operators, we give some conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra.

Let (L, ∧, ∨, ⊙, →, 0,1) be an MTL-algebra and τ be a very true operator on L. Then the following conditions are equivalent:

  1. (L, ∧, ∨, ⊙, →, 0,1) is an MV-algebra,

  2. every very true operator τ satisfies τ (xy) = (τ(x) → τ(y)) → τ(y) = (τ(y) → τ(x)) → τ(x) for all x, yL.

(1) ⇒ (2) We note that an MV-algebra satisfies (xy) → y = (yx) → x for all x, yL. By Proposition 2.2 (11) and 3.3(7), we have τ(xy) = τ(x) ∨ τ (y) = ((τ(x) → τ(y)) → τ (y). In the similar way, we can prove τ(xy) = τ(x) ∨ τ(y) = ((τ(y) → τ(x)) → τ (x). Thus τ(xy) = (τ(x) → τ(y)) → τ(y) = (τ(y) → τ(x)) → τ(x).

(2) ⇒ (1) Suppose that every very true operator τ satisfies τ(xy) = (τ(x) → τ (y)) → τ(y) = (τ(y) → τ(x)) → τ (x) for all x, yL. Taking τ = idL, we have xy = (xy) → y = (yx) → x for all x, yL. Therefore, L is an MV-algebra.

Let (L, ∧, ∨, ⊙, →, 0,1) be an MTL-algebra and τ be a very true operator on L. Then the following conditions are equivalent:

  1. (L, ∧, ∨, ⊙, →, 0,1) is a Gödel algebra,

  2. every very true operator τ satisfies τ(xy) = τ(x) ⊙ τ(y) for all x, yL,

  3. every very true operator τ satisfies τ(xy) = τ(x) ⊙ τ(xy) for all x, yL.

(1) ⇒ (2) Suppose that L is a Gödel algebra. Then one can obtain that xy = xy = x ⊙(xy) for any x, yL. By Proposition 3.3 (6), we have τ(xy) = τ(x) ∧ τ(y) = τ(x) ⊙ τ (y). Thus τ (xy) = τ(x) ⊙ τ(y).

(2) ⇒ (1) Suppose that every very true operator τ satisfies τ(xy) = τ(x) ∧ τ(y) for all x, yL. Taking τ = idL, we have xy = xy for any x, yL. Taking x = y, we get xx = x for all xL. Therefore, L is a Gödel algebra.

(1) ⇒(3) From (1) ⇒ (2), one can obtain that τ(xy) = τ (x) ⊙ τ (y) for any x, yL. Hence τ(xy) = τ(x ⊙ (xy) = τ(x) ⊙ τ (xy).

(3) ⇒(1) Suppose that every very true operator τ satisfies τ(xy) = τ(x) ⊙ τ (xy) for all x, yL. Taking τ = idL, we have xy = x ⊙(xy) for any x, yL. Taking x = y, we get xx = x for all xL. Therefore, L is a Gödel algebra.

4 Very true filters of very true MTL-algebras

In this section, we introduce very true filters of very true MTL-algebras. In particular, we focus on algebraic structures of VF(L) of all very true filters in the very true MTL-algebras and obtain that VF(L) forms a complete Heyting algebra. Moreover, we characterize subdirectly irreducible very true MTL-algebras and prove a representation theorem for very true MTL-algebras via very true filters.

Let (L, τ) be a very true MTL-algebra and F be a filter of L. Then F is called a very true filter of (L, τ) if xF implies τ (x) ∈ F for all xL.

We will denote the set of all very true filters of (L, τ) by VF[L].

Considering Example 3.2 (c), one can easily check that the very true filters of (L, τ) are {a, b, 1} and {1} and L. However, {b, 1} is a filter of L but not a very true filter of (L, τ).

Let (L, τ) be a very true MTL-algebra. For any nonempty set X of L, we denote by 〈Xτ the very true filter of (L, τ) generated by X, that is, 〈Xτ is the smallest very true filter of (L, τ) containing X. If F is a very true filter of (L, τ) and xF, we put 〈F, xτ := 〈F ∪ {x}〉τ.

The next theorem gives a concrete description of the very true filter generated by a subset of very true MTL-algebra (L, τ).

Let (L, τ) be a very true MTL-algebra and X be a nonempty set of L. ThenXτ = {xL|x≥ τ(y1) ⊙ … τ (yn), yiX, n ≥ 1}.

The proof is easy, and we hence omit the details.

Considering Example 3.2(c), one can easily obtain that 〈0, aτ = 〈0, bτ = 〈0, a, bτ = {a, b, 1}, 〈Lτ = 〈a, b, 1〉τ = 〈0, a, 1〉τ = 〈0, b, 1〉τ = {1} and 〈0〉τ = L.

Let F, F1, F2 be very true filters of (L, τ) and aF. Then:

  1. aτ = {xL|x ≥ (τa)n, n ≥ 1},

  2. Faτ = {xL|xf ⊙(τ a)n, fF} = F ∨ [τ a),

  3. F1F2τ = {xL|xf1f2, f1F1, f2F2},

  4. if ab, thenbτ ⊆ 〈aτ,

  5. 〈τ(a)〉τ = 〈aτ,

  6. aτ ∨ 〈bτ = 〈abτ = 〈abτ,

  7. aτ ∩ 〈bτ = 〈τ(a)∨ τ(b)〉τ.

The proof of (1) – (5) are obvious.

(6) Since ababa, b, we deduce that 〈aτ, 〈bτ ⊆ 〈abτ ⊆ 〈abτ. It follows from that 〈aτ ∨ 〈bτ⊆ 〈abτ ⊆ 〈abτ. Conversely, let a ∈ 〈abτ. Then for some natural number n≥ 1, a ≥(τ(ab))n ≥( τ a ⊙ τ b)n = (τ a)n ⊙ (τ b)n. Hence a ∈ 〈aτ ∨ 〈bτ, we deduce that 〈 abτ ⊆ 〈aτ ∨ 〈bτ. Therefore 〈aτ ∨ 〈bτ = 〈abτ = 〈abτ.

(7) Since τ(a)≤ τ(a) ∨ τ(b), we deduce that 〈 τ(a) ∨ τ(b)〉τ ⊆ 〈 τ(a)〉τ = 〈aτ. Analogously, 〈 τ(a) ∨ τ(b) 〉τ⊆ 〈 τ(b) 〉τ = 〈bτ. It follows that 〈 τ(a) ∨ τ(b) 〉τ ⊆ 〈aτ ∩ 〈bτ. Moreover, let t ∈ 〈aτ ∩ 〈bτ. Then for some natural number n, m ≥ 1, t ≥(τ(a))m and t ≥(τ(b))n. Hence t≥(τ(a))m∨ (τ(b))n ≥ (τ(a) ∨ τ(b))mn = (τ(ab))mn, we deduce that a∈ 〈 τ(a)∨ τ(b)〉τ, that is, 〈aτ ∩ 〈bτ ⊆ 〈τ(a) ∨ τ(b) 〉τ. Therefore 〈aτ ∩ 〈bτ = 〈 τ(a)∨ τ(b)〉τ.

The next results shows that the algebraic structure of VF(L) of all very true filters in very true MTL-algebras forms a complete Heyting algebra.

Let(L, τ) be a very true MTL-algebra. Define binary operations ∧, ∨, ↦ on VF(L) as follows : for all F1, F2VF(L), F1F2 = F1F2, F1F2 = 〈 F1F2τ, F1F2 = {xL|τ(x)∨ f1F2 for any f1F1}. Then (VF(L), ∧, ∨, ↦, 1, L) is a complete Heyting algebra.

Suppose that {Fi}i∈ I is a family of very true filters of (L, τ) . From Theorem 4.5, it is easy to check that the infimum of {Fi}iI = ∩iIFi and the supermum is ∨iIFi= {xL|xfi1fi2 ⊙ … fim, f1jFij, ijI, 1 ≤ jm}. Therefore, (VF(L), ∧, ∨, 1, L) is a complete lattice under the inclusion order ⊆. Next, we define F1F2 = {xL|τ(x)∨ f1F2 for any f1F1 〉 for any F1, F2VF(L). And, we shall prove that F1F2F3 if and only if F2F1F3 for all F1, F2, F3VF(L), that is, (VF(L), ∧, ∨, ↦, 1, L) is a complete Heyting algebra. In order to do this, we first show that F1F2 is a very true filter of (L, τ).

Now, we will show that F1F2 is a very true filter of (L, τ). Clearly 1∈ F1F2. Let xF1F2 and xy, then for any f1F1 such that τ(x)∨ f1F2. Since τ(x)∨ f1 ≤ τ(y)∨ f1F2 and hence yF1F2. Assume that x, yF1F2, then for any f1F1, τ(x)∨ f1, τ(y)∨ f1F2 and hence f1 ∨ τ(xy)∈ F2. So xyF1F2. Obviously, if xF1F2, then τ(x)∈ F1F2 and thus F1F2 is a very true filter of (L, τ).

Next, we will prove that F1F2F3 if and only if F1F2F3. Assume that F1F2F3. Let f1F1. Then τ(f1)∈ F1 and for any f2F2, we have τ(f2)∨ f1f1, τ(f2) ∨ f1 ≥ τ(f2). Hence τ(f2)∨ f1F1F2F3 and hence f1F1F2. Conversely, assume that F1F2F3. Let xF2F3, then xF2F3. For any f3F3, we have τ(x) ∨ f3F3. Taking f3 = xF3, we have x ∨ τ(x) = xF3. Thus F1F2F3.

Therefore, (VF(L), ∧, ∨, ↦, 1, L) is a complete Heyting algebra.

Let (L, τ) be a very true MTL-algebra and FVF(L). Then thefollowing conditions are equivalent:

  1. F is a compact element of VF(L),

  2. F is a principal very true filter of (L, τ).

(1) ⇒ (2) Suppose that F is the compact element of VF(L). Since F = ∨xFxτ, then there exist x1, x2xn such that F = 〈x1τ ∨ 〈x2τ ∨…∨ 〈xnτ. By Proposition 4.5 (6), we have F = 〈x1x2 ⊙ … ⊙xnτ. Therefore, F is a principal very true filter of (L, τ).

(2) ⇒(1) Let F be a principal very true filter of (L, τ) . Then there exists xL such that F = 〈xτ. Suppose that {Fi}iIVF(L) and F = 〈xτ ⊆ ∨iI {Fi}. Then x ∈ ∨iI Fi= 〈∪iI Fiτ. It follows that there exist ijI, fijFij for all 1 ≤ jm such that xfi1fi2 ⊙ … fim, that is, x ∈ 〈Fi1Fi2 ∪ … ∪ Finτ = fi1Fi2 ∨ … ∨ Fin. Hence F= 〈xτFi1Fi2 ∨ … ∨ Fin. Therefore, F is a compact element of VF(L).

Let (L, τ) be a very true MTL-algebra and θ be a congruence on L. Then θ is called a very true congruence on (L, τ) if (x, y)∈ θ implies (τ(x), τ(y)) ∈ θ, for any x, yL.

Considering Example 3.2 (c), one can see that R = {{0,0}, {a, a}, {b, b}, {1,1}, {a, b}, {b, a}, {a, 1}, {1, a}, {b, 1}, {1, b}} is a very true congruence on (L,τ).

For any very true MTL-algebra there exists a one to one correspondence between its very true filters and its very true congruences.

The proof is easy, and we hence omit the details.

Let (L, τ) be a very true MTL-algebra and F be a very true filter. We define the mapping τF : L/FL/F such that τF([x])= [τ(x)] for any xL.

Let (L, τ) be a very true MTL-algebra and F a very true filter of (L, τ). Then (L/F, τF) is a very true MTL-algebra.

The proof is easy, and we hence omit the details.

Let (L, τ) be a very true MTL-algebra. A proper very true filter F of (L, τ) is called a prime very true filter of (L, τ), if for all very true filter F1, F2 of (L, τ) such that F1F2F, then F1F or F2F.

Considering Example 3.2(c), one can easily obtain that {a, b, 1} is a prime very true filter of (L, τ).

Let (L, τ) be a very true MTL-algebra and F be a proper very true filter of (L, τ). Then the following are equivalent:

  1. F is a prime very true filter of (L, τ),

  2. if τ(x) ∨ τ(y) ∈ F for some x, yL, then xF or yF,

  3. (L/F, τF) is a chain.

(1) ⇒(2) Let τ(x) ∨ τ(y) ∈ F for some x, yL. Then 〈xτ ∩ 〈yτ = 〈 τ(x) ∨ τ(y) 〉τF. Since F is a prime very true filter of (L, τ), then 〈xτF or 〈yτF. Therefore, xF or yF.

(2) ⇒ (1) Suppose that F1,F2MF[L] such that F1F2F and F1F and F2F. Then there exist xF1 and yF2 such that x, yF. Since F1, F2 are very true filters of (L, τ), then τ(x)∈ F1 and τ(y) ∈ F2. From τ(x), τ(y) ≤ τ(x) ∨ τ(y), we obtain that τ(x)∨ τ (y) ∈ F1F2 = F. By (2), we get xF or yF, which is a contradiction. Therefore, F is a prime very true filter of (L, τ).

(1) ⇔(3) From (2), one can obtain that every prime very true filter of (L, τ) must be a prime filter of L. Based on this, the equivalence of (1) and (3) is clear.

For proving the subdirect representation theorem of very true MTL-algebras we will need the following theorem.

Let (L, τ) be a very true MTL-algebra and aL. If a≠ 1, then there exists a prime very true filter P of (L, τ) such that aP.

Denote Fa= {F|F is a proper very true filter of (L, τ) such that FF, aF}. Then fa ≠  since F is a very true filter not containing a and Fa is a partially set under inclusion relation. Suppose that {Fi|iI} is a chain in Fa, then ∪ {Fi|iI} is a very true filter of (L, τ) and it is the upper bounded of this chain. By zorn’s Lemma, there exists a maximal element P in Fa. Now, we shall prove that P is the desire prime very true filter of ours. Since PFa, then P is a proper very true filter and aP.

Let xyP for some x, yL. Suppose that xP and yP. Since P is strictly contained in 〈P, xτ and 〈P, yτ and by the maximality of P, we deduce that 〈P, xτFa and 〈P, yτFa. Then a∈ 〈P, xτ = P ∨[τ x) and a ∈ 〈P, yτ = P ∨[τ(y)) . Then we have a ∈(P ∨[τ(x)))∧(P ∨[τ(y))) = P ∨([τ(x))∧[τ(y))) = P∨[τ(x)∨τ(y)) = P ∨[τ(xy))∈ P, which implies that aP, a contradiction. Therefore, P is a prime very true filter such that FP and aP. Put F = { 1 }, the result is easy to obtain.

Now, we will prove that every very true MTL-algebra is a subdirect product of linearly ordered very true MTL-algebras.

Each very true MTL-algebra is a subalgebra of the direct product of a system of linearly ordered very true MTL-algebras.

The proof of this theorem is as usual and the only critical point is the above Theorem 4.15.

The next results shows that MTL-algebra is representable if and only if very true MTL-algebra is representable.

Let (L, τ) be a very true MTL-algebra. Then the following conditions are equivalent:

  1. L is representable;

  2. (L, τ) is a subdirect product of linearly ordered very true MTL-algebras.

(1) ⇒(2) Suppose that the MTL-algebra L is representable. Then by Theorem 2.9, there exists a system S of prime filter of L such that ∩ S= {1}. Since every prime filter of L contains a minimal prime filter, we get that in our case the intersection of all minimal prime filter is equal to {1}. Moreover, we will show that every minimal prime filter in (L, τ). Let P be a minimal prime filter of L. Then by Theorem 2.8, P = ∪ {a|aP}. If xP, then there is aP such that xa = 1, hence 1 = τ(xa) = τ(x)∨ τ(a). Since aP, we get τ(a)∉ P, therefore τ(x)∈ P, that means that P is a very true filter in (L, τ). Therefore, (L, τ) is a subdirect product of linearly ordered very true NM-algebras.

(2) ⇒ (1) The converse is trivial and we hence omit this.

Let L be an MTL-algebra. Then the following condition are equivalent:

  1. L is representable,

  2. L can be embedded in a very true MTL-algebra (L, τ).

(1) ⇒(2) If L is representable, then L is a subdirect product of MTL-chain. By Example 3.2 (b), any MTL-chain has a structure of very true MTL-algebra. Moreover, the class of very true MTL-algebras is a variety, so a direct product of very true MTL-algebra is still a very true MTL-algebra.

(2) ⇒(1) is straightforward, since any very true MTL-algebra is a subdirect product of very true MTL-chains.

A very true MTL-algebra (L, τ) is said to be a subdirectly irreducible if it has the least nontrivial very true congruence.

Let (L, τ) be subdirectly irreducible. Then by Theorem 4.10, there is a very true filter F of (L, τ) such that θF = F, that means, F is the least very true filter of (L, τ) such that F ≠ {1}. Thus, we can conclude that a very true MTL-algebra (L, τ) is said to be subdirectly irreducible if among the nontrivial very true filters of (L, τ) there exists the least one, i.e., ∩ {FVF(L)|F ≠ {1}} ≠ {1}.

Considering Example 3.2 (c), one can easily check that the very true MTL- algebra (L, τ) is subdirectly irreducible.

Next, we will show that every subdirectly irreducible very true MTL-algebra is linearly ordered. To prove this important result, we need the following several propositions and theorems.

Let(L, τ) be a subdirectly irreducible very true MTL-algebra and F1, F2VF(L). If F1F2= {1}, then F1= {1} or F2= {1}.

Suppose F1 ≠ {1} and F2 ≠ {1}, i.e., F1, F2 ∈ ∩ {FVF(L)|F ≠ {1}}≠ {1}, then ∩{FVF(L)|F ≠ {1}} ≠ {1} ⊆ F1F2. By F1F2 = {1}, we can get ∩ {FVF(L)|F ≠ {1}} = {1}, which contradicts the fact that (L, τ) is subdirectly irreducible. Hence, F1 = {1} or F2 = {1}.

Let (L, τ) be a very true MTL-algebra. Then the following conditions are equivalent:

  1. (L, τ) is a subdirectly irreducible very true MTL-algebra,

  2. there exists an element aL,a < 1, such thatfor any xL, x < 1 and a ∈ 〈xτ.

(1) ⇒ (2) Suppose that (L, τ) is a subdirectly irreducible very true MTL-algebra, i.e., ∩ {FVF(L)|F ≠ {1}} ≠ {1}, then ∩ {〈xτ|x <1} ≠ {1}. Take a ∈ ∩ {〈xτ|x<1} satisfying a ≠ 1, then for any xL, x ≠ 1, a ∈ 〈xτ, by Theorem 4.5 (1), there exists mN, such that a ≥(τ(x))m. Clearly, a is the element that we need.

(2) ⇒(1) Conversely, we need to prove that for any FVF(L), if F ≠ {1}, then aF. In fact, by F ≠ {1}, it follows that there exists xF, x <1, and then, by the known condition, a∈ 〈xτ, furthermore, aF, so a ∈ ∩ {FVF(L)|F≠ {1}}. Hence, ∩ {FVF(L)|F ≠ {1}} ≠ {1}, i.e., (L, τ) is a subdirectly irreducible very true MTL-algebra.

We recall that a non-unit element aL is said to be a coatom of L if ab, then b ∈ {a, 1}, i.e. b = a or b = 1 ([25]). In the following proposition, we will show that every subdirectly irreducible very true MTL-algebra has at most one coatom.

Let (L, τ) be a subdirectly irreducible very true MTL-algebra. For any x,yL, if xy = 1, then x = 1 or y = 1.

For any x, yL, if xy = 1, then by Theorem 4.5, we have 〈xτ ∩ 〈yτ = 〈τ(x)∨ τ(y)〉τ = 〈τ(xy)〉τ= 〈1〉τ= {1}. By Proposition 4.21, we have 〈xτ = {1} or 〈yτ= {1}, hence x = 1 or y = 1.

The following Theorem shows that the subdirectly irreducible very true MTL-algebra (L, τ) is linearly ordered, that is to say, the fuzzy truth value of all propositions in very true MTL logic are comparable. This is of key importance from the logical point of view.

Let (L, τ) be a very true MTL-algebra. Then the following conditions are equivalent:

  1. (L, τ) is a subdirectly irreducible very true MTL-algebra,

  2. (L, τ) is a chain.

(1) ⇒(2) Suppose (L, τ) is a subdirectly irreducible very true MTL-algebra. Applying Definition 2.1, we have (xy)∨(yx) = 1 for any x, yL, then by Proposition 4.23, we have xy = 1 or yx = 1, i.e., xy or yx, so (L, τ) is a chain.

(2) ⇒ (1) Since (L, τ) is a chain and nontrivial, there exists a unique dual atom, denoted as a. Suppose F is any very true filter of (L, τ) satisfying F ≠ {1}, then aF. Since F is chosen arbitrarily from VF(L), then a ∈ ∩ {FVF(L)|F≠ {1}}. Hence ∩ {FVF(L)|F ≠ {1}} ≠ {1}, i.e., (L, τ) is a subdirectly irreducible very true MTL-algebra.

In what follows, we introduce the stabilizer of a nonempty subset set X with respect to a very true operator τ and study some properties of them. Let (L, τ) be a very tue MTL-algebra. Given a nonempty subset X of L, we put Rτ(X) = {aL|τ(a)→ x = x, ∀ xX}, and Lτ(X) = {aL|x → τ (a) = τ(a), ∀ xX}, which are called right and left stabilizer of X with respect to τ. Clearly, Rτ (X), Lτ (X) ≠ Ø. In fact, 1 ∈ Rτ (X) ∩ Lτ (X). In particular, if τ = idL, which is a right and left stabilizer of X (see [26]).

Considering Example 3.2 (c). Let X = {a, b}. Then Lτ(X) = {a, b, 1} and Rτ(X) = {1}.

Let L be a MTL-algebra and X, YL. Then the following conditions hold:

  1. 1 ∈ Rτ(X)∩ Lτ(X),

  2. If XY, then Lτ(X) ⊆ Lτ(Y) and Rτ(X) ⊆ Rτ(Y),

  3. XLτ (Rτ(X)) ∩ Rτ(Lτ(X)),

  4. Rτ(X) = Rτ(Lτ (Rτ(X))) and Lτ (X) = Lτ (Rτ (Lτ(X))),

  5. Lτ(L) = Rτ (L) = {1} and Lτ (1) = Rτ (1) = L,

  6. If Ø ≠ XL, then Rτ (X) is a very true filter of (L, τ).

The proof of (1)-(5) is easy by Proposition 2.2.

(6) Let a, bRτ(X). Then τ(a)→ x = x and τ(b) → x = x for all xX. Hence by Proposition 2.2 (12), (τ(a) ⊙ τ(b))→ x = τ(a) → (τ(b) → x) = τ(a) → x = x for all xX and so τ(ab) → x ≤(τ(a) ⊙ τ(b)) → x = x. On the other hand, we have x ≤ τ(ab) → x. Therefore x = τ(ab) → x for all xX and hence abRτ(X). Now, let ab and aRτ (X). Then τ(a) → x = x, for all xX. Hence by Proposition 3.3, τ(b) → x ≤ τ(a) → x = x. Since by x ≤ τ(b) → x, then τ(b)→ x = x and bRτ(X). Finally, one can easy check that if aRτ(X) then τ(a)∈ Rτ(X). Therefore Rτ(X) is a very true filter of (L, τ).

Let F be a very true filter of (L, τ). Then Rτ(F) is a pseudocomplemented of F in the complete Heyting algebra (VF(L), ∧, ∨, ↦, 1, L).

First, we prove that FRτ (F) = {1}. Let xFRτ (F). Since xRτ (F), then for any aF, τ(x) → a = a. Now, since xF, put a = x we have x = 1. Therefore, FRτ (F) = {1}. Now, let G be a very true filter of (L, τ) such that FG = {1}. Let aG, then τ(a) ∈ G. Then for any xF, since τ(a), x ≤ τ(a) ∨ x, xF, then τ(a) ∨ xF and τ(a) ∨ xG and so τ(a) ∨ x ∈ {1}. Hence τ(a) ∨ x = 1, that is ((τ(a) → x) → x) ∧ ((x → τ(a)) → τ(a)) = 1 and so ((τ(a) → x) → x) = 1. Hence, τ(a) → xx. On the other hand, x ≤ τ(a) → x, then τ(a) → x = x and aRτ (F). Thus GRτ (F). Therefore, Rτ (F) is a pseudocomplemented of F in the complete Heyting algebra (VF(L), ∧, ∨, ↦, 1, L).

Let (L, τ) be a very true MTL-algebra. Then (VF(L), ∧, ∨, ↦, 1, L) is a complete pseudocomple-mented Heyting lattice.

It follows from Theorem 4.6 and Theorem 4.27

Now, we use of the right and left stabilizers of a very true MTL-algebra to produce a basis for a topology on it. Then we show that the generated topology by this basis is Baire, connected, locally connected and separable.

Let (L, τ) be a very true MTL-algebra and XL. Define a mapping α : P( L,≪) → P( L,≪)such that α(X) = Rτ(Lτ(X)) for all XP( L). Then the following conditions hold,

  1. α is a closure map on (L, τ),

  2. X ⊆ α(Y) if and only if α(X) ⊆ α(Y), for all YL,

  3. βα = {XP(L)|α(X) = X} is a basis for a topology on (L, τ).

  1. It follows from Proposition 4.26 (2),(3),(4).

  2. The proof is easy.

  3. Let βα = {XP(L) |α(X) = X}. It is clear that Ø ∈ βα. Also, by Proposition 4.26 (5), α(L)= Rτ (Lτ(L)) = Rτ ({1}). Thus, α(L) = L, and L ∈ βα. Now, suppose that X, Y ∈ βα. Then α(X) = X and α(Y) = Y. We prove that XY ∈ βα. Since XYX, Y, by (1), α(XY) ⊆ α(X), α (Y). Thus, α(XY) ⊆ α(X) ∩ α(Y). Also, since X, Y ∈ βα, we have α(XY) ⊆ XY. Moreover, by Proposition 4.26 (3), XY ⊆ α(XY). Then α(XY) = XY, and so XY ∈ βα. Therefore, βα is a basis.

Based on Theorem 4.29, we introduce the topological space, (L,T) is called stabilizer topology on very true MTL-algebras.

From Proposition 4.26(6), one can obtain that Rτ(X) ∈ VF(L), for any XL and hence every element of βα is a very true filter of (L, τ).

The stabilizer topology (L,T) is connected and locally connected.

The proof is easy, and we hence omit the details.

The stabilizer topology (L,T) is Hausdorffspace if and only if L = {1}.

The proof is easy, so we hence omit the details.

The stabilizer topology (L,T) is separable.

First, we show that if Ø ≠ XL such that 1 ∈ X, then X¯ = L. Let Ø ≠ XL such that 1 ∈ X. We only show that LX¯. Let xL. If x = 1, then xX¯. Hence X¯ = L. Now, suppose that 1 ≠ xL. Then there exists an open subset U ∈ βα such that xU. Since UVF(L) and 1 ∈ U, we have U ∩(X − {x}) ≠ Ø. Hence xX¯, and so X¯ = L. Since 1 ∈ βα, thus 1¯ = L. Hence (L,T) is separable.

The stabilizer topology (L,T) is a Baire space.

Let UT). Since UVF(L), we have 1 ∈ U. Then by Theorem 4.32, U¯ = L. Thus, every open set of (L,T) is dense. On the other hand, for each collection of open set Un, ∩ UnVF(L), so 1 ∈ ∩ Un. Thus, by Theorem 4.32, ∩ Un is dense. Therefore, (L,T) is a Baire space.

5 Very true MTL-logic

In this section, we translate the defining properties of very true MTL-algebras into logical axioms, and show that the resulting logic, i.e. very true MTL logic (MTLvt, for short) is sound and complete with respect to the variety of very true MTL-algebras.

Now, we deal with propositional calculus and define the axioms of the logic MTLvt to be those of MTL [2] plus the following ones:

  1. (vtl) vt(φ) ⇒ φ,

  2. (vt2) vt(φψ) ⇒ (vtφvtψ),

  3. (vt3) vt(φ) ⇒ vt(vtφ),

  4. (vt4) vt(φψ) ⊔ vt(ψφ).

The deduction rules are modus ponens (MP, from ϕ and ϕψ infer ψ), and Generalization (G, from ϕ infer vtϕ).

To prove the completeness theorem, we need some definitions and results about MTLvt logic.

The consequence relation ⊢ is defined in the usual way. Let T be a theory, i.e., a set of formulas in MTLvt. A (formula) proof of a formula ϕ in T is a finite sequence of formulas with ϕ at its end, such that every formula in the sequence is either an axiom of MTLvt, a formula of T, or the result of an application of an deduction rule to previous formulas in the sequence. If a proof of ϕ exists in T, we say that ϕ can be deduced from T and we denote this by Tϕ. Moreover T is complete if for each pair ϕ,ψ, Tϕψ or Tψϕ.

Let (L, τ) be a very true MTL-algebra and T be a theory. An L-evaluation is a mapping e from the set of formulas of MTLvt to L that satisfies, for each two formulas ϕ and ψ : e(ϕψ) = e (ϕ) → e(ψ), e(ϕψ) = e(ϕ) ∨ e(ψ), e(ϕψ)= e(ϕ)∧ e(ψ), e (ϕ&ψ) = e(ϕ) ⊙e (ψ), e(vtϕ) = τ e(ϕ), e (0¯)=0 and e(1¯)=1. If an L- evaluation e satisfies e(χ) = 1 for every χ in T, it is called an L-model of T.

Now, we stress our attention to the Lindenbaum-Tarski algebra of MTLvt

Let T be a fixed theory over MTLvt. For each formula ϕ, let [ϕ]T be the set of all formulas ψ such that Tϕψ ( where ϕψ stands for (ϕψ) & (ψφ)) and LT be the set of all the class [ϕ]T. We define: 0 = [0]T, 1 = [1]T, [ϕ]T ⇒ [ψ]T = [ϕψ]T, [ϕ]τ & [ψ]T = [ϕ&ψ]T, [ϕ]T ⊔[ψ]T = [ϕψ]T, [ϕ]T ⊓[ψ]T = [ϕψ]T, τT[ϕ]T = [vtϕ]T. This algebra is denoted by (LT, τT).

(LT, τT) is a very true MTL-algebra.

It follows similarly from the proof of Proposition 4.11.

Let T be a theory over MTLvt. Then T is complete if and only if the very true MTL-algebra (LT, τT) is linearly ordered.

It follows similarly from the proof of Theorem 4.14.

It is easy to check that MTLvt is sound with respect to the variety of very true MTL-algebras, i.e., that if a formula ϕ can be deduced from a theory T in MTLvt, then for every very true MTL-algebra (L, τ) and for every L-model e of T, e(ϕ) = 1. Indeed, we need to verify the soundness of the new axioms and deduction of MTLvt (for the axioms and rules of MTL, the reader can check [2]). For the axioms this is easy, as they are straightforward generalizations of axioms of very true MTL-algebras. We will now verify the soundness of the new deduction rules.

The deduction rules of MTLvt are sound in the following sense, for any formula ϕ and ψ.

  1. If for all very true MTL-algebra (L, τ) andfor all L-model e for T, e(ϕ) = 1, then for all very true MTL-algebra (L, τ) and for all L-model e for T, e(vtϕ) = 1.

  2. If for all very true MTL-algebra (L, τ) and for all L-model e for T, e(ϕ) = 1 and e(ϕψ) = 1, then for all very true MTL-algebra (L, τ) and for all L-model e for T, e(ψ) = 1.

  1. Take such a very true MTL-algebra (L, τ) and such a model e. Then e(ϕ) = 1, and e(vtϕ)= τ e(ϕ) = τ(1)=1.

  2. Take such very true MTL-algebra and a model e. Then e(ϕ)→ e(ψ)= e(ϕ)→ e (ψ) = 1, which means e (ϕ) ≤ e(ψ). If e(ϕ) = 1, we have 1 = e(ϕ)≤ e(ψ) and thus e(ψ)= 1.

Let T be a theory over MTLvt. If T is a theory and T ϕ, then there is a consistent complete supertheory TT such that T ϕ.

It follows similarly with the proof of Theorem 4.15.

The next result in the sequence is the completeness of very true MTL-logic is proved based on the variety of very true MTL-algebras.

Let T be a theory over MTLvt. For each formula φ, the following are equivalent:

  1. Tϕ;

  2. for each very true MTL-algebra (L, τ) and for every L-model e of T, e(ϕ) = 1;

  3. for each linearly ordered very true MTL-algebra (L, τ) and for every L-model e of T, e(ϕ) = 1.

It follows from Theorems 4.16, 5.4, 5.6 and Proposition 5.5.

MTLvt is a conservative extension of MTL.

Suppose MTL φ then there is a linearly ordered MTL-algebra L such that φ is not an L-model. Expand an MTL-algebra L to a very true MTL-algebra follows from Example 3.2 (b). By Theorem 5.7, we have MTLvt φ. Therefore, MTLvt is a conservative extension of MTL.

6 Conclusion

In this paper, motivated by the previous research of very true operators on BL-algebras, we extended the concept of very true operators to MTL-algebras. Also, we gave s characterizations of MV-algebras and Gödel algebras via this operator. Moreover, we investigated very true filters of very true MTL-algebras and focus on an analogous of representation theorem for very true MTL-algebras and characterize subdirectly irreducible very true MTL-algebras by very true filters. Then, we introduced the left and right stabilizers of very true MTL-algebras and constructed stabilizer topology via it. Finally, the corresponding logic very true MTL-logic was constructed and the soundness and completeness of this logic were proved based on very true MTL-algebras. Our further work on this topic will focus on the varieties of very true MTL-algebras. In particular, we will investigate semisimple, locally finite, finitely approximated and splitting varieties of very true MTL-algebras as well as varieties with the disjunction and the existence properties.

Acknowledgements

The authors thank the editors and the anonymous reviewers for their valuable suggestions in improving this paper. This research was partially supported by the National Natural Science Foundation of China (11571281).

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About the article

Received: 2016-05-27

Accepted: 2016-10-18

Published Online: 2016-12-10

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0086.

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© 2016 Wang et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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