Very true operators on MTL-algebras

Jun Tao Wang; Xiao Long Xin; Arsham Borumand Saeid

doi:10.1515/math-2016-0086

Article Open Access

Very true operators on MTL-algebras

Jun Tao Wang , Xiao Long Xin and Arsham Borumand Saeid

Published/Copyright: December 10, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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 x ∧ y = x ⊙ (x → y). 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 x ∧ y = x ⊙ (x → y). 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.

Definition 2.1

([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:

(L, ∧, ∨, 0,1) is a bounded lattice,
(L, ⊙, 1) is a commutative monoid,
x ⊙ y ≤ zif and only ifx ≤ y → z,
(x → y) ∨ (y → x) = 1,
for anyx, y, z ∈ L.

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

Proposition 2.2

([5]). In any MTL-algebra L, the following properties hold: for allx, y, z ∈ L,

x ≤ yif and only ifx → y = 1,
x ⊙ y ≤ x ∧ y,
1 → x = x,
x → (y ∧ z) = (x → y) ∧ (x → z),
(x ∨ y) → z = (x → z) ∧ (y → z),
x ≤ yimpliesx ⊙ z ≤ y ⊙ z,
x → y = x → (x ∧ y),
x → y = (x ∨ y) → y,
x ∧ y → z = (x → z) ∨ (y → z),
⋀_i∈I(x_i → y) = ∨_i∈Ix_i → y, provided that both infimum as well as supremum exist,
x ∨ y = ((x → y) → y) ∧ ((y → x) → x),
(x ⊙ y) → z = x → (y → z).

Definition 2.3

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

a BL-algebra if x ∧ y = x ⊙ (x → y) for anyx, y ∈ L.
an MV-algebra if (x → y) → y = (y → x) → xfor anyx, y ∈ L.
a Gödel algebra ifx ⊙ x = xfor anyx ∈ L.

A nonempty subset F of L is called a filter of L if it satisfies: (1) x, y ∈ F implies x ⊙ y ∈ F;(2)x ∈ F, y ∈ L and x ≤ y implies y ∈ F. We denote by F[L] the set of all filers of L. A filter F of L is called a proper filter if F≠ L. A proper filter F of L is called a prime filter if for each x, y ∈ F and x ∨ y ∈ F, implies x ∈ F or y ∈ F. For any filter F of L we can associate a congruence on L defined by x ~ Fy if and only if (x → y) ∧ (y → x) ∈ 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]).

Definition 2.4

([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 fora, b ∈ H, a → bis the relative pseudocomplement of a with respect to b, i.e., a ∧ c ≤ b if and only ifc ≤ a → b.

Definition 2.5

([20]). Let L be a complete lattice anda ∈ L. Then element a is said to be compact if for every subset S of L, a ≤ ∨ Simplies thata ≤ ∨ Ffor some finite subset F of S.

Definition 2.6

([21]). A topological space (X, T) is calleda 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.

Definition 2.7

([22]). An element b of a lattice L is meet irreducible if ∧ X = b implies b ∈ X, 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 X ⊆ L. The set

X⊥={a∈L|a∨x=1,for eachx∈X}

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

Theorem 2.8

([24]). ForP ∈ F[L], the following conditions are equivalent:

P is a minimal prime,
P = ∪{a^⊥|a ∈ P}.

Theorem 2.9

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

L is representable,
There exists a set S of prime filters such that ∩ S = {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.

Definition 3.1

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

(∨l) τ(1)=1,

(∨2) τ(x) ≤ x,

(∨3) τ(x → y) ≤ τ (x) → τ(y),

(∨4) τ(x) ≤ τ τ(x),

(∨5) τ(x → y) ∨ τ (y → x)=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 φ → ψ 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).

Example 3.2

Let L be an MTL-algebra. One can easily check that id_L is a very true operator on L, that is to say, every MTL-algebra can be seen as a very true MTL-algebra.
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.
Let L = {0, a, b, 1} with 0 ≤ a ≤ b ≤ 1. Consider the operation ⊙ and → given by the following tables:
⊙ 0 a b 1 → 0 a b 1
0 0 0 0 0 0 1 1 1 1
a 0 0 0 a a b 1 1 1
b 0 0 b b b a a 1 1
1 0 a b 1 1 0 a b 1
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 onLsince τ(b ⊙ b) = a ≠ 0= τ(b) ⊙ τ (b) and τ (b → a) = a ≠ 1 = τ(b) → τ(a).

Proposition 3.3

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

τ(0) = 0,
τ(x) = 1 if and only if x = 1,
x ≤ y implies τ (x) ≤ τ (y),
τ (¬ x) ≤ ¬ (τ x),
τ(x) ⊙ τ (y) ≤ τ (x ⊙ y),
τ (x ∧ y) = τ(x) ∧ τ (y),
τ (x ∨ y) = τ(x) ∨ τ (y),
τ (x → y) ∨ (y → x ) = 1,
τ² (x) = τ (x),
τ (x) ≤ yif and only if τ(x) ≤ τ(y),
τ (L) = Fix_τ (L), whereFix_τ (L) = {x ∈ L | τ (x) = x},
Fix_τ (L) is closed under ⊙, ∧, ∨,
If τ (L) = L, then τ = id_L,
Ker(τ) = {1}, where Ker(τ) = {x ∈ L|τ(x)=1},
Ker(τ) is a filter of L.

Proof

Applying (V2), we have τ(0) ≤ 0 and hence τ(0) = 0.
If τ(x) = 1 for some x ∈ L then by (V2), 1 = τ(x) ≤ x giving x = 1. The converse follows by (Vl).
If x ≤ y, then x → y = 1. It follows from (Vl) and (V3) that τ(x) → τ (y) = 1. Thus, τ(x) ≤ τ (y).
It follows from (1) and (V3) that τ (¬ x) = τ (x→ 0) ≤ τ(x) → 0 = ¬ τ(x).
From x ⊙ y ≤ x ⊙ y, we get y ≤ x →(x ⊙ y). By (V3) and (3), we have τ(y) ≤ τ (x →(x ⊙ y)) ≤ τ(x) → τ(x ⊙ y) and hence τ(x) ⊙ τ (y) ≤ τ (x ⊙ y).
On one hand, it is easy to see that τ(x ∧ y) ≤ τ(x) ∧ τ (y). On the other hand, by (V2),(V4) and Proposition 2.2 (4), we obtain, for all x, y ∈ L, 1 = τ (x → y) ∨ τ (y → x) = τ(x→ (x ∧ y)) ∨ τ (y→(x∧ y))≤ (τ(x) → τ (x∧ y))∨(τ(y) → τ (x ∧ y) and hence by Proposition 2.2 (9), we get 1 = (τ(x) ∧ τ (y)) → τ (x ∧ y). Therefore, τ(x ∧ y) = τ(x) ∧ τ (y).
By(6) and Proposition 2.2 (11), we obtain, for all x, y ∈ L, τ (x∨ y) = τ((x → y) → y) ∧ τ ((y → x) → x). By (3) and (V3), we get τ(x ∨ y) ≤ τ((x → y) →(τ(x) ∨ τ (y))) ∧ (τ(y→ x) → (τ(x) ∨ τ(y))). Hence by Proposition 2.2 (10), we obtain τ(x∨ y) ≤ (τ(x→ y)) ∨ (τ(y → x)) →(τ(x)∨ τ(y)) and hence τ(x ∨ y) ≤ τ (x) ∨ τ(y). The other inequality follows easily from (3).
Applying (V2) and (V5), we get τ (x→ y)∨(y→ x)≥ τ(x → y) ∨ τ (y→ x) = 1.
By (V2) and (V4), we have τ²(x) = τ(x).
For all x, y ∈ L, 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.
Let y ∈ τ(L), so there exists x ∈ L such that y = τ(x). Hence τ(y) = τ τ (x) = τ(x) = y. This follows that y ∈ Fix_τ(L). Conversely, if y ∈ Fix_τ(L), we have y ∈ τ(L). Therefore, τ(L) = Fix_τ(L).
It follows from (5)-(7).
For any x ∈ L, we have x = τ(x₀) for some x₀ ∈ L. By (9), we have τ(x) = τ (τ (x₀)) = τ(x₀) = x. Therefore, τ = id_L.
Assume that x ∈ L 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}.
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 : L → L satisfying (1) – (4) and the following axiom (5′) which replaces the axiom (5): (5′) τ(x ∨ y) = 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 : L → L 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 Fix_τ(L) is not necessary a subalgebra of an MTL-algebra in general (in Example 3.2 (c), one can check that Fix_τ(L) is not a subalgebra of L since it is not closed under →), while it forms an MTL-algebra after redefined its fuzzy implication.

Theorem 3.4

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

Proof

First, we show that (Fix_τ(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 Fix_τ(L) is closed under ∨ and ∧. Thus (Fix_τ (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 Fix_τ(L), respectively. Therefore (Fix_τ(L), ∧, ∨, 0,1) is a bounded lattice.

Next, we prove that (Fix_τ(L), ⊙, 1) is a commutative monoid with 1 as neutral element. By Proposition 3.3 (5), we have Fix_τ(L) is closed under ⊙. It follows that (Fix_τ(L), ⊙) is a commutative semigroup. For all x ∈ Fix_τ(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, y ∈ Fix_τ(L), we define X ⇝ y = τ(x → y). Now, we will show that x ⊙ y ≤ z if and only if y ≤ X ⇝ z for all x, y, z ∈ Fix_τ(L). From Proposition 3.3 (10), we have τ(x) ≤ y if and only if τ(x) ≤ τ(y). Hence we have x ⊙ y ≤ z if and only if y ≤ x → z if and only if τ(y) ≤ x → z if and only if τ(y) ≤ τ (x → z) if and only if τ(y) ≤ x ⇝ y if and only if y ≤ x ⇝ z for all x, y, z ∈ Fix_τ(L).

Finally, we prove that the prelinearity condition holds. For all x, y ∈ Fix_τ(L), by (∨5), we have (x ⇝ y) ∨ (Y ⇝ x) = τ (x → y) ∨ τ(y → x) = 1.

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

The result of Theorem 3.4 shows that the fixed point set Fix_τ(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.

Theorem 3.5

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

(L, ∧, ∨, ⊙, →, 0,1) is an MV-algebra,
every very true operator τ satisfies τ (x ∨ y) = (τ(x) → τ(y)) → τ(y) = (τ(y) → τ(x)) → τ(x) for all x, y ∈ L.

Proof

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

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

Theorem 3.6

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

(L, ∧, ∨, ⊙, →, 0,1) is a Gödel algebra,
every very true operator τ satisfies τ(x ∧ y) = τ(x) ⊙ τ(y) for allx, y ∈ L,
every very true operator τ satisfies τ(x ⊙ y) = τ(x) ⊙ τ(x→ y) for all x, y∈ L.

Proof

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

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

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

(3) ⇒(1) Suppose that every very true operator τ satisfies τ(x ⊙y) = τ(x) ⊙ τ (x → y) for all x, y ∈ L. Taking τ = id_L, we have x ⊙ y = x ⊙(x→ y) for any x, y ∈ L. Taking x = y, we get x ⊙ x = x for all x ∈ L. 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.

Definition 4.1

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) ifx ∈ Fimplies τ (x) ∈ Ffor allx ∈ L.

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

Example 4.2

ConsideringExample 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 〈X〉_τ the very true filter of (L, τ) generated by X, that is, 〈X〉_τ is the smallest very true filter of (L, τ) containing X. If F is a very true filter of (L, τ) and x ∉ F, we put 〈F, x〉_τ := 〈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).

Theorem 4.3

Let (L, τ) be a very true MTL-algebra and X be a nonempty set of L. Then 〈 X 〉_τ = {x ∈ L|x≥ τ(y₁) ⊙ … τ (y_n), y_i ∈ X, n ≥ 1}.

Proof

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

Example 4.4

ConsideringExample 3.2(c), one can easily obtain that 〈0, a〉_τ = 〈0, b〉_τ = 〈0, a, b 〉_τ = {a, b, 1}, 〈L〉_τ = 〈a, b, 1〉_τ = 〈0, a, 1〉_τ = 〈0, b, 1〉_τ = {1} and 〈0〉_τ = L.

Theorem 4.5

LetF, F₁, F₂be very true filters of (L, τ) anda ∉ F. Then:

〈a〉_τ = {x ∈ L|x ≥ (τa)ⁿ, n ≥ 1},
〈F ∪ a〉_τ = {x ∈ L|x ≥ f ⊙(τ a)ⁿ, f ∈ F} = F ∨ [τ a),
〈F₁ ∪ F₂ 〉_τ = {x ∈ L|x ≥ f₁ ⊙ f₂, f₁ ∈ F₁, f₂ ∈ F₂},
ifa ≤ b, then 〈b〉_τ ⊆ 〈a〉_τ,
〈τ(a)〉_τ = 〈a〉_τ,
〈a〉_τ ∨ 〈b〉_τ = 〈a ∧ b〉_τ = 〈a ⊙ b〉_τ,
〈a〉_τ ∩ 〈_b〉_τ = 〈τ(a)∨ τ(b)〉_τ.

Proof

The proof of (1) – (5) are obvious.

(6) Since a ⊙ b ≤ a ∧ b ≤ a, b, we deduce that 〈a〉_τ, 〈b〉_τ ⊆ 〈a∧ b〉_τ ⊆ 〈a ⊙ b〉_τ. It follows from that 〈a 〉_τ ∨ 〈b〉_τ⊆ 〈a∧ b〉_τ ⊆ 〈a ⊙b〉_τ. Conversely, let a ∈ 〈a ⊙b〉_τ. Then for some natural number n≥ 1, a ≥(τ(a ⊙b))ⁿ ≥( τ a ⊙ τ b)ⁿ = (τ a)ⁿ ⊙ (τ b)ⁿ. Hence a ∈ 〈a〉_τ ∨ 〈b〉_τ, we deduce that 〈 a ⊙ b〉_τ ⊆ 〈a〉_τ ∨ 〈b〉_τ. Therefore 〈a〉_τ ∨ 〈b〉_τ = 〈a ∧ b〉_τ = 〈a ⊙ b〉_τ.

(7) Since τ(a)≤ τ(a) ∨ τ(b), we deduce that 〈 τ(a) ∨ τ(b)〉_τ ⊆ 〈 τ(a)〉_τ = 〈a 〉_τ. Analogously, 〈 τ(a) ∨ τ(b) 〉_τ⊆ 〈 τ(b) 〉_τ = 〈b 〉_τ. It follows that 〈 τ(a) ∨ τ(b) 〉_τ ⊆ 〈a〉_τ ∩ 〈b〉_τ. Moreover, let t ∈ 〈a〉_τ ∩ 〈b〉_τ. 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(a∨ b))^mn, we deduce that a∈ 〈 τ(a)∨ τ(b)〉_τ, that is, 〈a 〉_τ ∩ 〈b〉_τ ⊆ 〈τ(a) ∨ τ(b) 〉_τ. Therefore 〈a 〉_τ ∩ 〈b 〉_τ = 〈 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.

Theorem 4.6

Let(L, τ) be a very true MTL-algebra. Define binary operations ∧, ∨, ↦ on VF(L) as follows : for allF₁, F₂ ∈ VF(L), F₁∧ F₂ = F₁ ∩ F₂, F₁ ∨ F₂ = 〈 F₁ ∪ F₂ 〉_τ, F₁↦ F₂ = {x∈ L|τ(x)∨ f₁∈ F₂for anyf₁∈ F₁}. Then (VF(L), ∧, ∨, ↦, 1, L) is a complete Heyting algebra.

Proof

Suppose that {F_i}_{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 {F_i}_i ∈ I = ∩_{i∈ I}F_i and the supermum is ∨_{i ∈ I}F_i= {x ∈ L|x ≥ f_i₁ ⊙ f_i₂ ⊙ … f_{i_m}, f_{1_j} ∈ F_{i_j}, i_j ∈ I, 1 ≤ j ≤ m}. Therefore, (VF(L), ∧, ∨, 1, L) is a complete lattice under the inclusion order ⊆. Next, we define F₁ ↦ F₂ = {x ∈ L|τ(x)∨ f₁ ∈ F₂ for any f₁ ∈ F₁ 〉 for any F₁, F₂ ∈ VF(L). And, we shall prove that F₁∩ F₂⊆ F₃ if and only if F₂⊆ F₁↦ F₃ for all F₁, F₂, F₃ ∈ VF(L), that is, (VF(L), ∧, ∨, ↦, 1, L) is a complete Heyting algebra. In order to do this, we first show that F₁ ↦ F₂ is a very true filter of (L, τ).

Now, we will show that F₁↦ F₂ is a very true filter of (L, τ). Clearly 1∈ F₁↦ F₂. Let x∈ F₁ ↦ F₂ and x≤ y, then for any f₁ ∈ F₁ such that τ(x)∨ f₁ ∈ F₂. Since τ(x)∨ f₁ ≤ τ(y)∨ f₁ ∈ F₂ and hence y∈ F₁ ↦ F₂. Assume that x, y∈ F₁ ↦ F₂, then for any f₁∈ F₁, τ(x)∨ f₁, τ(y)∨ f₁ ∈ F₂ and hence f₁ ∨ τ(x ⊙y)∈ F₂. So x ⊙ y ∈ F₁ ↦ F₂. Obviously, if x∈ F₁ ↦ F₂, then τ(x)∈ F₁ ↦ F₂ and thus F₁ ↦ F₂ is a very true filter of (L, τ).

Next, we will prove that F₁ ∧ F₂ ≤ F₃ if and only if F₁ ≤ F₂↦ F₃. Assume that F₁ ∧ F₂ ≤F₃. Let f₁ ∈ F₁. Then τ(f₁)∈ F₁ and for any f₂ ∈ F₂, we have τ(f₂)∨ f₁≥ f₁, τ(f₂) ∨ f₁ ≥ τ(f₂). Hence τ(f₂)∨ f₁ ∈ F₁ ∧ F₂ ≤ F₃ and hence f₁ ∈ F₁ ∧ F₂. Conversely, assume that F₁ ≤ F₂ ↦ F₃. Let x ∈ F₂ ∧ F₃, then x ∈ F₂ ↦ F₃. For any f₃ ∈ F₃, we have τ(x) ∨ f₃ ∈ F₃. Taking f₃ = x ∈ F₃, we have x ∨ τ(x) = x ∈ F₃. Thus F₁ ≤ F₂ ↦ F₃.

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

Theorem 4.7

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

F is a compact element of VF(L),
F is a principal very true filter of (L, τ).

Proof

(1) ⇒ (2) Suppose that F is the compact element of VF(L). Since F = ∨_{x ∈ F} 〈x 〉_τ, then there exist x₁, x₂ … x_n such that F = 〈x₁ 〉_τ ∨ 〈x2 〉_τ ∨…∨ 〈x_n〉_τ. By Proposition 4.5 (6), we have F = 〈x₁ ⊙x₂ ⊙ … ⊙x_n〉_τ. 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 x ∈ L such that F = 〈x〉_τ. Suppose that {F_i}_{i ∈ I} ⊆ VF(L) and F = 〈x〉_τ ⊆ ∨_{i ∈ I} {F_i}. Then x ∈ ∨_{i ∈ I}F_i= 〈∪_{i ∈I}F_i〉_τ. It follows that there exist i_j ∈ I, f_{i_j} ∈ F_{i_j} for all 1 ≤ j ≤ m such that x ≥f_i₁ ⊙ f_i₂ ⊙ … f_{i_m}, that is, x ∈ 〈F_i₁ ∪ F_i₂ ∪ … ∪ F_{i_n} 〉_τ = f_i₁ ∨ F_i₂ ∨ … ∨ F_{i_n}. Hence F= 〈x〉_τ ⊆ F_i₁ ∨ F_i₂ ∨ … ∨ F_{i_n}. Therefore, F is a compact element of VF(L).

Definition 4.8

Let (L, τ) be a very true MTL-algebra andθ be a congruence on L. Then θ is called a very true congruence on (L, τ) if (x, y)∈ θimplies (τ(x), τ(y)) ∈ θ, for anyx, y ∈ L.

Example 4.9

ConsideringExample 3.2 (c), one can see thatR = {{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).

Theorem 4.10

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

Proof

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 τ_F : L/F → L/F such that τ_F([x])= [τ(x)] for any x ∈ L.

Proposition 4.11

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

Proof

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

Definition 4.12

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 filterF₁, F₂of (L, τ) such thatF₁ ∩ F₂ ⊆ F, then F₁ ⊆ F orF₂ ⊆ F.

Example 4.13

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

Theorem 4.14

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

F is a prime very true filter of (L, τ),
if τ(x) ∨ τ(y) ∈ Ffor somex, y ∈ L, thenx ∈ For y ∈ F,
(L/F, τ_F) is a chain.

Proof

(1) ⇒(2) Let τ(x) ∨ τ(y) ∈ F for some x, y ∈ L. Then 〈x〉_τ ∩ 〈y〉_τ = 〈 τ(x) ∨ τ(y) 〉_τ ∈ F. Since F is a prime very true filter of (L, τ), then 〈x〉_τ ⊆ F or 〈y〉_τ ⊆ F. Therefore, x∈ F or y∈ F.

(2) ⇒ (1) Suppose that F₁,F₂ ∈ MF[L] such that F₁ ∩ F₂ ⊆ F and F₁ ⊈ F and F₂ ⊈ F. Then there exist x ∈ F₁ and y ∈ F₂ such that x, y ∉ F. 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) ∈ F₁ ∩ F₂ = F. By (2), we get x ∈ F or y ∈ F, 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.

Theorem 4.15

Let (L, τ) be a very true MTL-algebra anda∈ L. If a≠ 1, then there exists a prime very true filterPof (L, τ) such thata∉ P.

Proof

Denote F_a= {F^′|F^′ is a proper very true filter of (L, τ) such that F ⊆ F^′, a ∉ F^′}. Then f_a ≠  since F is a very true filter not containing a and F_a is a partially set under inclusion relation. Suppose that {F_i|i ∈ I} is a chain in F_a, then ∪ {F_i|i ∈ I} 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 F_a. Now, we shall prove that P is the desire prime very true filter of ours. Since P ∈ F_a, then P is a proper very true filter and a ∉ P.

Let x ∨ y ∈ P for some x, y ∈ L. Suppose that x ∉ P and y ∉ P. Since P is strictly contained in 〈P, x 〉_τ and 〈P, y〉_τ and by the maximality of P, we deduce that 〈P, x 〉_τ ∉ F_a and 〈P, y 〉_τ ∉ F_a. Then a∈ 〈P, x〉_τ = P ∨[τ x) and a ∈ 〈P, y〉_τ = 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(x∨ y))∈ P, which implies that a ∈ P, a contradiction. Therefore, P is a prime very true filter such that F⊆ P and a∉ P. 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.

Theorem 4.16

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

Proof

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.

Theorem 4.17

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

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

Proof

(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^⊥|a ∈ P}. If x ∈ P, then there is a ∉ P such that x ∨ a = 1, hence 1 = τ(x∨a) = τ(x)∨ τ(a). Since a ∉ P, 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.

Theorem 4.18

LetLbe an MTL-algebra. Then the following condition are equivalent:

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

Proof

(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.

Definition 4.19

A very true MTL-algebra (L, τ) is said to bea 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 θ_F = 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., ∩ {F ∈ VF(L)|F ≠ {1}} ≠ {1}.

Example 4.20

ConsideringExample 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.

Proposition 4.21

Let(L, τ) be a subdirectly irreducible very true MTL-algebra andF₁, F₂ ∈ VF(L). IfF₁ ∩ F₂= {1}, thenF₁= {1} orF₂= {1}.

Proof

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

Theorem 4.22

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

(L, τ) is a subdirectly irreducible very true MTL-algebra,
there exists an elementa ∈ L,a < 1, such thatfor anyx ∈ L, x < 1 anda ∈ 〈x〉_τ.

Proof

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

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

We recall that a non-unit element a ∈ L is said to be a coatom of L if a ≤ b, 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.

Proposition 4.23

Let (L, τ) be a subdirectly irreducible very true MTL-algebra. For anyx,y ∈ L, ifx ∨ y = 1, thenx = 1 ory = 1.

Proof

For any x, y ∈ L, if x∨y = 1, then by Theorem 4.5, we have 〈x〉_τ ∩ 〈y〉_τ = 〈τ(x)∨ τ(y)〉_τ = 〈τ(x ∨ y)〉_τ= 〈1〉_τ= {1}. By Proposition 4.21, we have 〈x〉_τ = {1} or 〈y〉_{τ= {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.

Theorem 4.24

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

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

Proof

(1) ⇒(2) Suppose (L, τ) is a subdirectly irreducible very true MTL-algebra. Applying Definition 2.1, we have (x → y)∨(y → x) = 1 for any x, y ∈ L, then by Proposition 4.23, we have x→ y = 1 or y → x = 1, i.e., x ≤ y or y ≤ x, 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 a ∈ F. Since F is chosen arbitrarily from VF(L), then a ∈ ∩ {F ∈ VF(L)|F≠ {1}}. Hence ∩ {F ∈ VF(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 R_τ(X) = {a ∈ L|τ(a)→ x = x, ∀ x ∈ X}, and L_τ(X) = {a ∈ L|x → τ (a) = τ(a), ∀ x ∈ X}, which are called right and left stabilizer of X with respect to τ. Clearly, R_τ (X), L_τ (X) ≠ Ø. In fact, 1 ∈ R_τ (X) ∩ L_τ (X). In particular, if τ = id_L, which is a right and left stabilizer of X (see [26]).

Example 4.25

ConsideringExample 3.2 (c). LetX = {a, b}. ThenL_τ(X) = {a, b, 1} andR_τ(X) = {1}.

Proposition 4.26

LetLbe a MTL-algebra andX, Y ⊆ L. Then the following conditions hold:

1 ∈ R_τ(X)∩ L_τ(X),
IfX ⊆ Y, thenL_τ(X) ⊆ L_τ(Y) and R_τ(X) ⊆ R_τ(Y),
X ⊆ L_τ (R_τ(X)) ∩ R_τ(L_τ(X)),
R_τ(X) = R_τ(L_τ (R_τ(X))) andL_τ (X) = L_τ (R_τ (L_τ(X))),
L_τ(L) = R_τ (L) = {1} andL_τ (1) = R_τ (1) = L,
If Ø ≠ X ⊆ L, thenR_τ (X) is a very true filter of (L, τ).

Proof

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

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

Theorem 4.27

LetFbe a very true filter of (L, τ). ThenR_τ(F) is a pseudocomplemented ofFin the complete Heyting algebra (VF(L), ∧, ∨, ↦, 1, L).

Proof

First, we prove that F ∩ R_τ (F) = {1}. Let x ∈ F ∩ R_τ (F). Since x ∈ R_τ (F), then for any a ∈ F, τ(x) → a = a. Now, since x ∈ F, put a = x we have x = 1. Therefore, F ∩ R_τ (F) = {1}. Now, let G be a very true filter of (L, τ) such that F ∩ G = {1}. Let a ∈ G, then τ(a) ∈ G. Then for any x ∈ F, since τ(a), x ≤ τ(a) ∨ x, x ∈ F, then τ(a) ∨ x ∈ F and τ(a) ∨ x ∈ G 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) → x ≤ x. On the other hand, x ≤ τ(a) → x, then τ(a) → x = x and a ∈ R_τ (F). Thus G ⊆ R_τ (F). Therefore, R_τ (F) is a pseudocomplemented of F in the complete Heyting algebra (VF(L), ∧, ∨, ↦, 1, L).

Theorem 4.28

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

Proof

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.

Theorem 4.29

Let (L, τ) be a very true MTL-algebra andX ⊆ L. Define a mapping α : P( L,≪) → P( L,≪)such that α(X) = R_τ(L_τ(X)) for allX ∈ P( L). Then the following conditions hold,

α is a closure map on (L, τ),
X ⊆ α(Y) if and only if α(X) ⊆ α(Y), for allY ⊆ L,
β_α = {X ⊆ P(L)|α(X) = X} is a basis for a topology on (L, τ).

Proof

It follows from Proposition 4.26 (2),(3),(4).
The proof is easy.
Let β_α = {X ∈ P(L) |α(X) = X}. It is clear that Ø ∈ β_α. Also, by Proposition 4.26 (5), α(L)= R_τ (L_τ(L)) = R_τ ({1}). Thus, α(L) = L, and L ∈ β_α. Now, suppose that X, Y ∈ β_α. Then α(X) = X and α(Y) = Y. We prove that X ∩ Y ∈ β_α. Since X ∩ Y ⊆ X, Y, by (1), α(X ∩ Y) ⊆ α(X), α (Y). Thus, α(X ∩ Y) ⊆ α(X) ∩ α(Y). Also, since X, Y ∈ β_α, we have α(X ∩ Y) ⊆ X ∩ Y. Moreover, by Proposition 4.26 (3), X ∩ Y ⊆ α(X ∩ Y). Then α(X ∩ Y) = X ∩ Y, and so X ∩ Y ∈ β_α. Therefore, β_α is a basis.

Definition 4.30

Based onTheorem 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 R_τ(X) ∈ VF(L), for any X ⊆ L and hence every element of β_α is a very true filter of (L, τ).

Theorem 4.31

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

Proof

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

Theorem 4.32

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

Proof

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

Theorem 4.33

The stabilizer topology (L,T_⊘) is separable.

Proof

First, we show that if Ø ≠ X ⊆ L such that 1 ∈ X, then X¯ = L. Let Ø ≠ X ⊆ L such that 1 ∈ X. We only show that L ⊆ X¯. Let x ∈ L. If x = 1, then x ∈ X¯. Hence X¯ = L. Now, suppose that 1 ≠ x ∈ L. Then there exists an open subset U ∈ β_α such that x ∈ U. Since U ∈ VF(L) and 1 ∈ U, we have U ∩(X − {x}) ≠ Ø. Hence x ∈ X¯, and so X¯ = L. Since 1 ∈ β_α, thus 1¯ = L. Hence (L,T_⊘) is separable.

Theorem 4.34

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

Proof

Let U ∈ T_⊘). Since U ∈ VF(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 U_n, ∩ U_n ∈ VF(L), so 1 ∈ ∩ U_n. Thus, by Theorem 4.32, ∩ U_n 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 (MTL_vt, 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 MTL_vt to be those of MTL [2] plus the following ones:

(vtl) vt(φ) ⇒ φ,
(vt2) vt(φ ⇒ ψ) ⇒ (vtφ ⇒ vtψ),
(vt3) vt(φ) ⇒ vt(vtφ),
(vt4) vt(φ ⇒ ψ) ⊔ vt(ψ ⇒ φ).

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

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

The consequence relation ⊢ is defined in the usual way. Let T be a theory, i.e., a set of formulas in MTL_vt. 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 MTL_vt, 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 T ⊢ϕ. Moreover T is complete if for each pair ϕ,ψ, T ⊢ ϕ ⇒ ψ or T ⊢ ψ → ϕ.

Definition 5.1

Let (L, τ) be a very trueMTL-algebra andTbe a theory. AnL-evaluation is a mappingefrom the set of formulas ofMTL_vttoLthat satisfies, for each two formulasϕandψ : e(ϕ ⇒ ψ) = e (ϕ) → e(ψ), e(ϕ ⊔ ψ) = e(ϕ) ∨ e(ψ), e(ϕ ⊓ ψ)= e(ϕ)∧ e(ψ), e (ϕ&ψ) = e(ϕ) ⊙e (ψ), e(vtϕ) = τ e(ϕ), e (0¯)=0 ande(1¯)=1. If anL- evaluationesatisfiese(χ) = 1 for every χ inT, it is called anL-model ofT.

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

Definition 5.2

LetTbe a fixed theory overMTLvt. For each formulaϕ, let [ϕ]Tbe the set of all formulasψsuch thatT ⊢ ϕ ≡ ψ ( whereϕ ≡ ψstands for (ϕ → ψ) & (ψ → φ)) andL_Tbe the set of all the class [ϕ]_T. We define: 0 = [0]_T, 1 = [1]_T, [ϕ]_T ⇒ [ψ]_T = [ϕ ⇒ ψ]_T, [ϕ]τ & [ψ]_T = [ϕ&ψ]_T, [ϕ]_T ⊔[ψ]_T = [ϕ⊔ψ]_T, [ϕ]_T ⊓[ψ]_T = [ϕ⊓ψ]_T, τ_T[ϕ]_T = [vtϕ]_T. This algebra is denoted by (L_T, τ_T).

Proposition 5.3

(L_T, τ_T) is a very trueMTL-algebra.

Proof

It follows similarly from the proof of Proposition 4.11.

Theorem 5.4

Let T be a theory overMTL_vt. Then T is complete if and only if the very true MTL-algebra (L_T, τ_T) is linearly ordered.

Proof

It follows similarly from the proof of Theorem 4.14.

It is easy to check that MTL_vt 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 MTL_vt, 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 MTL_vt (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.

Proposition 5.5

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

If for all very true MTL-algebra (L, τ) andfor allL-modeleforT, e(ϕ) = 1, then for all very trueMTL-algebra (L, τ) and for allL-modeleforT, e(vtϕ) = 1.
If for all very true MTL-algebra (L, τ) and for allL-modeleforT, e(ϕ) = 1 ande(ϕ → ψ) = 1, then for all very true MTL-algebra (L, τ) and for allL-modeleforT, e(ψ) = 1.

Proof

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.
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.

Theorem 5.6

LetTbe a theory overMTL_vt. If T is a theory andT∀ϕ, then there is a consistent complete supertheoryT^′ ⊇ Tsuch thatT^′∀ϕ.

Proof

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.

Theorem 5.7

Let T be a theory overMTL_vt. For each formulaφ, the following are equivalent:

T ⊢ ϕ;
for each very true MTL-algebra (L, τ) and for everyL-modeleofT, e(ϕ) = 1;
for each linearly ordered very trueMTL-algebra (L, τ) and for everyL-modeleofT, e(ϕ) = 1.

Proof

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

Theorem 5.8

MTL_vt is a conservative extension of MTL.

Proof

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 MTL_vt∀φ. Therefore, MTL_vt 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).

References

[1] Hájek P., Metamathematics of Fuzzy logic, Kluwer Academic Publishers, Dordrech, 1998.10.1007/978-94-011-5300-3Search in Google Scholar

[2] Esteva F., Godo L., Monoidal t-norm based logic: towards a logic for left-continuoust-norms, Fuzzy Sets Syst., 2001, 124, 271-288.10.1016/S0165-0114(01)00098-7Search in Google Scholar

[3] Jenei S., Montagna F., A proof of standard completeness for Esteva and Godo’s logic MTL, Stud Log., 2002, 70, 183-192.10.1023/A:1015122331293Search in Google Scholar

[4] Vetterlein T., MTL-algebras arising from partially ordered groups, Fuzzy Sets Syst., 2010, 161, 433-443.10.1016/j.fss.2009.09.008Search in Google Scholar

[5] Noguera C., Esteva F., Gispert J., On Some Varieties of MTL-algebras, Log. J. IGPL., 2004, 13, 443-466.10.1093/jigpal/jzi034Search in Google Scholar

[6] Haveshki M., Saeid A. B., Eslami E., Some type of filters in BL-algebras, Soft Comput., 2006, 10, 657-664.10.1007/s00500-005-0534-4Search in Google Scholar

[7] Borzooei R. A., Shoar S. Khosravi., Ameri. R., Some type of filters in MTL-algebras, Fuzzy Sets Syst., 2012, 187, 92-102.10.1016/j.fss.2011.09.001Search in Google Scholar

[8] Haveshki M., A note on some types of filters in MTL-algebras, Fuzzy Sets Syst., 2014, 16, 135-137.10.1016/j.fss.2013.08.014Search in Google Scholar

[9] Hájek P., On very true, Fuzzy Sets Syst., 2001, 124, 329-333.10.1016/S0165-0114(01)00103-8Search in Google Scholar

[10] Zadeh L., Fuzzy logic and approximate reasoning, Synthese., 1975, 30, 407-428.10.1142/9789814261302_0016Search in Google Scholar

[11] Ganter B., Wille R., Formal concept analysis, Mathematical Foundations, Springer-Verlag, Berlin, 1999.10.1007/978-3-642-59830-2Search in Google Scholar

[12] Bělohlávek R., Vyhodil V., Reducing the size of fuzzy concept lattices by hedges, In: FUZZ-IEEE 2005, the IEEE internationnal conference on fuzzy systems (25-25 May 2005), Reno, NV, USA, 2005, 65-74.Search in Google Scholar

[13] Chajda I., Hedges and successors in basic algebras. Soft Comput., 2011, 15, 613-618.10.1007/s00500-010-0570-6Search in Google Scholar

[14] Hájek P., Harmancová D., A hedge for Gödel fuzzy logic, lnternational Journal of Uncertainly, Fuzziness and Knowledge-Based Systems., 2000, 8, 495-498.10.1142/S0218488500000332Search in Google Scholar

[15] Leuştean l., Non-commutative Łukasiewicz propositional logic, Arch Math Logic., 2006, 45, 191-213.10.1007/s00153-005-0297-8Search in Google Scholar

[16] Rachunek J., Šalounová D., Truth values on generalizations of some commutative fuzzy structures, Fuzzy Sets Syst., 2006,157,3159-3168.10.1016/j.fss.2006.08.007Search in Google Scholar

[17] Botur M., Švrček F., Very true on CBA fuzzy logic, Math Slovaca., 2010, 60, 435-446.10.2478/s12175-010-0023-9Search in Google Scholar

[18] Wang J T., Xin X L., Young B J., Very true operators on equality algebras, submitted for publication.Search in Google Scholar

[19] Chajda I., Kolařík M., Very true operators in effect algebras, Soft Comput., 2012, 16, 1213-1218.10.1007/s00500-012-0807-7Search in Google Scholar

[20] Grätzer G., Lattice theory, W H Freeman and Company, San Franciso, 1979.10.1007/978-3-0348-7633-9Search in Google Scholar

[21] Joshi K. D., lntroduction to general topology, New Age lnternational Publisher, lndia, 1983.Search in Google Scholar

[22] Blyth T. S., Lattices and Ordered Algebraic Structures, Springer, London, 2005.Search in Google Scholar

[23] Turunen E., BL-algebras of basic fuzzy logic, Mathware Soft Comput., 1998, 6, 49-61.Search in Google Scholar

[24] Zhang J. L., Topological properties of prime filters in MTL-algebras and fuzzy set representations for MTL-algebras, Fuzzy Sets Syst., 2011, 178, 38-53.10.1016/j.fss.2011.03.002Search in Google Scholar

[25] Abbasloo M., Saeid A. B., Coatoms and comolecules of BL-algebras, J Intell Fuzzy Syst., 2014, 27, 351-360.10.3233/IFS-131002Search in Google Scholar

[26] Saeid A. B., Mohtashamnia N., Stabilizer in residuated lattices, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 2012, 74, 65-74.Search in Google Scholar

Received: 2016-5-27

Accepted: 2016-10-18

Published Online: 2016-12-10

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.