Very true operators on MTL-algebras

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.


Introduction
Basic fuzzy logic (BL for short) is the many-valued residuated logic introduced by H K ajek [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 D xˇ.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][7][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 HK ajek [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, B M elohl K avek 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][13][14], it has been extended to other logical algebras such as MV-algebras [15], R`-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, BLalgebras, R`-monoids and commutative basic algebras, etc. All the above-mentioned algebraic structures satisfy the divisibility condition x^y D xˇ.x ! y/. In this case, the conjunctionˇon 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.

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.
In what follows, by L we denote the universe of an MTL-algebra .L;^; _;ˇ; !; 0; 1/. For any x 2 L, we define 5]). In any MTL-algebra L, the following properties hold: for all x; y; z 2 L, provided that both infimum as well as supremum exist, 5]). Let L be an MTL-algebra. Then L is called: A nonempty subset F of L is called a filter of L if it satisfies: (1) x; y 2 F implies xˇy 2 F ; (2) x 2 F ,y 2 L and x Ä y implies y 2 F . We denote by F OEL 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 2 F and x _ y 2 F , implies x 2 F or y 2 F . For any filter F of L we can associate a congruence on L defined by x F y if and only if .x ! y/^.y ! x/ 2 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]).   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 V X D b implies b 2 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 OEL. 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 Â L. The set is called the co-annihilator of X in L [23]. For any a 2 L, we write a ? instead of fag ? .

Very true operators on MTL-algebras
In this section, inspired by HK ajek [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 W L ! L is called a very true operator if it satisfies the following conditions: The pair .L; / is said to be a very true MTL-algebra. Such a proliferation of conditions deserves some explanation. Then "1" seen in (V1) is considered as the logical value absolutely true. First note that (V1) means that absolutely true is very true, which is sound for each natural interpretation in many valued logic system. (V2) means that if ' is very true then it is true. (V3) means that if both ' and ' ! are very true then so is , that means the connective preserve modus ponens. (V4) says that if ' is very true then .'/ 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 (V5).

Example 3.2.
(a) 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. (b) Any linearly ordered MTL-algebra L can admit a very true operator; i.e., .1/ D 1 and .x/ D 0 for any Consider the operationˇand ! given by the following tables: Then .L;^; _;ˇ; !; 0; 1/ is an MTL-algebra. Now, we define as follows: .0/ D 0, .a/ D .b/ D a, .1/ D 1. One can easily check that is a very true operator on L. However, is not a homomorphisms on Proposition 3.3. Let be a very true operator on L. Then for any x; y 2 L we have, (4) It follows from (1) and (V3) that .: (3), we have .y/ Ä .x ! .xˇy// Ä .x/ ! .xˇy/ and hence .x/ˇ .y/ Ä .xˇy/.
Assume that x 2 L but x ¤ 1 such that .x/ D 1. Applying (V2), we have 1 D .x/ Ä x and hence x D 1, which is a contradiction. Therefore, Ker. / D f1g.
(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 W L ! L satisfying .1/ .4/ and the following axiom .5 0 / which replaces the axiom (5): From this point, one can check that the very true MTL-algebra essentially generalize very true BL-algebra, which was introduced by H K ajek in 2001. A very true operator on an MV-algebra L was introduced in Leuştean (2006) as a mapping W L ! L satisfying conditions (V1)-(V3) 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 F ix .L/ is not necessary a subalgebra of an MTL-algebra in general (in Example 3.2 (c), one can check that F ix .L/ is not a subalgebra of L since it is not closed under !), while it forms an MTL-algebra after redefined its fuzzy implication. Proof. First, we show that .F ix .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 F ix .L/ is closed under _ and^. Thus .F ix .L/;^; _/ is a lattice. For all x 2 .L/, one can easily check that x _ 1 D 1 and x^0 D 0. Thus, 0 is the smallest element and 1 is the greatest element in F ix .L/, respectively. Therefore .F ix .L/;^; _; 0; 1/ is a bounded lattice. Next, we prove that .F ix .L/;ˇ; 1/ is a commutative monoid with 1 as neutral element. By Proposition 3.3 (5), we have F ix .L/ is closed underˇ. It follows that .F ix .L/;ˇ/ is a commutative semigroup. For all x 2 F ix .L/, we obtain that xˇ1 D x, that is, 1 is a unital element.
The result of Theorem 3.4 shows that the fixed point set F ix .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. Proof. .1/ ) .2/ We note that an MV-algebra satisfies .x ! y/ ! y D .y ! x/ ! x for all x; y 2 L. By Proposition 2.2 (11) and 3.3(7), we have .
.2/ ) .1/ Suppose that every very true operator satisfies .xˇy/ D .x/^ .y/ for all x; y 2 L. Taking D id L , we have xˇy D x^y for any x; y 2 L. Taking x D y, we get xˇx D x for all x 2 L. Therefore, L is a Gödel algebra.
.3/ ) .1/ Suppose that every very true operator satisfies .xˇy/ D .x/ˇ .x ! y/ for all x; y 2 L. Taking D id L , we have xˇy D xˇ.x ! y/ for any x; y 2 L. Taking x D y, we get xˇx D x for all x 2 L. Therefore, L is a Gödel algebra.

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; / be a very true MTL-algebra. For any nonempty set X of L, we denote by hX i the very true filter of .L; / generated by X , that is, hXi is the smallest very true filter of .L; / containing X . If F is a very true filter of .L; / and x … F , we put hF; xi WD hF [ fxgi .
The next theorem gives a concrete description of the very true filter generated by a subset of very true MTLalgebra .L; /. Theorem 4.3. Let .L; / be a very true MTL-algebra and X be a nonempty set of L. Then hX i D fx 2 Ljx .y 1 /ˇ .y n /; y i 2 X; n 1g.
Proof. Suppose that fF i g i 2I is a family of very true filters of .L; /. From Theorem 4.5, it is easy to check that the infimum of fF i g i 2I D \ i2I F i and the supermum is Therefore, .VF .L/;^; _; 1; L/ is a complete lattice under the inclusion order Â. Next, we define F 1 7 ! F 2 D fx 2 Lj .x/ _ f 1 2 F 2 for any f 1 2 F 1 g for any F 1 ; F 2 2 VF .L/. And, we shall prove that F 1 \ F 2 Â F 3 if and only if F 2 Â F 1 7 ! F 3 for all F 1 ; F 2 ; F 3 2 VF .L/, that is, .VF .L/;^; _; 7 !; 1; L/ is a complete Heyting algebra. In order to do this, we first show that F 1 7 ! F 2 is a very true filter of .L; /. Now, we will show that F 1 7 ! F 2 is a very true filter of .L; /. Clearly 1 2 F 1 7 ! F 2 . Let x 2 F 1 7 ! F 2 and x Ä y, then for any f 1 Assume that x; y 2 F 1 7 ! F 2 , then for any f 1 2 F 1 , .x/ _ f 1 ; .y/ _ f 1 2 F 2 and hence f 1 _ .xˇy/ 2 F 2 . So xˇy 2 F 1 7 ! F 2 . Obviously, if x 2 F 1 7 ! F 2 , then .x/ 2 F 1 7 ! F 2 and thus F 1 7 ! F 2 is a very true filter of .L; /.
Therefore, .VF .L/;^; _; 7 !; 1; L/ is a complete Heyting algebra. Proof. .1/ ) .2/ Suppose that F is the compact element of VF .L/. Since F D _ x2F hxi , then there exist x 1 ; x 2 x n such that F D hx 1 i _hx 2 i _ _hx n i . By Proposition 4.5 (6), we have F D hx 1ˇx2ˇ ˇx n i . Therefore, F is a principal very true filter of .L; /. .2/ ) .1/ Let F be a principal very true filter of .L; /. Then there exists x 2 L such that F D hxi . Suppose that fF i g i2I Â VF .L/ and F D hxi Â _ i2I fF i g. Then    Proof. The proof is easy, and we hence omit the details.
Let .L; / be a very true MTL-algebra and F be a very true filter. We define the mapping F W L=F ! L=F such that F .OEx/ D OE .x/ for any x 2 L. Proposition 4.11. Let .L; / be a very true MTL-algebra and F a very true filter of .L; /. Then .L=F; F / is a very true MTL-algebra.
Proof. The proof is easy, and we hence omit the details.   Proof. .1/ ) .2/ Let .x/ _ .y/ 2 F for some x; y 2 L. Then hxi \ hyi D h .x/ _ .y/i 2 F . Since F is a prime very true filter of .L; /, then hxi Â F or hyi Â F . Therefore, x 2 F or y 2 F . .2/ ) .1/ Suppose that F 1 ,F 2 2 MF OEL such that F 1 \ F 2 Â F and F 1 ª F and F 2 ª F . Then there exist x 2 F 1 and y 2 F 2 such that x; y … F . Since F 1 , F 2 are very true filters of .L; /, then .x/ 2 F 1 and .y/ 2 F 2 .
.1/ , .3/ From (2), one can obtain that every prime very true filter of .L; / 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. Proof. Denote F a D fF 0 jF 0 is a proper very true filter of .L; / such that F Â F 0 , a … F 0 g. 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 fF i ji 2 I g is a chain in F a , then [fF i ji 2 I g is a very true filter of .L; / 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 2 F a , then P is a proper very true filter and a … P . Let x _ y 2 P for some x; y 2 L. Suppose that x … P and y … P . Since P is strictly contained in hP; xi and hP; yi and by the maximality of P , we deduce that hP; xi … F a and hP; yi … F a . Then a 2 hP; xi D P _OE x/ and a 2 hP; yi D P _ OE .y//. Then we have a 2 .P _ OE .x///^.P _ OE .y/// D P _ .OE .x//^OE .y/// D P _ OE .x/ _ .y// D P _ OE .x _ y// 2 P , which implies that a 2 P , a contradiction. Therefore, P is a prime very true filter such that F Â P and a … P . Put F D f1g, 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 MTLalgebras.
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; / be a very true MTL-algebra. Then the following conditions are equivalent: .1/ L is representable; .2/ .L; / 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 T S D f1g. 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 f1g. Moreover, we will show that every minimal prime filter in .L; /. Let P be a minimal prime filter of L. Then by Theorem 2.8, P D [fa ? ja 2 P g. If x 2 P , then there is a … P such that x _ a D 1, hence 1 D .x _ a/ D .x/ _ .a/. Since a … P , we get .a/ … P , therefore .x/ 2 P , that means that P is a very true filter in .L; /. Therefore, .L; / is a subdirect product of linearly ordered very true NM-algebras.
.2/ ) .1/ The converse is trivial and we hence omit this. Proof. .1/ ) .2/ If L is representable, then L is a subdirect product of MTL-chain. By Example 3.2 (b), any MTLchain 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. Let .L; / be subdirectly irreducible. Then by Theorem 4.10, there is a very true filter F of .L; / such that Â F D F , that means, F is the least very true filter of .L; / such that F ¤ f1g. Thus, we can conclude that a very true MTLalgebra .L; / is said to be subdirectly irreducible if among the nontrivial very true filters of .L; / there exists the least one, i.e., \fF 2 VF .L/jF ¤ f1gg ¤ f1g.
Example 4.20. Considering Example 3.2 (c), one can easily check that the very true MTL-algebra .L; / 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.
We recall that a non-unit element a 2 L is said to be a coatom of L if a Ä b, then b 2 fa; 1g, i.e. b D a or b D 1  ([25]). In the following proposition, we will show that every subdirectly irreducible very true MTL-algebra has at most one coatom. The following Theorem shows that the subdirectly irreducible very true MTL-algebra .L; / 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. In what follows, we introduce the stabilizer of a nonempty subset set X with respect to a very true operator and study some properties of them. Let .L; / be a very tue MTL-algebra. Given a nonempty subset X of L, we put R .X/ D fa 2 Lj .a/ ! x D x; 8x 2 X g; and L .X / D fa 2 Ljx ! .a/ D .a/; 8x 2 X g; which are called right and left stabilizer of X with respect to . Clearly, R .X /; L .X / ¤ ;. In fact, 1 2 R .X / \ L .X /. In particular, if D id L , which is a right and left stabilizer of X (see [26]).    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.   Proof. The proof is easy, and we hence omit the details. Proof. The proof is easy, so we hence omit the details. Proof. First, we show that if ; ¤ X Â L such that 1 2 X , then X D L. Let ; ¤ X Â L such that 1 2 X . We only show that L Â X . Let x 2 L. If x D 1, then x 2 X . Hence X D L. Now, suppose that 1 ¤ x 2 L. Then there exists an open subset U 2ˇ˛such that x 2 U . Since U 2 VF .L/ and 1 2 U , we have U \ .X fxg/ ¤ ;. Hence x 2 X , and so X D L. Since 1 2ˇ˛, thus 1 D L. Hence .L; T / is separable.

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: The deduction rules are modus ponens (MP, from and ) infer ), and Generalization(G,from infer vt ).
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 in T is a finite sequence of formulas with 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 exists in T , we say that can be deduced from T and we denote this by T` . Moreover T is complete if for each pair , , T` ) or T` ) . 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 can be deduced from a theory T in MTL vt , then for every very true MTL-algebra .L; / and for every L-model e of T , e. / D 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. Theorem 5.6. Let T be a theory over MTL vt . If T is a theory and T° , then there is a consistent complete supertheory T 0 Ã T such that T 0° .
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. Proof. Suppose M T L°' then there is a linearly ordered MTL-algebra L such that ' 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 M T L vt°' . Therefore, MTL vt is a conservative extension of MTL.

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.