Some weaker versions of topological residuated lattices

Accepted Manuscript Some weaker versions of topological residuated lattices

Jiang Yang, Xiongwei Zhang

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S0165-0114(19)30188-5 https://doi.org/10.1016/j.fss.2019.03.014 FSS 7627

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Fuzzy Sets and Systems

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21 January 2018 9 March 2019 14 March 2019

Please cite this article in press as: J. Yang, X. Zhang, Some weaker versions of topological residuated lattices, Fuzzy Sets Syst. (2019), https://doi.org/10.1016/j.fss.2019.03.014

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Some weaker versions of topological residuated lattices Jiang Yanga,∗ , Xiongwei Zhangb a

School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an, 710021, China b School of Mathematics and Statistics, Yulin University, Yulin, 719000, China

Abstract The aim of this paper is to investigate some weaker versions of topological residuated lattices, which are para-∧-topological residuated lattices(∧ continuous), para-∨-topological residuated lattices(∨ continuous), para--topological residuated lattices( continuous) and para-→-topological residuated lattices(→ continuous), respectively. This paper mainly focuses on the latter two para-topological residuated lattices. We prove the independence of these para-topological residuated lattices by using their algebraic properties. Moreover, we establish the correspondence between algebraic properties and topologies in a residuated lattice L, i.e., there is an injective mapping from M Eq(L) (IEq(L), AEq(L), Con(L), resp.) to M T RL(L) (IT RL(L), AT RL(L), T RL(L), resp.). Finally, some interactions between the topologies and algebraic properties in para(semi)-topological residuated lattices are studied. Keywords: Residuated lattice; Para(semi)-topological residuated lattice; M -system; (, →, adjoint) compatible; Filter 1. Introduction It is well known that non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information. Various logical algebras have been proposed as the semantical systems of nonclassical logic systems, for example, residuated lattices[35], BL-algebras[20], M T Lalgebras[12] and R0 -algebras (N M -algebras)[31, 34], M V -algebras[8], etc. Among these logic algebras, residuated lattices are very basic and important algebraic structures because the other logic algebras are particular cases of residuated lattices. According to the viewpoint of the School of Bourbaki, there are three mother structures in mathematics from which all other mathematical structures can be generated, and which are not reducible one to the other. These are precisely: algebraic ∗

Corresponding author. Email addresses: [email protected],[email protected](J. Yang), zhangxw1019 @163.com(X.W. Zhang) Preprint submitted to Fuzzy Sets and Systems

March 15, 2019

structures, topological structures, and order structures. The interaction between topological structures and order structures is a stimulating topic in mathematics and computer science, e.g., the theory of domains [18] and the theory of locales [23]. In representation theory [25] and topological groups [1, 22, 32], topologies and algebras come in contact naturally. In recent decades, several researchers have endowed a number of algebraic structures associated with logical systems equipped with topologies. For more about them, we refer to [5, 6, 7, 9, 10, 14, 15, 16, 17, 19, 21, 25, 26, 27, 29, 36, 37, 38]. As all mentioned above, in a certain sense, are particular types of residuated lattices equipped with topologies. Therefore, it is meaningful to establish the topological theory of residuated lattices which will be universal properties of above-mentioned logical algebras. It is well known that the characteristical operations on residuated lattices are the multiplication  and implication →, which are closely tied by adjointness property. From the algebraic point of view, most important properties of residuated lattices are determined by the multiplication  and its adjointness →. Topological residuated lattices (which are residuated lattices with topologies with respect to that all binary operations are continuous) have been introduced and studied [14, 15, 16, 17, 24, 29]. Obviously, if (L, ∧, ∨, , →, T ) is a topological residuated lattice, then (L, ∧, T ) is a para-∧-topological residuated lattice, (L, ∨, T ) is a para-∨-topological residuated lattice, (L, , T ) is a para--topological residuated lattice and (L, →, T ) is a para-→-topological residuated lattice. On the one hand, some properties of topological residuated lattices require only one or two of their operations to be jointly continuous or separately continuous. For example, the separation property of topological BL-algebras only needs the continuity of →(see, [37] Theorem 3.12.), it is easy to show that this property also holds in topological residuated lattices. And, the connected component of 1 on the topological residuated lattice (L, T ) is the greatest closed filter of L, which actually only needs separate continuity of  and ∨. On the other hand, given a residuated lattice, how does the continuity of operations interact each other on different para-topological residuated lattices? Based on above viewpoints, it is therefore necessary to investigate independently the continuity of each operation and the combinations among them. Luckily, lattice structures of residuated lattices endowed topologies, namely topological lattices or continuous lattices, have been more detailed studied in [23] and [18]. Therefore, in this paper, we mainly investigate para--topological residuated lattices, para-→-topological residuated lattices and their combination, i.e., adjoint topological residuated lattices(both  and → are continuous). Given a residuated lattice (L, ∧, ∨, , →, 0, 1) and a system of filters F on it. We prove that L with the topology TF induced by F is a topological residuated lattice. Particularly, when F is a singleton set {F }, we briefly write T{F } for TF . We not only show that T  = sup{TF : F ∈ F} is a topological residuated lattice but also prove that the equation T  = TF holds. Thus, all results [15, 16, 19, 21, 27, 29, 37] are special cases for these topologies. From the algebraic point of view, the operations ∧, ∨, , → on L are independent each other. A question of interest arises: How about independence of para-∧-topological residuated lattices, para-∧-topological residuated lattices, para--topological residuated lat2

tices and para-→-topological residuated lattices? Inspired by this question, we give the concepts of  compatibility and → compatibility by using universal algebraic language, and prove that the later two para-topological residuated lattices are mutually independent. Similarly, we conclude that all the para-topological residuated lattices are independent each other. The rest of the paper is organized as follows. In Section 2, we recall some facts about residuated lattices and topologies, used in the sequel. In Section 3, we introduce the concepts of para--topological residuated lattices and para-→topological residuated lattices, in order to construct these para-topological algebras, the notions of  compatible relations and → compatible relations are defined on residuated lattices. Particularly, by using a system of filters and a M -system, we construct topological residuated lattices and para--topological residuated lattices, respectively. In Section 4, the relationships between topological properties and algebraic properties in (para-, para-→, adjoint, semi) topological residuated lattices are investigated. 2. Preliminaries In this section, we summarize some definitions and results about residuated lattices and topologies, which will be used in the following sections. Recall that a set A with a family T of subsets of A is called a topological space, denoted by (A, T ), if A, ∅ ∈ T , the intersection of any finite members of T is in T , and the arbitrary union of members of T is in T . The members of T are called open sets of A, and the complement of an open set U , A \ U , is a closed set. A subfamily {Uα }α∈I of T is called a base of T if for each x ∈ U ∈ T there is an α ∈ I such that x ∈ Uα ⊆ U . A subset P of A is a neighbourhood of x ∈ A, if there exists an open set U such that x ∈ U ⊆ P . Let Tx denote the totality of all neighbourhoods of x in A, then subfamily Vx of Tx is a fundamental system of neighbourhoods of x, if for each Ux in Tx , there exists a Vx in Vx such that Vx ⊆ Ux . Let (A, T ) and (B, V) be two topological spaces, a mapping f from A to B is continuous if f −1 (U ) ∈ T for any U ∈ V. A topological space (A, T ) is compact if each open covering of A is reducible to a finite open covering. More concepts about topologies can be found in [11]. Definition 2.1. (See [22, 28].) Let (L, ∗) be an algebra of type 2 and T be a topology on L. Then (L, ∗, T ) is called a (i) left(right) topological algebra, if for all a ∈ L the map L → L defined by x → a ∗ x(x → x ∗ a) is continuous, or equivalently, if for any x in L and any open set U of a ∗ x(x ∗ a), there exists an open set V of x such that a ∗ V ⊆ U (V ∗ a ⊆ U ), (ii) semitopological algebra, or operation ∗ is separately continuous, if (L, ∗, T ) is a right and left topological algebra. Note that if (L, ∗) is a commutative algebra of type 2 and T is a topology on L, then right and left topological algebras are equivalent. Moreover, (L, ∗, T ) is a semitopological algebra if and only if it is a right or left topological algebra. 3

Remark 2.2. (See [2].) Let {τγ : γ ∈ Γ} be a non-empty family of topologies on X and τ = γ∈Γ τγ . Then (X, τ ) is a topological space. The topology τ is the greatest lower bound of the family of topologies {τγ : γ ∈ Γ} on X and is denoted  by inf{τγ : γ ∈ Γ}. It is clear that τ ≤ τγ for any γ ∈ Γ. Let {τω : ω ∈ Ω} be a  family of all topologies on X such that τγ ≤ τω for any γ ∈ Γ and any ω ∈ Ω. Since  the discrete topology belongs to {τω : ω ∈ Ω}, then this family is non-empty and,  hence, we can consider the topology τ  = inf{τω : ω ∈ Ω}. The topology τ  is called the least upper bound of the family {τγ } and is denoted by sup{τγ : γ ∈ Γ}. Clearly, τγ ≤ τ  for any γ ∈ Γ. For each x ∈ X, consider the family Bx (X, τ  ) of subsets U of X such that there exist finite sets of topologies τγ1 , . . . , τγn and neighbourhoods U1 , . . . , Unof x in the topological spaces (X, τγ1 ), . . . , (X, τγn ), respectively, such that U = ni=1 Ui . The family Bx (X, τ  ) is a fundamental system of neighbourhoods of the element x in (X, τ  ). Now, we recall some definitions and properties about residuated lattices. Definition 2.3. (See [13, 35].) An algebraic structure L = (L, ∧, ∨, , →, 0, 1) of type (2, 2, 2, 2, 0, 0) is called a residuated lattice if it satisfies the following conditions: (1) (L, ∧, ∨, 0, 1) is a bounded lattice, (2) (L, , 1) is a commutative monoid, (3) x  y ≤ z if and only if x ≤ y → z, for all x, y, z ∈ L, where ≤ is the partial order of the lattice (L, ∧, ∨, 0, 1). Throughout this paper we will slightly abuse notation L the universe of a residuated lattice L = (L, ∧, ∨, , →, 0, 1), when there is no chance to confusion. Let L be a residuated lattice and A, B ⊆ L. We write A ∗ B for {x ∗ y : x ∈ A, y ∈ B}, and when dealing with singleton sets we shall simply write a ∗ B and A ∗ b rather than {a} ∗ B and A ∗ {b}, where ∗ ∈ {∧, ∨, , →}. For convenience of readers, we provide some basic properties of residuated lattices in the following proposition. Proposition 2.4. (See [13, 30, 33].) In any residuated lattice (L, ∧, ∨, , →, 0, 1), the following properties hold: for any x, y, z ∈ L, (R1 ) (R2 ) (R3 ) (R4 ) (R5 ) (R6 ) (R7 )

1 → x = x, x → 1 = 1, x ≤ y if and only if x → y = 1, if x ≤ y, then y → z ≤ x → z, z → x ≤ z → y and x  z ≤ y  z, x  (x → y) ≤ y, x  y ≤ x ∧ y, x ≤ y → x, x → (y → z) = (x  y) → z = y → (x → z), x → y = ((x → y) → y) → y.

Definition 2.5. (See [33].) Let (L, ∧, ∨, , →, 0, 1) be a residuated lattice. A filter is a nonempty set F ⊆ L such that for each x, y ∈ L, 4

(i) x, y ∈ F implies x  y ∈ F , (ii) if x ∈ F and x ≤ y, then y ∈ F . Note that in a residuated lattice L, a filter F of L is equivalent to a deductive system, that is, F satisfies the following conditions: (i) 1 ∈ F , and (ii) x, x → y ∈ F implies y ∈ F . With any filter of L we can associate a congruence relation θF on L by defining (x, y) ∈ θF if and only if x → y, y → x ∈ F . For any x ∈ L, let x/F be the equivalence class x/θF . If we denote by L/F the quotient set L/θF , then L/F becomes a residuated lattice with the operations induced from those of L. 3. On (para-, para-→, adjoint, semi) topological residuated lattices The main purpose of this section is to endow a residuated lattice with topologies such that the operations ∧, ∨, , → are semicontinuous or continuous with regard to these topologies. Definition 3.1. Let (L, ∧, ∨, , →, 0, 1) be a residuated lattice and T be a topology on it. Then (L, T ) is called (i) a para--topological residuated lattice if , as a mapping of L × L to L, is continuous when L×L is endowed with the product topology, or equivalently, if for any x, y ∈ L and any open neighbourhood W of x  y there exist two open neighbourhoods U and V of x and y, respectively, such that U V ⊆ W , in notation, (L, , T ), (ii) a para-→-topological residuated lattice if →, as a mapping of L × L to L, is continuous when L×L is endowed with the product topology, or equivalently, if for any x, y ∈ L and any open neighbourhood W of x → y there exist two open neighbourhoods U and V of x and y, respectively, such that U → V ⊆ W , in notation, (L, →, T ), (iii) an adjoint topological residuated lattice if both  and → are continuous, in notation, (L, , →, T ). Let (L, ∧, ∨, , →, 0, 1) be a residuated lattice and T be a topology on L. We say that (L, ∧, ∨, , →, T ) is a topological residuated lattice if the operations ∧, ∨, , →, as mappings of L × L to L, are continuous, (see [24]), and simply write (L, ∧, ∨, , →, T ) for (L, T ) if no confusion will arise. Given a residuated lattice L. We denote the set of all topological residuated lattices of L, the set of all para--topological residuated lattices of L, the set of all para-→-topological residuated lattices of L, the set of all adjoint-topological residuated lattices of L by T RL(L), M T RL(L), IT RL(L) and AT RL(L), respectively. Then we conclude that T RL(L) ⊆ AT RL(L) ⊆ M T RL(L), IT RL(L). 5

Example 3.2. Let L be a residuated lattice. It is easy to verify that L equipped with the discrete topology is a topological residuated lattice. In this manner, any residuated lattice could be considered as a topological residuated lattice. Also, any residuated lattice could be considered as a topological residuated lattice in the antidiscrete topology. Example 3.3. Let  and → on the real unit interval I = [0, 1] be defined as follows:  1, x ≤ y, x  y = min{x, y} and x → y = y, otherwise. Then I = (I, min, max, , →, 0, 1) is a residuated lattice [20]. Consider the interval topology T on I, then (I, T ) is a para--topological residuated lattice but not para-→-topological residuated lattice. It is easy to prove that  is continuous. Now, we prove that → is not continuous. For this, let 0 → 0 = 1 ∈ (1/2, 1] and U and V be arbitrary open neighbourhoods of 0. But 0 ∈ U → V and 0 ∈ / (1/2, 1]. Therefore, U → V  (1/2, 1]. Proposition 3.4. Let L be a residuated lattice. Then there is a topology T on L such that (L, T ) is a topological residuated lattice. Proof. Let F be a filter of a residuated lattice L. We construct a topology on L by T = {U ⊆ L : ∀ x ∈ U, x/F ⊆ U }. Indeed, the topology T is generated by the congruence classes {x/F : x ∈ L} = {{y ∈ L : (x, y) ∈ θF } : x ∈ L} = {{y ∈ L : x → y, y → x ∈ F } : x ∈ L} as the system of neighbourhoods. Now, we prove the continuity of ∗, where ∗ ∈ {∧, ∨, , →}. Since θF is a congruence, one can easily verify (x/F ) ∗ (y/F ) = (x ∗ y)/F . Note that x/F is a least open neighbourhood of x, hence, (L, T ) is a topological residuated lattice. For simplicity, the topology T constructed in Proposition 3.4 will be denoted by TF . In what follows, by Proposition 3.4, we obtain some interesting results. Remark 3.5. Let L be a residuated lattice. Consider filters F1 = {1} and F2 = L, one can easily verify that TF1 is a discrete topology and TF2 is an anti-discrete topology. Suppose that F is a proper filter of L, note that x/F is a least open neighbourhood of x with respect to TF , hence, we deduce that TF is a nontrivial topology and then (L, TF ) is a nontrivial topological residuated lattice. Assume that F1 and F2 are two different filters of L, we conclude that TF1 = TF2 , hence, (L, TF1 ) and (L, TF2 ) are two different topological residuated lattices. In fact, by F1 = F2 , then there exists x ∈ L such that x/F1 = x/F2 , it follows that TF1 = TF2 . In view of above facts, we conclude that there is an injective mapping from F(L) the set of all filters of L to T RL(L). Therefore, |F(L)| ≤ |T RL(L)|, where |F(L)| is the cardinality of F(L). 6

Theorem 3.6. Let L be a residuated lattice and let Σ be a family of topologies on L such that (L, τ ) is a topological residuated lattice for any τ ∈ Σ. Then (L, τ  ) is a topological residuated lattice, where τ  = sup{τ : τ ∈ Σ}. Proof. Let us denote by Ba (τ ) the family of all neighbourhoods of the element a in the topological space (L, τ ). Then, by Remark 2.2, 

Ba (τ ) = {U ⊆ L : U =

n 

Ui , Ui ∈ Ba (τi ), τi ∈ Σ, i = 1, 2, . . . , n, n ∈ N}.

i=1

We only prove that the operation  is continuous with respect to τ  , and the proofs  of continuity about rest operations are similar. Let a, b ∈ L and n let U ∈ Bab (τ ). Then there exist topologies τ1 , τ2 , . . . , τn ∈ Σ such that U = i=1 Ui , where Ui ∈ Bab (τi ). Since (L, τi ) is a para--topological residuated lattice for i = 1, . . . , n, then there exist Vi ∈ Ba (τi ) and Wi ∈ Bb (τi ) such that Vi  Wi ⊆ Ui for every i = 1, . . . , n. Then n  V = Vi ∈ Ba (τi ) i=1

and W =

n 

Wi ∈ Bb (τi ),

i=1

besides, V W =

n  i=1

Vi 

n 

Wi ⊆

i=1

n 

(Vi  Wi ) ⊆

i=1

n 

Ui = U.

i=1

This means that  is continuous. Therefore, the theorem is proved. Recall that a family ξ of non-empty subsets of a set X is called a prefilter  on X if X ∈ ξ and for elements A1 , . . . , Ak of ξ, there exists B ∈ ξ such that B ⊆ ki=1 Ai (see [1]). We will denote by F(L) the set of all filters in a residuated lattice L, then it is a prefilter. Definition 3.7. Let L be a residuated lattice. For any a ∈ L and a non-empty subset V of L, we define V (a) = {x ∈ L : Ra (x) and La (x) ∈ V }, where Ra (x) = a → x and La (x) = x → a for all x ∈ L. Proposition 3.8. Let V be a non-empty subset of a residuated lattice L. Then following properties hold: for any x, y ∈ L, (i) if 1 ∈ V , then x ∈ V (x), in particular, V (x) = ∅, (ii) y ∈ V (x) if and only if x ∈ V (y), 7

(iii) if 1 ∈ V , then V (1) = V , (iv) if F is a filter of L, then F (x) = x/F , (vi) if V ⊆ U ⊆ L, then V (x) ⊆ U (x). Proof. The proofs are straightforward. Theorem 3.9. Let F be a prefilter on a residuated lattice L such that for every p, q ∈ V ∈ F: (i) 1 ∈ ∩F, (ii) Rq ◦ Rp (x) = 1 implies x ∈ V . Then there exists a topology T on L such that (L, T ) is a topological residuated lattice. Proof. Let T = {U ⊆ L : ∀ a ∈ U, ∃ V ∈ F s.t. V (a) ⊆ U }. We prove the following claims. Claim 1. T is a topology on L, and F is a fundamental system of neighbourhoods of 1.  Clearly, ∅, L ∈ T . Let {Uα } be a subfamily of T , and a ∈ U α . Then, a ∈ Uα for some  α. Hence, there exists V ∈ F such that V (a) ⊆ Uα ⊆ Uα . It follows that Uα ∈ T . Let Uα , Uβ ∈ T and a ∈ Uα ∩ Uβ . Then, there exist V1 ∈ F and V2 ∈ F such that V1 (a) ⊆ Uα , and V2 (a) ⊆ Uβ . Since F is a prefilter, there exists V ∈ F such that V ⊆ V1 ∩ V2 . Hence V (a) ⊆ (V1 ∩ V2 )(a) ⊆ V1 (a) ∩ V2 (a) ⊆ Uα ∩ Uβ , i.e., Uα ∩Uβ ∈ F. Now, we show that F is a fundamental system of neighbourhoods of 1. Suppose that a ∈ V ∈ F. By (i), we have 1 ∈ V . Let x ∈ V (a). Then, a → x ∈ V . It follows that Ra→x ◦ Ra (x) = (a → x) → (a → x) = 1. By (ii), we have x ∈ V . Then V (a) ⊆ V and V ∈ T . Therefore, V is an open neighbourhood of 1. Let U be an open neighbourhood of 1. Then, there exists V ∈ F such that V (1) ⊆ U . It follows that 1 ∈ V = V (1) ⊆ U which implies that F is a fundamental system of 1. Claim2. If V ∈ F, then V is a filter. Let x ∈ V and x ≤ y. It is obvious that R1 ◦Rx (y) = 1 → (x → y) = x → y = 1. By (ii) and x, 1 ∈ V , we can get that y ∈ V . Now suppose that x, y ∈ V . Then Rx ◦ Ry (x  y) = x → (y → (x  y)) = (x  y) → (x  y) = 1. According to (ii), it implies that x  y ∈ V . Hence V is a filter.

8

Claim 3. {V (a)}V ∈F ,a∈L is a base of T . Firstly, we show that V (a) is an open set of L for any V ∈ F and a ∈ L. Let x ∈ V (a). We show that V (x) ⊆ V (a). Suppose that y ∈ V (x). Then, x → y ∈ V and y → x ∈ V . From x ∈ V (a), we have x → a ∈ V and a → x ∈ V . Since V is a filter, we get (y → x)  (x → a) ∈ V and (a → x)  (x → y) ∈ V . By (R4 ), we get that (y → x)  (x → a) ≤ y → a and (a → x)  (x → y) ≤ a → y. Hence, y → a ∈ V and a → y ∈ V . Therefore, y ∈ V (a) and V (x) ⊆ V (a). Finally, we prove that {V (a)}V ∈F ,a∈A is a base of T . In fact, for any x ∈ L and any open neighbourhood U of x, there exists V ∈ F such x ∈ V (x) ⊆ U . Claim 4. (L, T ) is a topological residuated lattice. By Claim 3, it suffices to show that V (x) ∗ V (y) ⊆ V (x ∗ y) for any V ∈ F and any x, y ∈ L, where ∗ ∈ {∧, ∨, , →}. Combining Proposition 3.8 (iv) with Claim 2, it follows that V (x) = x/F , hence, we deduce that x/V ∗ y/V = (x ∗ y)/F , that is, V (x) ∗ V (y) = V (x ∗ y). Example 3.10. Let L = {0, a, b, c, 1} be a chain (0 < a < b < c < 1). Define operations  and → as follows:  0 a b c 1

0 0 0 0 0 0

a 0 0 0 a a

b c 0 0 0 a b b b c b c

→ 0 a b c 1

1 0 a b c 1

0 a b c 1 1 1 1 b 1 1 1 a a 1 1 0 a b 1 0 a b c

1 1 1 1 1 1

It is easily checked that (L, ∧, ∨, , →, 0, 1) is a residuated lattice. Let F = {V1 , V2 }, where V1 = {c, 1} and V2 = {0, a, b, c, 1}. One can easily show that F is a prefilter and satisfies the conditions of Theorem 3.9. Then, T = {U ⊆ A : ∀ a ∈ U, ∃ V ∈ F s.t. V (a) ⊆ U } is a topology on L. Routine calculation shows that V1 (0) = {0}, V1 (a) = {a}, V1 (b) = {b}, V1 (c) = {c, 1}, V1 (1) = {c, 1} and V2 (x) = {0, a, b, c, 1} for any x ∈ L. We get {{0}, {a}, {b}, {c, 1}, {0, a, b, c, 1}} is a base of T . Therefore, by Theorem 3.9, (L, T ) is a topological residuated lattice. Remark 3.11. The prefilter F = {Fi : i ∈ Λ} satisfies the conditions of Theorem 3.9 if and only if F is a family of filters and (F, ⊆) is a down-directed set if and only if F is a family of filters and the pair (Λ, ≤) is an upward directed set, and it satisfies if i ≤ j implies Fj ⊆ Fi for any i, j ∈ Λ. Note that the order ≤ on Λ is defined by i ≤ j if and only if Fj ⊆ Fi . The above concept is called a system of filters in [37], or an inductive family of filters in [29], or a directed family of ideals in [21], or a family of filters which is closed under intersection in [15, 19], or a family of fuzzy filters F which is closed under intersection and f (1) = t for 9

all f ∈ F where t ∈ [0, 1] in [16]. All of these concepts are either equivalent to a system of filters or a special system of filters. Applying these concepts to Theorem 3.9, a special kind of topological BL-algebras [19, 37], topological residuated lattices [15, 16, 29] and topological M V -algebras [21, 27] were studied, respectively. In view of above remark, we give the following definition. Definition 3.12. Let F be a family of filters of a residuated lattice L. Then F is called a system of filters of L if (F, ⊆) is a down-directed set. A topology T of L is called a linear topology on L if there exists a base β for T , such that for any element B of β containing 1, B is a filter of L. For a system F of filters of a residuated lattice L, we denote by TF = {U ⊆ L : ∀ x ∈ U, ∃ Fi ∈ F s.t. x/Fi ⊆ U } the topology induced by F. Particularly, when F = {F }, we briefly write F{F } for FF and thus the notation coincides with in Proposition 3.4. In light of Remark 3.11 and Theorem 3.9, one can easily obtain (L, TF ) is a topological residuated lattice. Theorem 3.13. Let F = {Fi : i ∈ Λ} be a system of filters of a residuated lattice L. Then TF = sup{TFi : i ∈ Λ}. Proof. Let T  = sup{TFi : i ∈ Λ}. According to Proposition 3.4, (L, TF ) is a topological residuated lattice for any Fi ∈ F. Thus, by Theorem 3.6, (L, T  ) is a topological residuated lattice. We shall prove that T  = TF . Suppose that x ∈ . . , Uin of x in topological U ∈ T  , by Remark 2.2, there exist neighbourhoods Ui1 , .  n spaces (L, Ui1 ), . . . , (L, Uin ), respectively, such that U = n j=1 Uij . Since (F, ⊆) is a down-directed Uij . Thus, x ∈ x/F ⊆ n nset, there exists n F ∈ F such that F ⊆ j=1  x/ j=1 Fij = j=1 x/Fij ⊆ j=1 Uij = U . Therefore, T ⊆ TF . The converse is obvious. Furthermore, we have the following remark. Remark 3.14. In light of Remark 2.2 and the above facts, the topologies in [15, 16, 19, 21, 27, 29, 37] are special cases of Remark 2.2. We give only topological BL-algebras [37] as an example. Let F = {Fi : i ∈ Λ} be a system of filters of a BLalgebra L. Obviously, the topology T induced by the base β = {x/Fi : x ∈ L, i ∈ Λ} (see, [37], Theorem 3.4) is equivalent to TF . The rest of proof is similar to Theorem 3.13. Note that the topology TF induced by F = {Fi : i ∈ Λ} a system of filters of a residuated lattice L is a linear topology. In fact, β = {x/Fi : x ∈ L, i ∈ Λ} is a base for TF , hence, suppose that 1 ∈ x/Fi ∈ β, it follows that x/Fi = 1/Fi = Fi is a filter. Particularly, if F is a filter of L, then TF is a linear topology. We shall denote by LT RL(L) the set of all linearly topological residuated lattices in L. Thus, by Remark 3.5, we deduce that there exists an injective mapping from F(L) to LT RL(L), hence, |F(L)| ≤ |LT RL(L)|. A question of interest arises as follows. 10

Open problem 3.15. Whether the sets F(L) and LT RL(L) are equipotent? Theorem 3.16. Let L be a residuated lattice and F be a prefilter on L. Then there exists a topology T on L such that (L, , T ) is a semitopological residuated lattice. Proof. Let T = {U ⊆ L : ∀ a ∈ U, ∃ P ∈ F s.t. P a ⊆ U }. One can easily prove that T is a topology on L. We need to prove that (L, , T ) is a semitopological residuated lattice. For this, it is sufficient to show that for each a ∈ L, the mapping Ta (x) = a  x of L is continuous. We prove that inverse images of all members of T are open. Suppose that a ∈ L and V ∈ T . If b ∈ Ta−1 (V ), then c = a  b ∈ V . Hence, there exists P ∈ F such that P c ⊆ V . Since Ta (P b) = P ab = P c, we have P  b ⊆ Ta−1 Ta (P  b) = Ta−1 (P  c) ⊆ Ta−1 (V ). Hence, Ta−1 (V ) is an open set. Example 3.17. In Example 3.10, considering a prefilter {{a, b}, L} of L. By Theorem 3.16, routine calculation shows that T = {∅, {0}, {0, a}, {0, b}, {0, a, b}, {0, a, b, c}, L}. One can easily verify that (L, , T ) is a semitopological residuated lattice. Definition 3.18. Let N be a family of subsets in  a residuated lattice L. We call N a M -system of 1 (M -system for short) if 1 ∈ N and (i) for any x ∈ U ∈ N , there is V ∈ N such that x  V ⊆ U , (ii) for any U ∈ N , there is V ∈ N such that V  V ⊆ U , (iii) for any U, V ∈ N , there is W ∈ V such that W ⊆ U ∩ V . Example 3.19. It is easy to check that {{1}} and F(L) are M -systems of any residuated lattice  L. In Example 3.3, {(x, 1]}1≥x>0 is a M -system of I. In fact, clearly, 1 ∈ 1≥x>0 (x, 1]. For y ∈ (x, 1], then y  (x, 1] = (x, y] ⊆ (x, 1]. For any (y, 1] ∈ {(x, 1]}1≥x>0 , we have (y, 1]  (y, 1] = (y, 1]. Suppose (x, 1] , (y, 1] ∈ {(x, 1]}1≥x>0 , then (x, 1] ∩ (y, 1] = (x  y, 1] ∈ {(x, 1]}x>0 . Therefore, {(x, 1]}1≥x>0 is a M -system. Proposition 3.20. Let N be a fundamental system of open neighbourhoods of 1 in a para--topological residuated lattice (L, , T ). Then N is a M -system of 1 in L.  Proof. Clearly, 1 ∈ N . Let x ∈ U ∈ N . As  is continuous and x  1 = x ∈ U , there exists an open neighbourhood W of 1 such that x  W ⊆ U . Since N is a fundamental system of open neighbourhoods of 1, there is V ∈ N such that 1 ∈ V ⊆ W . Hence, x  V ⊆ x  W ⊆ U . Thus (i) of Definition 3.18 is correct. Let U ∈ N . Since 1  1 = 1 ∈ U and the mapping  is continuous, there are open neighbourhoods W0 and W1 of 1 such that W0  W1 ⊆ U . Take W = W0 ∩ W1 . As N is a fundamental system of neighbourhoods of 1, there is V ∈ N such that 1 ∈ V ⊆ W . Hence V  V ⊆ W  W ⊆ W0  W1 ⊆ U , and then Definition 3.18 (ii) holds. Since N is a fundamental system of neighbourhoods of 1, it is closed under finite intersection. Then, it satisfies (iii) of Definition 3.18. 11

Theorem 3.21. Let N be a M -system of 1 in a residuated lattice L. Then there is a topology T on L such that (L, , T ) is a para--topological residuated lattice, and N is a fundamental system of open neighbourhoods of 1. Proof. Let T = {W ⊆ L : ∀ x ∈ W, ∃ U ∈ N s.t. x  U ⊆ W }. We prove the following claims. Claim 1. x  U ∈ T for each x ∈ L and U ∈ N . Let y ∈ x  U . Then y = x  a, for some a ∈ U . Since N is a M -system, there is V ∈ N such that a  V ⊆ U . Hence y  V = x  a  V ⊆ x  U which implies that x  U ∈ T . Claim 2. T is a topology on L, and N is a fundamental system of open neighbourhoods of 1.  Clearly, ∅, L ∈ T . Let {Wi : i ∈ I} be a subfamily of T . Then Wi ∈ T , because,   x∈ Wi ⇒ ∃Wi s.t. x ∈ Wi ⇒ ∃U ∈ N s.t. x  U ⊆ Wi ⇒ x  U ⊆ Wi . Let W1 , W2 ∈ T , and W = W1 ∩ W2 . We prove that W ∈ T . Let x ∈ W . Then there exist U1 , U2 ∈ N such that x  U1 ⊆ W1 and x  U2 ⊆ W2 . Since N is a system of 1, there is U ∈ N such that U ⊆ U1 ∩ U2 . Then W ∈ T because x  U ⊆ x  (U1 ∩ U2 ) ⊆ (x  U1 ) ∩ (x  U2 ) ⊆ W1 ∩ W2 = W. Therefore, T is a topology on L. Now, we show that N is a fundamental system of neighbourhoods of 1. By Claim 1, for any U ∈ N , we have 1 ∈ U = 1  U ∈ T . Let W be an open neighbourhood of 1. Then, by Definition 3.18 (i), there exists V ∈ N such that 1 ∈ 1  V = V ⊆ W which implies that N is a fundamental system of 1. Claim 3. (L, , T ) is a para--topological residuated lattice. Let z = x  y and W is an open neighbourhood of z. Then there is U ∈ N such that z  U ⊆ W . According to Definition 3.18 (ii), there exists V ∈ T such that V  V ⊆ U . Now x  V and y  V are two open neighbourhoods of x and y such that (x  V )  (y  V ) = x  y  V  V ⊆ x  y  U = z  U ⊆ W. Therefore, (L, , T ) is a para--topological residuated lattice. Theorem 3.22. Let (L, , T ) be a para--topological residuated lattice and the mapping Ta : L → L defined by Ta (x) = a  x is open for all a ∈ L. Then there exists a M -system of 1 in L which can induce a para--topological residuated lattice (L, , T ) such that (L, , T ) = (L, , T ). 12

Proof. Let (L, , T ) be a para--topological residuated lattice. We denote N (1) the set of all open neighbourhoods of 1. It is easy to prove that N (1) is a fundamental system of open neighbourhoods of 1. By Proposition 3.20, we get that N (1) is a M-system of 1 in L. According to Theorem 3.21, we get a topology T = {W ⊆ A : ∀ x ∈ W, ∃ U ∈ N s.t. x  U ⊆ W } such that (L, , T ) is a para--topological residuated lattice. To complete the proof, we need only show that T = T . Let x ∈ U ∈ T . As  is continuous and x  1 = x ∈ U , there exists an open neighbourhood V of 1 such that x  V ⊆ U . It follows that U ∈ T . Conversely, let x ∈ U ∈ T . Since  is continuous andx  1 = x ∈ U , there exists Vx ∈ N (1) such that x  Vx ⊆  U . It follows that x∈U x  Vx ⊆ U . Clearly, x ∈ x  Vx . This implies that x∈U x  Vx = U . Since Tx is open and Vx ∈ T , it  follows that U = x∈U x  Vx ∈ T . Therefore, T = T . At the end of this section, in order to discuss the independence of para-topological residuated lattices and para-→-topological residuated lattices, we shall introduce the concepts of  compatibility and → compatibility, respectively. We will denote by Eq(L) the set of all equivalence relations on L. Definition 3.23. Let L be a residuated lattice and let θ ∈ Eq(L). Then θ is said to be (i)  compatible, if (x, y) ∈ θ implies (x  z, y  z) ∈ θ for all x, y, z ∈ L, (ii) → compatible, if (x, y) ∈ θ implies (x → z, y → z), (z → x, z → y) ∈ θ for all x, y, z ∈ L, (iii) adjoint compatible, if both  and → are compatible with θ. Note that in a residuated lattice L if an adjoint compatible relation θ is also compatible with ∧ and ∨, then it is a congruence on L. And if we denote the set of all  compatible relations, the set of all → compatible relations and the set of all adjoint compatible relations by M Eq(L), IEq(L) and AEq(L), respectively, then Con(L) ⊆ AEq(L) ⊆ M Eq(L), IEq(L), where Con(L) is the set of all congruences of L. Example 3.24. Let L = {0, a, b, 1} be a lattice such that (0 < a < b < 1) and , → be defined as follows:  0 a b 1

0 0 0 0 0

a 0 0 0 a

b 0 0 b b

→ 0 a b 1

1 0 a b 1

13

0 a b 1 1 1 1 1 b 1 1 1 a a 1 1 0 a b 1

Then (L, ∧, ∨, , →, 0, 1) is a residuated lattice. Let θ1 = Δ ∪ {(0, b), (b, 0)} and θ2 = Δ ∪ {(0, a), (a, 0), (b, 1), (1, b)}, where Δ = {(0, 0), (a, a), (b, b), (1, 1)}. We can easily check that θ1 satisfies  compatibility property and θ2 satisfies → compatibility property. But, θ1 doesn’t satisfy → compatibility property, since (0, b) ∈ θ1 and (a → 0, a → b) = (b, 1) ∈ / θ1 . Since θ2 is also  compatible, then θ2 is adjoint compatible. Note that if θ ∈ Eq(L), then T = {U ⊆ L : ∀ x ∈ U, [x]θ ⊆ U } is a topology on L. To be more precise, the topology T is induced by the base β = {[x]θ : x ∈ L}, and it will be denoted by Tθ . Moreover, if θ is a special equivalence relation, then we have following results. Theorem 3.25. Let L be a residuated lattice. If θ ∈ M Eq(L) (IEq(L), AEq(L), Con(L)), then (L, , Tθ ) is a para--topological residuated lattice ((L, →, Tθ ) is a para-→-topological residuated lattice, (L, , →, Tθ ) is an adjoint topological residuated lattice, (L, Tθ ) is a topological residuated lattice). Proof. We only prove the case of θ ∈ IEq(L), the proofs for the rest cases are left to the reader. Let θ ∈ IEq(L). We claim that (L, →, Tθ ) is a para-→-topological residuated lattice. Suppose that x → y ∈ U ∈ Tθ . According to the → compatibility property, we can get [x]θ → [y]θ = [x → y]θ . From [x → y]θ ⊆ U , it follows that [x]θ → [y]θ ⊆ U , which completes the proof. Remark 3.26. By Theorem 3.25, [x]θ is a least open neighbourhood of x with respect to Tθ . Similar to Remark 3.5, there is an injective mapping from M Eq(L) (IEq(L), AEq(L), Con(L)) to M T RL(L) (IT RL(L), AT RL(L), T RL(L)). Example 3.27. Let L = {0, a, b, 1} be a lattice such that (0 < a < b < 1) and , → be defined as follows:  0 a b 1

0 0 0 0 0

a 0 a a a

b 0 a b b

→ 0 a b 1

1 0 a b 1

0 1 0 0 0

a 1 1 a a

b 1 1 1 1 1 1 1 b 1

Then (L, ∧, ∨, , →, 0, 1) is a residuated lattice. Let θ1 = Δ∪{(0, a), (a, 0)} and θ2 = Δ∪{(b, 1), (1, b)}, where Δ = {(0, 0), (a, a), (b, b), (1, 1)}. One can easily check that θ1 satisfies  compatibility property and θ2 satisfies → compatibility property. By Theorem 3.25, we have Tθ1 = {∅, L, {0, a}, {b}, {1}, {0, a, b}, {0, a, 1}, {b, 1}} and Tθ2 = {∅, L, {0}, {a}, {b, 1}, {0, a}, {0, b, 1}, {a, b, 1}}. Then (L, , Tθ1 ) is a para--topological residuated lattice and (L, →, Tθ2 ) is a para-→-topological residuated lattice. Theorem 3.28. Let L be a residuated lattice. If θ ∈ IEq(L) but θ ∈ / M Eq(L), then (L, →, Tθ ) is a para-→-topological residuated lattice, but it is not a para-topological residuated lattice. 14

Proof. Let θ ∈ IEq(L) and θ ∈ / M Eq(L). Then, by Theorem 3.25, (L, →, Tθ ) is a para-→-topological residuated lattice. Now, we prove that  is not continuous with respect to Tθ . By assumption, there exist (x, y) ∈ θ and z ∈ L such that (x  z, y  z) ∈ / θ. Note that [y  z]θ is a least open neighbourhood of y  z. Thus, in order to show that  is continuous, [y]θ  [z]θ ⊆ [y  z]θ must be true. Actually, [y]θ  [z]θ  [y  z]θ because x ∈ [y]θ and z ∈ [z]θ but x  z ∈ / [y  z]θ . A simple theorem below complements Theorem 3.28. Theorem 3.29. Let L be a residuated lattice. If θ ∈ M Eq(L) but θ ∈ / IEq(L), then (L, , Tθ ) is a para--topological residuated lattice, but it is not a para-→topological residuated lattice. Proof. The proof is similar to that of Theorem 3.28. Example 3.30. In Example 3.27, θ1 satisfies the condition of Theorem 3.29, since (0, a) ∈ θ1 , but (a → 0, a → a) ∈ / θ1 . Clearly, a → a ∈ {1} ∈ Tθ1 , but {0, a} → {0, a} = {0, 1}  {1}. Thus, → is not continuous with respect to Tθ1 . Interestingly, the examples satisfying Theorem 3.29 are easily constructed. However, can we give an example for Theorem 3.28? This is an open problem. 4. Some connections between topologies and filters In this section we prove several results on (para--, para-→-, semi, U) topological residuated lattices by considering the properties of filters. Proposition 4.1. Let F be a proper filter of a para-→-topological residuated lattice (L, →, T ). If 1 is an interior point of F , then F is a closed set. Proof. Let 1 be an interior point of a filter F . We show that L \ F is an open set. Let x ∈ L \ F . Since 1 is an interior point of F , there is an open neighbourhood W of 1 such that 1 ∈ W ⊆ F . From x → x = 1, there is an open neighbourhood U of x such that U → U ⊆ W ⊆ F . Now we prove that U ⊆ L \ F . Suppose that U  L \ F , then there is y ∈ F ∩ U . Hence, for each z ∈ U , y → z ∈ F . Since F is a filter and y ∈ F , we get that z ∈ F . Therefore, U ⊆ F which is a contradiction. Then x ∈ U ⊆ L \ F which implies that L \ F is an open set and so F is a closed set. Example 4.2. In Example 3.24, by Theorem 3.25, (L, →, Tθ2 ) is a para-→-topologi cal residuated lattice, where Tθ2 = {∅, L, {0, a}, {b, 1}}. It is easily checked that F = {b, 1} is a filter of L. Since F ∈ Tθ2 , 1 is an interior point of F . As a result, F is a closed set. Theorem 4.3. Let (L, →, T ) be a semitopological residuated lattice. If the filter {1} is an open set, then (L, T ) is a discrete space. 15

Proof. Let x ∈ L. Since (L, →, T ) is a semitopological residuated lattice, then the mapping → is separately continuous. By (R1 ), x → x = 1. Hence there are two open neighbourhoods V1 and V2 of x such that V1 → x ⊆ {1} and x → V2 ⊆ {1}.  Put V = V1 V2 . Then V is an open neighbourhood of x and V → x = {1} = x → V . Now, let y ∈ V . Since y → x = x → y = 1, then x = y. This implies that V = {x} and T is a discrete topology on L. Therefore, (L, T ) is a topological residuated lattice. Proposition 4.4. Let B and C be subsets of a para--topological residuated lattice (L, , T ). If B and C are compact subsets, then B  C is a compact subset,too. Proof. Let f : L × L → L defined by f (a, b) = a  b. Since  is continuous, hence its restriction onto subspace B × C of L × L is continuous, too. Thus, B  C = f (B × C) is a compact subset in L (as a continuous image of compact space B × C is compact and finite product of compact spaces is compact). Proposition 4.5. Let F and P be two disjoint subsets of a para--topological residuated lattice (L, , T ). Suppose that F is compact, P is closed and for each a ∈ L, Ta (Ta (x) = ax) is an open map. Then there exists an open neighbourhood V of 1 such that (F  V ) ∩ P = ∅. Proof. Suppose that x ∈ F . Since F ∩ P = ∅ and L \ P is an open set of x, then there exists an open neighbourhood Wx of x such that Wx ⊆ L \ P . According to  is continuous and x  1 = x ∈ Wx , then there exists an open neighbourhood Vx of 1 such that x  Vx ⊆ Wx . Also, 1  1 = 1 ∈ Vx , then there exists an open neighbourhood Ux of 1 such that Ux  Ux ⊆ Vx . Now, considering the set {x  Ux : x ∈ F }. Since Tx is an open map, {x  Ux : x ∈ F } is an open covering of compact subset F . As F is compact, there exists a finite set C ⊆ F such that F ⊆ ∪x∈C (x  Ux ). Let V = ∩x∈C Ux . Then V is an open neighbourhood of 1. We claim that (F  V ) ∩ P = ∅. It is enough to prove that for each y ∈ F , (y  V ) ∩ P = ∅. Let y ∈ F . Then there exists x ∈ C such that y ∈ x  Ux . Hence we have y  V ⊆ x  Ux  V ⊆ x  Ux  Ux ⊆ x  Vx ⊆ Wx ⊆ L \ P. This prove that the sets F  V and P are disjoint. Proposition 4.6. Let (L, , T ) be a para--topological residuated lattice and U be an open neighbourhood of 1. If for each a ∈ L the mapping Ta : Ta (x) = a  x is an open map, then F ⊆ F  U for each proper filter F of L. Proof. Let F be a proper filter of L. Considering B = {a ∈ L : (a  U ) ∩ F = ∅}. Since 0 ∈ B, then B = ∅. Put V = L \ (B  U ). Then B  U = ∪b∈B (b  U ) = ∪b∈B Tb (U ) is an open set and (BU )∩F = (∪b∈B (bU ))∩F = ∪b∈B ((bU )∩F ) = ∅. Thus V is a closed subset, and F ⊆ V . It follows that F ⊆ V . For any y ∈ V , (y  U ) ∩ F = ∅, then there exists h ∈ U such that y  h = z ∈ F . Since z = y  h ≤ y and F is a filter, then y ∈ F ⊆ F  U . Therefore, F ⊆ V ⊆ F  U . 16

Proposition 4.7. If B and C are subsets of a para--topological (para-→-topological) residuated lattice, then B  C ⊆ B  C (B → C ⊆ B → C). Proof. The proof is standard. Recall that in a topological space X, a subset C of X is called a connected component of x if it is a maximal connected set containing x and we simply write it for C(x). Lemma 4.8. The following statements hold: (i) if (L, , T ) is a semitopological residuated lattice and (L, ∨, T ) is a semitopological residuated lattice, then C(1) is the greatest closed filter of L, which is connected, (ii) if (L, →, T ) is a semitopological residuated lattice, then C(1) = {1} if and only if C(x) = {x}, for any x ∈ L. Proof. (i) Let x, y ∈ C(1). As Tx (z) = x  z is continuous, and so x  C(1) = Tx (C(1)) is connected. Since x ∈ C(1) ∩ (x  C(1)) = ∅, then C(1) ∪ (x  C(1)) is a connected subset of L containing 1. By C(1) is the connected component of 1, we have C(1) ∪ (x  C(1)) ⊆ C(1). It follows that x  C(1) ⊆ C(1). Hence x  y ∈ C(1). Suppose that x ∈ C(1) and x ≤ y. From separately continuity of ∨, we have y ∨ C(1) = Hy (C(1)) is connected, where Hy (z) = y ∨ z. It follows that y ∨ C(1) ⊆ C(1). Then, y = y ∨ x ∈ y ∨ C(1) ⊆ C(1), so y ∈ C(1). This proves that C(1) is a filter. Obviously, C(1) is a closed subset. (ii) Let C(1) = {1} and x ∈ L. Similar to the proof of (i), we can show that x → C(x) is a connected subset of L. Since x ∈ C(x), then 1 ∈ x → C(x) and so x → C(x) ⊆ C(1) = {1}. Hence x → a = 1, for any a ∈ C(x). By the similar way, it follows that a → x = 1, for any a ∈ C(x), so C(x) = {x}. The proof of the converse is clear. Note that in Lemma 4.8 (ii), if L is a finite set, then the conclusion of Lemma 4.8 implies that (L, T ) is a discrete space. In fact, for any x ∈ L, {x} = C(x) is closed, by finiteness of L, it follows that {x} is open. Therefore, (L, T ) is a discrete space. Recall that a topology space (X, T ) is a T1 -space if and only if {x} is closed for any x ∈ X. Hence we have the following proposition. Proposition 4.9. Let (L, →, T ) be a semitopological residuated lattice. If C(1) = {1}, then (L, T ) is a T1 -space. Proof. It directly follows from Lemma 4.8 (ii). Proposition 4.10. Let (L, T ) be a topological residuated lattice and θ be a binary relation on L defined by (x, y) ∈ θ if and only if C(x) = C(y), for any x, y ∈ L. Then the following assertions hold: 17

(i) Then θ is a congruence relation on L. (ii) θ = θC(1) , where θC(1) = {(x, y) : x → y, y → x ∈ C(1)} is a congruence relation on L induced by the filter C(1). Proof. (i) Clearly, θ is an equivalence relation on L. Let (x, y) ∈ θ and z ∈ L. Then C(x) = C(y). Since z → C(x) is a connected subset of L containing z → x and z → y, we get z → y ∈ z → C(x) ⊆ C(z → x) and so C(z → x) ⊆ C(z → y). Since z → C(y) is a connected subset of L containing z → x and z → y, we have z → x ∈ z → C(y) ⊆ C(z → y) and so C(z → y) ⊆ C(z → x). Hence C(z → y) = C(z → x) and (z → x, z → y) ∈ θ. By the similar way, we can show that (x → z, y → z) ∈ θ and (x ∗ z, y ∗ z) ∈ θ, for any ∗ ∈ {∧, ∨, }. Therefore, θ is a congruence relation on L. (ii) Let (x, y) ∈ θ. Then C(x) = C(y), so 1 ∈ y → C(x), 1 ∈ x → C(y). Since x → C(y) and y → C(x) are connected subsets of L, then we get that x → C(y), y → C(x) ⊆ C(1) and so x → y, y → x ∈ C(1). Hence (x, y) ∈ θC(1) . Conversely, let (x, y) ∈ θC(1) . Then x → y, y → x ∈ C(1) and so there exist t, s ∈ C(1) such that x → y = t and y → x = s. Hence t → (x → y) = 1 and s → (y → x) = 1 whence by Proposition 2.4 (R6 ), (t  x) → y = 1 and y → (s → x) = 1. Since t ∈ C(1), then t  x ∈ C(1)  x, x ∈ C(1)  x, and C(1)  x is a connected subset of L containing x, so t  x ∈ C(1)  x ⊆ C(x). It follows that 1 ∈ C(x) → y. Moreover, s ∈ C(1) implies that s → x ∈ C(1) → x ⊆ C(x), hence by y → (s → x) = 1, we conclude that 1 ∈ y → C(x), whence 1 ∈ (C(x) → y) ∩ (y → C(x)). By Proposition 2.4 (R2 ), there exist a, b ∈ C(x) such that a ≤ y and y ≤ b. Since (a, b) ∈ θ and θ is a congruence relation on L, then (a, y) = (a ∧ y, b ∧ y) ∈ θ, so C(x) = C(a) = C(y). Thus (x, y) ∈ θ. Therefore, θ = θC(1) . Example 4.11. In Example 3.10, one can easily verify that C(0) = {0}, C(a) = {a}, C(b) = {b}, C(c) = C(1) = {c, 1}, by Proposition 4.10, θ = Δ ∪ {(c, 1), (1, c)} is a congruence relation of L. According to Lemma 4.8 (i), C(1) is a filter of L. Routine calculation shows that θC(1) = {(0, 0), (a, a), (b, b), (c, c), (1, 1), (c, 1), (1, c)}. Remark 4.12. By Proposition 4.9, if a para-→-topological residuated lattice (L, → , T ) is not a T1 -space, then C(1) = {1}. Also, if (L, →, T ) is not a connected space, then C(1) = L. As a consequence, we conclude if (L, →, T ) is neither a T1 -space nor a connected space, then θC(1) is a non-trivial congruence relation on L(since C(1) = {1} and C(1) = L). Let F be a filter of a residuated lattice L. Then L/F is a quotient residuated lattice and πF : L → L/F defined by πF (x) = x/F is a natural homomorphism. Giving a topology T on L, we can induce a quotient topology T on L/F by T = {U ⊆ L/F : πF−1 (U ) ∈ T }. It is well known that the topology is the largest topology on L/F such that πF is continuous. Lemma 4.13. Let L be a residuated lattice and F be a filter of L. If T is a topology on L and T is the quotient topology on L/F , then for each x ∈ L, x/F is πF -saturated (i.e., πF−1 ◦ πF (x/F ) = x/F ). 18

Proof. Clearly, x/F ⊆ πF−1 ◦ πF (x/F ). It is enough to show that πF−1 ◦ πF (x/F ) ⊆ x/F . Suppose that y ∈ πF−1 ◦ πF (x/F ). Then, y/F ∈ πF (x/F ). So there exists z ∈ x/F such that y/F = z/F . It follows that y/F = z/F = x/F . Therefore, y ∈ x/F . Proposition 4.14. Let (L, T ) be a topological residuated lattice and F be a filter of L. If the natural homomorphism πF is an open map, then quotient residuated lattice L/F equipped with the quotient topology is also a topological residuated lattice. Proof. It is sufficient to prove that the map (x/F, y/F ) → x/F ∗ y/F = (x ∗ y)/F is continuous with respect to the quotient topology, where ∗ ∈ {∧, ∨, , →}. Let U be an open neighbourhood of x/F ∗ y/F . Then πF−1 (U ) is an open subset of L and x ∗ y ∈ πF−1 (U ). Since (L, T ) is a topological residuated lattice, so there exist two open subsets V1 and V2 containing x and y, respectively, such that V1 ∗V2 ⊆ πF−1 (U ). It follows that πF (V1 ∗ V2 ) ⊆ πF ◦ πF−1 (U ) = U . Put πF (V1 ) = U1 and πF (V2 ) = U2 . Obviously, x/F ∈ U1 , y/F ∈ U2 . Since πF is an open map, U1 and U2 are open subsets of L/F . It follows that U1 ∗ U2 = πF (V1 ) ∗ πF (V2 ) ⊆ πF (V1 ∗ V2 ) ⊆ U . Therefore, ∗ is continuous. Corollary 4.15. Let L be a residuated lattice and F be a filter of L. If (L, TF ) is a topological residuated lattice induced by a system of filters F, which satisfies  F ⊆ F, then (L/F, T ) is a topological residuated lattice, too. Proof. By Proposition 4.14, it suffices to show that the natural homomorphism πF is open. According to Theorem 3.9, {V (a)}a∈L,V ∈F is a base for TF . So it suffices to show that πF (V (a)) is an open subset in L/F , for any a ∈ L and V ∈ FF . Let a ∈ L and V ∈ FF . We shall show that πF−1 (πF (V (a))) ∈ TF . By Proposition 3.8 (iv) and Lemma 4.13, it follows that πF−1 (πF (V (a))) = V (a) ∈ TF . Therefore, πF is open. Definition 4.16. Let L be a residuated lattice with a topology T on it. We call (L, T ) satisfies an open condition if for any filter F of L the natural homomorphism πF is open. In the following, Proposition 4.18 provides an example of a residuated lattice with a topology satisfying open condition. To prove it, we need the following lemma. Lemma 4.17. Let F be a filter of residuated lattice L and x, y ∈ L. If x/F ≤ y/F , then for each a ∈ x/F there exists b ∈ y/F such that a ≤ b. Proof. Suppose that x/F ≤ y/F and a ∈ x/F . It follows that a → x, x → y ∈ F . Since F is a filter and (a → x)  (x → y) ≤ a → y, we get a → y ∈ F . Put b = (a → y) → y. By Proposition 2.4 (R7 ), we have b → y = (a → y) → y) → y = a → y ∈ F , and by Proposition 2.4 (R6 ), we obtain y → b = y → ((a → y) → y) = (a → y) → (y → y) = 1 ∈ F . Hence, b ∈ y/F . According to Proposition 2.4 (R6 ), we get a → b = a → ((a → y) → y) = (a → y) → (a → y) = 1. This implies that a ≤ b. 19

Proposition 4.18. Let L be a residuated lattice. For each a ∈ L, we denote the set {x ∈ L : a ≤ x} by a↑ . Then there exists a topology T on L with a base {a↑ : a ∈ L} such that (L, T ) satisfies the open condition. Proof. Let T = {U ⊆ L : ∀ x ∈ U, x↑ ⊆ U }. It is easy to prove that T is a topology on L with a base {a↑ : a ∈ L}. We prove that (L, T ) satisfies the open condition. Let F be a filter of L and the mapping πF : L → L/F be the natural homomorphism. Firstly, we need show that πF (a↑ ) = (a/F )↑ and πF−1 (a/F )↑ ∈ T . Suppose that y/F ∈ πF (a↑ ), then there exists x ∈ a↑ such that y/F = x/F . It follows that a/F ≤ x/F = y/F (since a ≤ x), and then y/F ∈ (a/F )↑ . This shows that πF (a↑ ) ⊆ (a/F )↑ . Conversely, suppose that y/F ∈ (a/F )↑ , i.e., a/F ≤ y/F , by Lemma 4.17, there exists z ∈ y/F such that a ≤ z. Hence y/F = πF (z) ∈ πF (a↑ ), which implies that (a/F )↑ ⊆ πF (a↑ ). To prove that πF−1 (a/F )↑ ∈ T is true, it suffices to prove that for any x ∈ πF−1 (a/F )↑ , x↑ ⊆ πF−1 (a/F )↑ . Suppose that y ∈ x↑ , it follows that a/F ≤ x/F ≤ y/F , which implies that y/F ∈ (a/F )↑ . Therefore, x↑ ⊆ πF−1 (a/F )↑ . Now, we prove that πF is an open map. For this, we show that {(a/F )↑ : a ∈ L} is a base for the quotient topology. Let U be an open subset of L/F and a/F ∈ U . Since a ∈ πF−1 (U ) ∈ T , a↑ ⊆ πF−1 (U ). It follows that a/F ∈ (a/F )↑ = πF (a↑ ) ⊆ πF ◦ πF−1 (U ) ⊆ U . Therefore, {(a/F )↑ : a ∈ L} is a base for the quotient topology. Definition 4.19. The topology in Proposition 4.18 will be denoted by TU , and it is called a U-topology induced by {a↑ : a ∈ L}. Example 4.20. In Example 3.24, one can easily verify that 0↑ = L, a↑ = {a, b, 1}, b↑ = {b, 1} and 1↑ = {1}. According to Proposition 4.18, routine calculation shows that TU = {∅, {1}, {b, 1}, {a, b, 1}, L}. Obviously, F = {b, 1} is a filter of L. It is easy to check that L/F = {0/F, 1/F } = {{0, a}, {b, 1}} and the natural homomorphism πF is an open map. Proposition 4.21. Let L be a residuated lattice. Then the operations ∧, ∨ and  are continuous with respect to TU . In particular, (L, TU ) is a para--topological residuated lattice. Proof. Let ∗ ∈ {∧, ∨, } and {a↑ : a ∈ L} be a base for TU . It suffices to show that x↑ ∗ y ↑ ⊆ (x ∗ y)↑ , for any x, y ∈ L. Suppose that t ∈ x↑ ∗ y ↑ , then there exist u ∈ x↑ and v ∈ (y)↑ such that t = u ∗ v. It follows that x ≤ u and y ≤ v, this implies that x ∗ y ≤ u ∗ v. Thus Lx ∗ Ly ⊆ Lx∗y . Therefore, ∗ is continuous with respect to TU . In general, x↑ → y ↑ ⊆ (x → y)↑ does’t hold, hence (L, TU ) is generally not a para-→-topological residuated lattice. Example 4.22. In Example 4.20, one can easily verify that (L, , TU ) is a para-topological residuated lattice, and the operations ∧, ∨ are continuous with respect to TU . Obviously, 0↑ → 0↑ = L → L  (0 → 0)↑ = 1↑ = {1}. Moreover, (L, →, TU ) is not a para-→-topological residuated lattice, since 0 → 0 = 1 ∈ {1} ∈ TU and L is an only open neighbourhood of 0 which does’t satisfy L → L ⊆ {1}. 20

5. Conclusion It is well known that residuated lattices play an important role in investigating the algebraic structures of logical systems. In this study we endowed residuated lattices with some kinds of topologies. We explored the existence of these topologies and their relationship. Interestingly, these topologies correspond to the corresponding compatibility properties on residuated lattices. In this paper, we used a new method to describe a special topological residuated lattice (L, TF ), which is induced by a system of filters F, the characterization is TF = T  (T  = sup{TF : F ∈ F}). Particularly, by Remark 2.2, the fundamental system of neighbourhoods of x in (L, T  ) can be simply depicted as 

Bx (L, T ) = {U ⊆ L : U =

n 

x/Fij , Fij ∈ F, n ∈ N}.

j=1

Given a residuated lattice (L, ∧, ∨, , →, 0, 1), then (L, ∧, ∨, , 1) is a lattice ordered Abelian semigroup. What is the relationship between a topological residuated (L, T ) and a topological lattice ordered Abelian semigroup (L, ∧, ∨, , T )? By Proposition 4.21, we constructed a topological lattice ordered Abelian semigroup (L, ∧, ∨, , TU ) by using lattice principal filters of it, and concluded that the former can imply the latter, but, in general, not vice-versa. How about compatible combination of order and implication both algebraic and topological properties? These questions would be our further research topics. Acknowledgments The authors are highly grateful to the editor and anonymous referees for their careful reading and valuable suggestions which helped to improve the paper. This research is supported by a grant of National Natural Science Foundation of China (11601302), the Postdoctoral Science Foundation of China (2016M602761) and the Natural Science Foundation of Shaanxi Province (2017JQ1005). References [1] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Press/World Scientific, Paris, 2008. [2] V.I. Arnautov, S.T. Glavatsky, A.V. Mikhalev, Introduction to the Theory of Topological Rings and Modules, Marcel Dekker, Inc. New York, 1996. [3] M. Bakhsi, Spectrum topology of a residuated lattice, Fuzzy Inf. Eng. 5(2) (2013) 159-172 [4] K. Blount, C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput. 13 (2003), 437-461. [5] R.A. Borzooei, G.R. Rezaei, Metrizability on (semi)topological BL-algebras, Soft Comput. 16(10) (2012) 1681-1690. 21

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