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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 15, Issue 1

# Uniform topology on EQ-algebras

Jiang Yang
/ Xiao Long Xin
/ Peng Fei He
Published Online: 2017-04-14 | DOI: https://doi.org/10.1515/math-2017-0032

## Abstract

In this paper, we use filters of an EQ-algebra E to induce a uniform structure (E, 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E. We prove that the pair (E, 𝒯) is a topological EQ-algebra, and some properties of (E, 𝒯) are investigated. In particular, we show that (E, 𝒯) is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, by using convergence of nets, the convergence of topological EQ-algebras is obtained.

MSC 2010: 06F99; 54E15

## 1 Introduction

EQ-algebras were proposed by Novák [1] with the introduction of developing an algebraic structure of truth values for fuzzy type theory (F T T). It has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. In Novák et al. [2], the study of EQ-algebras has been further deepened. Moreover, the axioms originally introduced in [1] have been slightly modified. Motivated by the assumption that the truth values in F T T were either an IMTL-algebra, a BL-algebra or an MV-algebra, all the algebras above are special kinds of residuated lattices with monoidal operation (multiplication) and its residuum. The latter is a natural interpretation of implication in fuzzy logic; the equivalence is then interpreted by the biresiduum, a derived operation. From the algebraic point of view, the class of EQ-algebras generalizes, in a certain sense, the class of residuated lattices and so, they may become an interesting class of algebraic structures as such. Some interesting consequences of EQ-algebras were obtained (see [3-5]).

The concept of a uniform space can be considered either as axiomatizations of some geometric notions, close to but quite independent of the concept of a topological space, or as convenient tools for an investigation of topological space. In the paper [6], Haveshki et al. considered a collection of filters and used the congruence relation with respect to filters to define a uniformity and turned the BL-algebra into a uniform topological space. Then Ghorbani and Hasankhani [7] defined uniform topology and quotient topology on a quotient residuated lattice and proved these topologies coincide. Furthermore, many mathematicians have endowed a number of algebraic structures associated with logical systems with topology and have found some of their properties. In [8], Hoo introduced topological MV-algebras and obtained some interesting results. Hoo’s mainly work reveals that the essential ingredients are the existence of an adjoint pair of operations and the fact that ideals of MV-algebras correspond to their congruences. Nganou and Tebu [9] generalized Hoo’s work to FLew-algebras. They considered a similar approach to study FLew-algebras. Ciungu [10] investigated some concepts of convergence in the class of perfect BL-algebras. Mi Ko and Kim [11] studied relationships between closure operators and BL-algebras. In [12, 13], Borzooei et al. studied metrizability on (semi)topological BL-algebras and the relationship between separation axioms and (semi)topological quotient BL-algebras. As EQ-algebras are the generalizations of residuated lattices which the adjoint property failed, our study of uniform topologies in EQ-algebras is meaningful.

This paper is organized as follows: In Section 2, we recall some facts about EQ-algebras and topologies, which are needed in the sequel. In Section 3, in order to induce uniform topology, we use the class of filters of EQ-algebras to construct uniform structures. In Section 4, using the given concept of topological EQ-algebras, we show that EQ-algebras with the uniform topology are topological EQ-algebras, and also some properties are obtained.

## 2 Preliminaries

In this section, we summarize some definitions and results about EQ-algebras, which will be used in the following sections of this paper.

#### Definition 2.1

([2, 14]). An EQ-algebra is an algebra E = (E, ∧, ⊗, ∼, 1) of type (2,2,2,0) satisfying the following axioms:

(E1) (E, ∧, 1) is a ∧-semilattice with top element 1. We put xy if and only if xy = x;

(E2) (E, ⊗, 1) is a monoid andis isotone in both arguments with respect to ≤;

(E3) xx = 1, (reflexivity axiom);

(E4) ((xy) ∼ z) ⊗ (tx) ≤ z ∼ (ty), (substitution axiom);

(E5) (xy) ⊗ (zt) ≤ (xz) ∼ (yt), (congruence axiom);

(E6) (xyz) ∼ x ≤ (xy) ∼ x, (monotonicity axiom);

(E7) xyxy. (boundedness axiom).

For the convenience of readers, we mention some basic properties of the operations on EQ-algebras in the following proposition.

#### Proposition 2.2

([2, 14]). Let E be an EQ-algebra, xy := (xy) ∼ x and := x ∼ 1. Then the following properties hold for all x, y, zE:

1. xyxyx, y;

2. z ⊗ (xy) ≤ (zx) ∧ (zy);

3. xyxy;

4. xx = 1;

5. (xy) ⊗ (yz) ≤ xz;

6. (xy) ⊗ (yz) ≤ xz;

7. x. 1̄ = 1;

8. x ⊗ (xy) ≤ ȳ;

9. (xy) ⊗ (yx) ≤ xy ≤ (xy) ∧ (yx);

10. if xyz, then xy;

11. if xy < z, then zxzy and xzxy.

#### Definition 2.3

([2, 14]). Let E be an EQ-algebra. We say that it is

1. separated if for all a, bE, ab = 1 implies a=b;

2. good if for all aE, a ∼ 1 = a.

#### Definition 2.4

([2, 14]). Let E be an EQ-algebra. A subset F of E is called an EQ-filter(filter for short) of E if for all a, b, cE we have that:

1. 1 ∈ F;

2. if a, abF; then bF,

3. if abF, then acbcF and cacbF.

#### Remark 2.5

Note that Definition 2.4 differs from the original definition of filters (see [2, Definition 4]). In Definition 2.4, we do not need this condition: (ii)′ if a, bF, then abF (see [2, Definition 4]) because it follows from the other conditions. In fact, let F be a filter of an EQ-algebra E. First, we show that F satisfies the condition that if xF and xy, then yF. From xy = x it follows that xy = 1. By Definition 2.4 (i) and (ii), it follows that yF. Let a, bF. From Proposition 2.2 (vii), it follows that b ≤ 1 → b. From Definition 2.4 (iii), it then follows that (a ⊗ 1) → (ab) ∈ F. Hence, by Definition 2.4 (i) and (ii), abF.

#### Proposition 2.6

([2, 14]). Let F be a filter of an EQ-algebra E. For all a, b, a′ ,b′, c, e, 𝑓 ∈ F such that ab and a′b′F, the following holds:

1. if eF and e ≤ 𝑓, then 𝑓 ∈ F;

2. if e, e ∼ 𝑓 ∈ F, then 𝑓 ∈ F;

3. abF, (ab) ⊗ (bc) ∈ F, where ab := (ab) ∧ (ba);

4. (aa′) ∼ (bb′) ∈ F;

5. (ac) ∼ (bc) ∈ F and (ca) ∼ (cb) ∈ F,

6. (aa′) ∼ (bb′) ∈ F.

As is usually done, given a filter F of an EQ-algebra E, we can define a binary relation on E by $a≡Fb if and only if a∼b ∈ F.$

From Proposition 2.6, we immediately have the following theorem.

#### Theorem 2.7

([2, 14]). Let F be a filter of an EQ-algebra E. The relationF is a congruence relation on E.

#### Definition 2.8

([15]). A poset (D, ≤) is called an upward directed set if for any x,yD there exists zD such that xz and yz.

We recall some basic notions of general topology which will be needed in the sequel.

Recall that a set A with a family 𝒯 of its subsets is called a topological space, denoted by (A, 𝒯), if A, ∅ ∈ 𝒯, the intersection of any finite number of the 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, AU, is a closed set. A subfamily {Uα }α∈I of 𝒯 is called a base of 𝒯 if for each xU ∈ 𝒯 there is an α ∈ I such that xUαU .A subset P of A is a neighborhood of xA, if there exists an open set U such that xUP. Let 𝒯x denote the totality of all neighborhoods of x in A, then subfamily 𝓥x of 𝒯x is a fundamental system of neighborhoods of x, if for each Ux in 𝒯x, there exists a Vx in Vx such that VxUx. If every point x in A has a countable fundamental system of neighborhoods, then we say that the space (A, 𝒯) satisfies the first axiom of countability or is first-countable. A topological space (A, 𝒯) is a zero-dimensional space if 𝒯 has a clopen base. A topological space (A, 𝒯) is called a regular space if for any closed subset C of A and xA such that xC, then there exist disjoint open sets U, V such that xU and CV, or equivalently, for any open subset U containing x, there exists open subset V such that xVU .A topological space (A, 𝒯) is called a completely regular space, if for every xX and every closed set FA such that xF there exists a continuous function 𝑓 : A → [0,1] such that 𝑓(x) = 0 and 𝑓(y) = 1 for yF. Let (A, 𝒯) and (B, 𝓥) be two topological spaces, a mapping 𝑓 of A to B is continuous if 𝑓−1(U) ∈ 𝒯 for any U ∈ 𝓥. The mapping 𝑓 from (A, 𝒯) to (B, 𝓥) is called a homeomorphism if 𝑓 is bijective, and 𝑓 and 𝑓−1 are continuous, or equivalently, if 𝑓 is bijective, continuous and open (closed). The mapping 𝑓 from (A, 𝒯) to (B, 𝓥) is called a quotient map if 𝑓 is surjective, and V ∈ 𝓥 if and only if 𝑓−1(V) ∈ 𝒯. A topological space (A, 𝒯) is compact if each open cover of A is reducible to a finite subcover, and locally compact if for every xA there exists a neighborhood U of x such that Ū is a compact subspace of A.

Let (X, 𝒯) be a topological space. We have following separation axioms in (X, 𝒯):

1. T0 : For each x, yX and xy, there is at least one of them has a neighborhood excluding the other.

2. T1 : For each x, yX and xy each has neighborhood not containing the other.

3. T2 : For each x, yX and xy both have disjoint neighborhoods U, V ∈ τ such that xU and yV.

A topological space satisfying Ti is called a Ti-space, for any i = 0, 1, 2. A T2-space is also known as a Hausdorff space.

Let (A, ∗) be an algebra of type 2 and 𝒯 be a topology on A. Then (A, ∗, 𝒯) is called a left (right) topological algebra, if for all aA the map ∗ : AA is defined by xax (xxa) is continuous, or equivalently, for any xA and any open subset V containing ax(xa) there exists an open subset W containing x such that aWV (WaV). A right and left topological algebra (A, ∗, 𝒯) is called a semitopological algebra. Moreover, if the operation ∗ is continuous, or equivalently, for each x, yA and each open subset W containing xy, there exist two open subsets V1 and V2 containing x and y respectively, such that V1V2W, then (A, ∗, 𝒯) is called a topological algebra.

## 3 Uniformity in EQ-algebras

From now on, we write E instead of the EQ-algebra < E, ∧, ⊗, ∼, 1 > for convenience, unless otherwise stated.

Let X be a nonempty set and U, V be any subsets of X × X. We have the following notation:

1. UV = {(x, y) ∈ X × X : (x, z) ∈ U, (y, z) ∈ V, for some zX};

2. U−1 = {(x, y) ∈ X × X : (y, x) ∈ U};

3. Δ = {(x, x) ∈ X × X : xX}.

#### Definition 3.1

([16]). By a uniformity on X we shall mean a nonempty collection 𝓚 of subsets of X × X which satisfies the following conditions:

(U1) Δ ⊆ U for any U ∈ 𝓚;

(U2) if U ∈ 𝓚, then U−1 ∈ 𝓚;

(U3) if U ∈ 𝓚, then there exists V ∈ 𝓚 such that VVU;

(U4) if U, V ∈ 𝓚, then UV ∈ 𝓚;

(U5) if U ∈ 𝓚 and UVX × X, then V ∈ 𝓚.

The pair (X, 𝓚) is then called a uniform structure(uniform space) on X.

In the following we use the filters of EQ-algebras to induce uniform structures.

#### Theorem 3.2

Letbe an arbitrary family of filters of E which is closed under intersection. If UF = {(x, y) ∈ E × E : xF y} and 𝓚 = {UF : F ∈ ∧}, then 𝓚 satisfies conditions (U1)-(U4).

#### Proof

(U1): Since F is a filter of E, we have xF x, for any xE. Hence Δ ⊆ UF for all UF ∈ 𝓚.

(U2): For any UF ∈ 𝓚, we have $(x,y)∈(UF)−1⇔(y,x)∈UF⇔y≡Fx⇔x≡Fy⇔(x,y)∈UF.$

(U3): For any UF ∈ 𝓚, the transitivity of ≡F implies that UFUFUF.

(U4): For any UF, UJ ∈ 𝓚, we claim that UFUJ = UFJ. If (x, y) ∈ UFUJ, then xF y and xJ y. Hence xyF and xyJ. Then xyFJ and so (x, y) ∈ UFJ. Conversely, let (x, y) ∈ UFJ. Then xFJ y, hence xyFJ, and thus xyF, xyJ. Therefore xF y and xJ y, which means that (x, y) ∈ UFUJ. So UFUJ = UFJ. Since F, J ∈ ∧, then FJ ∈ ∧ and so UFUJ ∈ 𝓚. □

#### Theorem 3.3

Let 𝓚 = {UE × E : ∃ UF ∈ 𝓚 s.t. UFU }, where 𝓚 comes from Theorem 3.2. Then 𝓚 satisfies a uniformity on E and the pair (E, 𝓚) is a uniform structure.

#### Proof

By Theorem 3.2, the collection 𝓚 satisfies the conditions (U1)-(U4). It suffices to show that 𝓚 satisfies (U5). Let U ∈ 𝓚 and UVE × E. Then there exists UFUV, which means that V ∈ 𝓚. □

Let xE and U ∈ 𝓚. Define U[x] := {yE : (x, y) ∈ U}. Clearly, if VU, then V[x] ⊆ U[x].

#### Theorem 3.4

Let E be an EQ-algebra. Then $T={G⊆E:(∀x∈G)(∃U∈K)s.t.U[x]⊆G}$ is a topology on E, where K comes from Theorem 3.3.

#### Proof

Clearly, ∅ and the set E belong to 𝒯. It is clear that 𝒯 is closed under arbitrary union. Finally to show that 𝒯 is closed under finite intersection, let G, H ∈ 𝒯 and suppose that xGH. Then there exist U, V ∈ 𝓚 such that U[x] ⊆ G and V[x] ⊆ H. If W = UV, then W ∈ 𝓚. Also W[x] ⊆ U[x] ∩ V[x] and so W[x] ⊆ GH, hence GH ∈ 𝒯. Thus 𝒯 is a topology on E. □

Note that for any x in E, U[x] is a neighborhood of x.

#### Definition 3.5

Letbe an arbitrary family of filters of an EQ-algebra E which is closed under intersection. Then the topology 𝒯 comes from Theorem 3.4 is called a uniform topology on E induced by ∧.

We denote the uniform topology 𝒯 obtained from an arbitrary family of filters ∧ by 𝒯Λ, and if ∧ = {F}, we denote it by 𝒯F.

#### Example 3.6

Let E = {0, a, b, 1} be a chain with Cayley tables as follows:

We can easily check that < E, ∧, ⊗, ∼, 1 > is an EQ-algebra. Consider the filter F = {b, 1}, and ∧ = {F}. Therefore as in Theorem 3.2, we construct 𝓚 = {UF} = {{(x, y) : xF y}} = {{(0,0), (a, a), (b, b), (b, 1), (1, b), (1, 1)}}. We can check that (E, 𝓚) is a uniform space, where 𝓚 = {U : UFUF}. Neighborhoods are UF [0] = {0}, UF[a] = {a}, UF [b] = {b, 1}, UF [1] = {b, 1}. From above we get that 𝒯F = {∅, {0}, {a}, {b, 1}, {0, a}, {0, b, 1}, {a, b, 1}, {0, a, b, 1}}. Thus (E, 𝓣F) is a uniform topological space.

## 4 Topological properties of the space (E, 𝒯Λ)

Note that from Theorem 3.4 giving the ∧ family of filters of an EQ-algebra E which is closed under intersection. We can induce a uniform topology 𝒯Λ on E. In this section we study topological properties on (E, 𝒯Λ).

Let E be an EQ-algebra and C, D be subsets of E. Then we define CD as follows: CD = {xy : xC, yD}, where ∗ ∈ {∧, ⊗, ∼}.

#### Definition 4.1

Let 𝓤 be a topology on E. Then (E, 𝓤) is called a topological EQ-algebra(TEQ-algebra for short) if the operations ∧, ⊗ andare continuous with respect to 𝓤.

Recall that a topological space (X, 𝓤) is a discrete space if for any xX, {x} is an open set.

#### Example 4.2

Every EQ-algebra with a discrete topology is a TEQ-algebra.

#### Theorem 4.3

The pair (E, 𝒯Λ) is a TEQ-algebra.

#### Proof

By Definition 4.1, it suffices to show that ∗ is continuous, where ∗ ∈ {∧, ⊗, ∼}. Indeed, assume that xyG, where x, yE and G is an open subset of E. Then there exist U ∈ 𝓚, U[xy] ⊆ G, and a filter F such that UF ∈ 𝓚 and UFU. We claim that the following relation holds: $UF[x]∗UF[y]⊆UF[x∗y]⊆U[x∗y].$

Let hkUF [x]∗UF [y]. Then hUF [x] and kUF [y] we get that xF h and yF k. Hence xyF hk. From that we obtain (xy, hk) ∈ UFU. Hence hkUF [xy] ⊆ U[xy]. Then hkG. Clearly, UF [x] and UF [y] are neighborhoods of x and y, respectively. Therefore, the operation ∗ is continuous. □

#### Example 4.4

In Example 3.6, it is easy to check that (E, 𝒯F) is a TEQ-algebra.

#### Theorem 4.5

Letbe a family of filters of E which is closed under intersection. Any filter in the collectionis a clopen subset of E for the topology 𝒯Λ.

#### Proof

Let F be a filter of E in ∧ and yFc. Then yUF [y] and we get Fc ⊆ ∪{UF[y] : yFc}. We claim that for all yFc, UF [y] ⊆ Fc. If zUF [y], then zF y. Hence zyF .If zF, by Lemma 2.6 (i), we get that yF, which is a contradiction. So zFc and we get ∪{UF [y] : yFc} ⊆ Fc. Hence Fc = ∪{UF [y] : yFc}. Since UF [y] is open for all yE, it follows that F is a closed subset of E. We show that F = ∪{UF [y] : yF}. If yF, then yUF [y] and we get F ⊆ ∪{UF [y] : yF}. Let yF .If zUF [y], then zF y and so yzF. Since yF, by Lemma 2.6 (i), zF, and we get ∪{UF [y] : yF} ⊆ F .So F is also an open subset of E. □

#### Theorem 4.6

Letbe a family of filters of E which is closed under intersection. For any xE and F ∈ ∧, UF [x] is a clopen subset of E for the topology 𝒯Λ.

#### Proof

First we show that (UF [x])c is open. If y ∈ (UF [x])c, then yxFc. We claim that UF [y] ⊆ (UF [x])c. If zUF [y], then z ∈ (UF [x])c, otherwise zUF [x], we get that zyF and zxF. Since F is a filter, we get that (xz) ⊗ (zy) ∈ F. By (xz) ⊗ (zy) < xy and F is a filter, it follows that xyF, which is a contradiction. Hence UF [y] ⊆ (UF [x])c for all y ∈ (UF [x])c, and so UF [x] is closed. It is clear that UF [x] is open. So UF [x] is a clopen subset of E. □

A topological space X is connected if and only if X has only X and ∅ as clopen subsets. Therefore we have the following corollary.

#### Corollary 4.7

The space (E, 𝒯Λ) is not, in general, a connected space.

#### Proof

It clearly follows from Theorem 4.6. □

#### Theorem 4.8

𝒯 = 𝒯J, where J = ∩{F : F ∈ ∧}.

#### Proof

Let 𝓚 and 𝓚 be as in Theorems 3.2 and 3.3. Now consider ∧0 = {J}, define (𝓚0) = {UJ} and 𝓚0 = {U : UJU}. Let G ∈ 𝒯Λ. So for each xG, there is U ∈ 𝓚 such that U[x] ⊆ G. From JF we get that UJUF for any filter F of ∧. Since U ∈ 𝓚, there exists F ∈ ∧ such that UFU. Hence UJ [x] ⊆ UF [x] ⊆ G. Since UJ ∈ 𝓚0, we get that G ∈ 𝒯J. So 𝒯Λ ⊆ 𝒯J. Conversely, let H ∈ 𝒯J. Then for any xH there is U ∈ 𝓚0 such that U[x] ⊆ H. Hence UJ [x] ⊆ H. Since ∧ is closed under intersection, so J ∈ ∧. Then we get UJ ∈ 𝓚 and so H ∈ 𝒯Λ. Therefore, 𝒯J ⊆ 𝒯Λ. □

#### Corollary 4.9

Let F and J be filters of E and FJ. Then J is clopen in the topological space (E, 𝒯F).

#### Proof

Consider ∧ = {F, J}. Then by Theorem 4.8, 𝒯Λ = 𝒯F. Hence by Theorem 4.5, J is clopen in the topological space (E, 𝒯F). □

#### Remark 4.10

Letbe a family of filters of E which is closed under intersection and J = ∩{F : F ∈ ∧}. We have the following statements:

1. By Theorem 4.8, we know that 𝒯Λ = 𝒯J. For any U ∈ 𝓚, xE, we can get that UJ [x] ⊆ U[x]. Hence 𝒯Λ is equivalent to {AE : ∀xA, UJ [x] ⊆ A}. So AE is open set if and only if for all xA, UJ [x] ⊆ A if and only if A = ∪xA UJ [x];

2. For all xE, by (i), we know that UJ [x] is the smallest neighborhood of x;

3. Let 𝓑J = {UJ [x] : xE}. By (i) and (ii), it is easy to check that 𝓑J is abase of 𝒯J;

4. For all xE, {UJ [x]} is a denumerable fundamental system of neighborhoods of x.

#### Lemma 4.11

If F is a filter of E, then for all xE, UF [x] is a clopen compact set in the topological space (E, 𝒯F).

#### Proof

By Theorem 4.6, it is enough to show that UF [x] is a compact set. Let UF [x] ⊆ ∪α∈I Oα, where each Oα is an open set of E. Since xUF [x], there exists α ∈ I such that xOα. Then UF [x] ⊆ Oα. Hence UF [x] is compact. Therefore UF [x] is a clopen compact set in the topological space (E, 𝒯F). □

#### Theorem 4.12

Letbe a family of filters of E which is closed under intersection. Then (E, 𝒯Λ) is a first-countable, zero-dimensional, disconnected and completely regular space.

#### Proof

By Theorem 4.8, it is suffices to show that (E, 𝒯J) is a first-countable, zero-dimensional, disconnected and completely regular space. Let xE. By Remark 4.10 (iv), {UJ [x]} is a denumerable fundamental system of neighborhoods of x, so (E, 𝒯J) is first-countable. Let 𝓑J = {UJ [x] : xE}. By Remark 4.10 (iii) and Theorem 4.6, we get that BJ is a clopen basis of (E, 𝒯J), hence (E, 𝒯J) is a zero-dimensional space. By Corollary 4.7, we get that (E, 𝒯J) is a disconnected space. By Lemma 4.11 and Remark 4.10 (ii), UJ [x] is a compact neighborhood of x. Hence (E, 𝒯J) is a locally compact space. Let xE and V be a neighborhood of x. By Remark 4.10 (ii) and Lemma 4.11, there exists closed neighborhood UJ [x] of x such that UJ [x] ⊆ V. Therefore, (E, 𝒯J) is a regular space. Since (E, 𝒯J) is a locally compact space, we get that it is completely regular. □

#### Theorem 4.13

Letbe a family of filters of E which is closed under intersection. Then (E, 𝒯Λ) is a discrete space if and only if there exists F ∈ ∧ such that UF [x] = {x} for all xE.

#### Proof

Let 𝒯Λ be a discrete topology on E. If for any F ∈ ∧, there exists xE such that UF [x] ≠ {x}. Let J = ∩∧. Then J ∈ ∧, there exists x0E such that UJ [x0] ≠ {x0}. It follows that there exists y0UF [x0] and x0y0. By Remark 4.10 (ii), UJ [x0] is the smallest neighborhood of x0. Hence {x0} is not an open subset of E, which is a contradiction. Conversely, for any xE, there exists F ∈ ∧ such that UF [x] = {x}. Hence {x} is an open set of E. Therefore, (E, 𝒯Λ) is a discrete space. □

#### Theorem 4.14

Letbe a family of filters of E which is closed under intersection, J = ∩∧ and E be a separated EQ-algebra. Then the following conditions are equivalent:

1. (E, 𝒯J) is a discrete space;

2. J = {1}.

#### Proof

1. ⇒ (ii): By Theorem 4.13, we have UJ [1] = {1}. We show that JUJ[1]. Let xJ. By Proposition 2.2 (vii), we get that xx ∼ 1. Since J is a filter and xJ, hence x ∼ 1 ∈ J. So xUJ [1]. It follows that JUJ [1]. Since UJ [1] = {1} and 1 ∈ J. Therefore, J = {1}.

2. ⇒ (i): Let J = {1}. Since E is separated, we can get that UJ [x] = {x}. It follows that (E, 𝒯J) is discrete. □

#### Corollary 4.15

Letbe a family of filters of E which is closed under intersection, J = ∩∧ and E be a separated EQ-algebra. Then (E, 𝒯J) is a Hausdorff space if and only if J = {1}.

#### Proof

Let (E, 𝒯J) be a Hausdorff space. First we show that for any xE, UJ [x] = {x}. If there exists xyUJ [x], then yUJ [x] ∩ UJ [y]. By Remark 4.10 (ii), UJ [x] and UJ [y] are the smallest neighborhoods of x and y, respectively. Hence for any neighborhood U of x and neighborhood V of y, we have that yUJ [x] ∩ UJ [y] ⊆ UV ≠ ∅, which is a contradiction. Hence by Theorems 4.13 and 4.14, J = {1}. The other side of the proof directly follows from Theorem 4.14. □

#### Definition 4.16

Let E1 and E2 be EQ-algebras. A mapping φ : E1E2 is called an EQ-morphism from E1 to E2 if $φ(x∗y)=φ(x)∗φ(y)$ for any ∗ ∈ {∧, ⊗, ∼}. If, in addition, the mapping φ is bijective, then we call φ an EQ-isomorphism. Note that φ(1) = 1 when φ is an EQ-morphism.

#### Proposition 4.17

Let φ : E1E2 be an EQ-morphism. Then the following properties hold:

1. if F is a filter of E2, then the set φ−1(F) is a filter of E1;

2. if φ is surjective and F is a filter of E1, then φ(F) is a filter of E2.

#### Proof

It is easy to prove by definition of filters. □

#### Lemma 4.18

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1E2 is an EQ-isomorphism, then $(a,b)∈Uφ−1(F) if and only if (φ(a),φ(b))∈UF,for any a,b∈E.$

#### Proof

For any (a, b) ∈ Uφ−1(F)abφ−1(F) ⇔ φ(a) ∼ φ(b) ∈ F ⇔ (φ(a), φ(b)) ∈ UF. □

#### Theorem 4.19

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1E2 is an EQ-isomorphism, then the following properties hold:

1. for any aE1, φ(Uφ−1(F) [a]) = UF [φ(a)];

2. for any bE2, φ−1(UF[b]) = Uφ−1(F)[φ−1(b)].

#### Proof

(i) Let bφUφ−1(F)[a]). Then there exists cUφ−1(F)[a] such that b = φ(c). It follows that acφ−1(F) ⇒ φ(a) ∼ φ(c) ∈ Fφ(a) ∼ bFbUF [φ(a)].

Conversely, bUF [φ(a)] ⇒ φ(a) ∼ bFφ−1(φ(a) ∼ b) ∈ φ−1(F) ⇒ aφ−1(b) ∈ φ−1(F) ⇒ φ−1(b) ∈ Uφ−1(F)[a] ⇒ bφ(Uφ−1(F)[a]).

(ii) aφ−1(UF[b]) ⇔ φ(a) ∈ UF[b] ⇔ φ(a) ∼ bFφ−1(φ(a) ∼ b) ∈ φ−1(F) ⇔ aφ−1(b) ∈ φ−1(F) ⇔ aUφ−1(F)[φ−1(b)]. □

#### Theorem 4.20

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1E2 is an EQ-isomorphism, then φ is a continuous map from (E1, 𝒯φ−1(F)) to (E2, 𝒯F).

#### Proof

Let A ∈ 𝒯F. By Remark 4.10 (i), we can get that A = ∪aA UF [a]. It follows that φ−1(A) = φ−1(∪aA UF [a]) = ∪aA φ−1(UF [a]). We claim that if bφ−1 (UF [a]), then Uφ−1(F)[b] ⊆ φ−1(UF [a]). Indeed, let cUφ−1(F)[b], we get that cbφ−1 (F), so φ(c) ∼ φ(b) ∈ F. Since φ(b) ∈ UF [a], we get that φ(b) ∼ aF. It follows that φ(c) ∼ aF. Thus we have that φ(c) ∈ UF [a]. So cφ−1(UF [a]). Hence φ−1(UF [a]) = ∪bφ−1(UF[a]) Uφ−1(F) [b] ∈ 𝒯φ−1(F). Therefore φ−1(A) = ∪aA φ−1 (UF [a]) ∈ 𝒯φ−1(F). So φ is a continuous map. □

#### Theorem 4.21

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1E2 is an EQ-isomorphism, then φ is a quotient map from (E1, 𝒯φ−1(F)) to (E2, 𝒯F).

#### Proof

From Theorem 4.20 we get that φ is a continuous surjective map. It is enough to show that φ is an open map. Let A be an open set of (E1, 𝒯φ−1(F)). We claim that φ(A) is an open set of (E2, 𝒯F). Let aφ(A). We shall show that UF [a] ⊆ φ(A). Indeed, for any bUF [a], we get that baF. By Lemma 4.15, we have φ−1(a) ∼ φ−1(b) ∈ φ−1(F). Hence φ−1(b) ∈ Uφ−1(F)[φ(a)]. Since aφ(A) and φ is injective we get that φ−1(a) ∈ A. By Remark 4.10 (i), it follows that Uφ−1(F)[φ−1(a)] ⊆ A. So φ−1(b) ∈ A, we get that bφ(A). Therefore, UF [a] ⊆ φ(A). So φ is a quotient map. □

#### Corollary 4.22

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1E2 is an EQ-isomorphism, then φ is a homeomorphism map from (E1, 𝒯φ−1(F)) to (E2, 𝒯F).

#### Proof

It clearly follows from Theorem 4.21. □

Recall that a uniform space (X, 𝓚) is totally bounded if for each U ∈ 𝓚, there exist x1,..., x1X such that $X={\cup }_{i=1}^{n}U\left[{x}_{i}\right]$.

#### Theorem 4.23

Let F be a filter of E. Then the following conditions are equivalent:

1. the topological space (E, 𝒯F) is compact;

2. the topological space (E, 𝒯F) is totally bounded;

3. there exists P = {x1, ... , xn} ⊆ E such that for all aE there exists xiP such that aF xi.

#### Proof

1. ⇒ (2): The proof is straightforward.

2. ⇒ (3): Since (E, 𝒯F) is totally bounded, there exist x1,..., xnE such that $E={\cup }_{i=1}^{n}{U}_{F}\left[{x}_{i}\right]$. Now let aE. Then there exists xi, such that aUF [xi], therefore axi, ∈ F i.e. aF xi.

3. ⇒ (1): For any aE, by hypothesis, there exists xiP such that axiF. We can get that a ∈ ∪F [xi], hence $E={\cup }_{i=1}^{n}{U}_{F}\left[{x}_{i}\right]$. Now let E = ∪α∈I Oα, where each Oα is an open set of E. Then for any xiE there exists αiI such that xi, ∈ Oαi. Since Oαi is an open set, UF [xi] ⊆ Oαi, so we have that $E=∪i∈IUF[xi]⊆∪i=1nOαi.$

Therefore $E={\cup }_{i=1}^{n}{O}_{{\alpha }_{i}}$, whence (E, 𝒯F) is compact.

□

#### Theorem 4.24

If F is a filter of E such that Fc is a finite set, then the topological space (E, 𝒯F) is compact.

#### Proof

Let E = ∪α∈I Oα, where each Oα is an open subset of E. Let Fc = {x1,... ,xn}. Then there exist α, α1,...,αnI such that 1 ∈ Oα, x1Oα1,...,xnOαn. Then UF [1] ⊆ Oα, but UF [1] = F. Hence $E={\cup }_{i=1}^{n}{O}_{{\alpha }_{i}}\cup {O}_{\alpha }$. □

#### Theorem 4.25

If F is a filter of E, then F is a compact set in the topological space (E, 𝒯F).

#### Proof

Let F ⊆ ∪αI Oα, where each Oα is open set of E. Since 1 ∈ F, there is α ∈ I such that 1 ∈ Oα. Then F = UF [1] ⊆ Oα. Hence F is a compact set in the topological space (E, 𝒯F). □

Our next target is to establish the convergence of EQ-algebras using the convergence of nets.

#### Definition 4.26

Let E be an EQ-algebra and (D, ≤) be an upward directed set. If for any α ∈ D we have aαE, then we call {aα}α∈D a net of E.

#### Definition 4.27

Let {aα}α∈D be a net of E. In the topological space (E, 𝒯F), say that {aα}α∈D

1. converges to the point a of E if for any neighborhood U of a, there exists d0D such that aαU for any α ≥ d0;

2. Cauchy sequence if there exists d0D such aαF aβ for any α, βd0.

A net {aα}iD, which converges to a is said to be convergent. For simplicity, we write limaα = a and we say that a is a limit of {aα}α∈D.

#### Example 4.28

Consider the TEQ-algebra (E, 𝒯F) in Example 4.4. Clearly, (ℕ, ≤) is an upward directed set, whereis a natural number set. We define {an}n∈ℕ as a0 = 0, a1 = a, a2 = b, an = 1, n ≥ 3. It is easy to check that {an}n∈ℕ is a net of E. Let n0 = 3. For any neighborhood U of 1, if n ≥ 3, then 1 ∈ U. Therefore, liman = 1.

#### Theorem 4.29

Let {aα}α∈D and {bα}α∈D be nets of E and F be a filter of E. Then in the topological space (E, 𝒯F) we have:

1. if limbα = b and limaα = a, for some a, bE, then the sequence {aαbα}α∈D is convergent and limaαbα = ab, for any operation ∗ ∈ {∧, ⊗, ∼};

2. any convergent sequence of E is a Cauchy sequence.

#### Proof

1. Let limaα = a, limbα = b and ∗ ∈ {∧, ⊗, ∼}, for some a, bE. For any neighborhood W of ab we get that UF [ab] ⊆ W. Clearly, UF [a] and UF [b] are neighborhoods of a and b, respectively. By hypothesis, there exist d1, d2D such that aαUF [a] and bαUF [b], for any α ≥ d1 and α ≥ d2. Since D is an upward directed set, then there exists d0D such that d0d1 and d0d2. By Theorem 4.3, we get that UF [a] ∗ UF [a] ⊆ UF [ab]. So aαUF [a] and bαUF [b], for any α ≥ d0. It follows that abUF [a] ∗ UF [a] ⊆ UF [ab] ⊆ U, for any α ≥ d0D. Therefore, lim aαbα = ab.

2. Let {aα}α∈D be a net of E and limaα = a. For the neighborhood UF [a] of a, there exists dD such that aαD, for any α ≥ d. So if α, βd, then aα, aβUF [a] that is aαF a and aβF a. It follows that aαF aβ. Therefore, {aα}α∈D is a Cauchy sequence.

□

## 5 Conclusion

It is well known that EQ-algebras play an important role in investigating the algebraic structures of logical systems. In this study, we endowed an EQ-algebra with uniform topology 𝒯Λ and proposed the concept of the topological EQ-algebra. We then stated and proved special properties of (E, 𝒯Λ). Especially, we proved that (E, 𝒯Λ) is a first-countable, zero-dimensional, disconnected and completely regular space. From the category point of view, the role of isomorphism in algebra is the same as the role of homeomorphism in topology. Hence we also studied the relationship between isomorphism(algebraic invariant) and homeomorphism(topological invariant) in topological EQ-algebras. Finally, we investigated the convergent properties of topological EQ-algebras.

## Acknowledgement

The authors thank the editors and the anonymous reviewers for their valuable suggestions in improving this paper. This research is supported by a grant of National Natural Science Foundation of China (11571281), Postdoctoral Science Foundation of China (2016M602761) and the Fundamental Research Funds for the Central Universities (GK201603004).

## References

• [1]

Novák V., EQ-algebras: primary concepts and properties, In: Proc. Czech-Japan Seminar, Ninth Meeting(August 2006, Kitakyushu and Nagasaki, Graduate School of Information, Waseda University, Japan), Kitakyushu and Nagasak, 2006, 18-22 Google Scholar

• [2]

Novák V., De Baets B., EQ-algebras, Fuzzy Sets Syst., 2009, 160, 2956-2978

• [3]

Borzooei R.A., Saffar B.G., States on EQ-algebras, J Intell Fuzzy Syst., 2015, 29, 209-211

• [4]

Dyba M., El-Zekey M., Novak V., Non-commutative first-order EQ-algebras, Fuzzy Sets Syst., 2016, 292, 215-241

• [5]

Liu L.Z., Zhang X.Y., Implicative and positive prefilters of EQ-algebras, J Intell Fuzzy Syst., 2014, 26, 2087-2097

• [6]

Haveshki M., Eslami E., Saeid A.B., A topology induced by uniformity on BL-algebras, Math. Log. Q., 2007, 53, 162-169

• [7]

Ghorbani S., Hasankhani A., Some properties of quotient topology on residuated lattices, Pure Math Appl., 2010, 21, 15-26 Google Scholar

• [8]

Hoo C.S., Topological MV-algebras, Topol. Appl., 1997, 81, 103-121 Google Scholar

• [9]

Nganou J.B., Tebu S.F.T., Topological FLew-algebras, J Appl Log., 2015, 13, 259-269

• [10]

Ciungu L.C., Convergences in perfect BL-algebras, Mathw Soft Comput., 2007, 14, 67-80 Google Scholar

• [11]

Ko J.M, Kim Y.C., Closure operators on BL-algebras, Commun. Korean Math. Soc., 2004, 19, 219-232

• [12]

Borzooei R.A., Rezaei G.R., Kuhestani N., Separation axioms in (semi)top-ological quotient BL-algebras, Soft Comput., 2012, 16, 1219-1227

• [13]

Borzooei R.A., Rezaei G.R., Metrizability on (semi)topological BL-algebras, Soft Comput., 2012, 16, 1681-1690

• [14]

El-Zekey M., Novák V., Mesiar R., On good EQ-algebras, Fuzzy Sets Syst., 2011, 178, 1-23

• [15]

Davey B.A., Priestley H.A., Introduction to Lattice and Order, Cambridge University Press, U.K., 2002 Google Scholar

• [16]

Joshi K.D., Introduction to General Topology, New Age International Publisher, India, 1983 Google Scholar

## About the article

Received: 2016-09-17

Accepted: 2017-01-26

Published Online: 2017-04-14

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 354–364, ISSN (Online) 2391-5455,

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© 2017 Yang et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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