## 1 Introduction

*EQ*-algebras were proposed by NovĆ”k [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 NovĆ”k 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. Hooās 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 Hooās work to *FL*_{ew}-algebras. They considered a similar approach to study *FL*_{ew}-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.

([2, 14]). *An* *EQ*-algebra *is an algebra* *E* = (*E*, ā§, ā, ā¼, 1) *of type (2,2,2,0) satisfying the following axioms*:

(E1) (*E*, ā§, 1) *is a* ā§-*semilattice with top element 1. We put* *x* ā¤ *y* *if and only if* *x* ā§ *y* = *x;*

(E2) (*E*, ā, 1) *is a monoid and* ā *is isotone in both arguments with respect to* ā¤;

(E3) *x* ā¼ *x* = 1, (*reflexivity axiom*);

(E4) ((*x* ā§ *y*) ā¼ *z*) ā (*t* ā¼ *x*) ā¤ *z* ā¼ (*t* ā§ *y*), (*substitution axiom*);

(E5) (*x* ā¼ *y*) ā (*z* ā¼ *t*) ā¤ (*x* ā¼ *z*) ā¼ (*y* ā¼ *t*), (*congruence axiom*);

(E6) (*x* ā§ *y* ā§ *z*) ā¼ *x* ā¤ (*x* ā§ *y*) ā¼ *x*, (*monotonicity axiom*);

(E7) *x* ā *y* ā¤ *x* ā¼ *y*. (*boundedness axiom*).

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

([2, 14]). *Let E be an EQ-algebra, x* ā *y* := (*x* ā§ *y*) ā¼ *x* *and* *xĢ* := *x* ā¼ 1. *Then the following properties hold for all x, y, z* ā *E*:

*x*ā*y*ā¤*x*ā§*y*ā¤*x*,*y*;*z*ā (*x*ā§*y*) ā¤ (*z*ā*x*) ā§ (*z*ā*y*);*x*ā¼*y*ā¤*x*ā*y*;*x*ā*x*= 1;- (
*x*ā¼*y*) ā (*y*ā¼*z*) ā¤*x*ā¼*z*; - (
*x*ā*y*) ā (*y*ā*z*) ā¤*x*ā*z*; *x*ā¤*xĢ*. 1Ģ = 1;*x*ā (*x*ā¼*y*) ā¤*yĢ*;- (
*x*ā*y*) ā (*y*ā*x*) ā¤*x*ā¼*y*ā¤ (*x*ā*y*) ā§ (*y*ā*x*); *if**x*ā¤*y*ā*z*,*then**x*ā*y*ā¤*zĢ*;*if**x*ā¤*y*<*z*,*then**z*ā¼*x*ā¤*z*ā¼*y**and**x*ā¼*z*ā¤*x*ā¼*y*.

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

- separated
*if for all a, b*ā*E*,*a*ā¼*b*= 1*implies a=b*; - good
*if for all a*ā*E*,*a*ā¼ 1 =*a*.

([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, c* ā *E we have that*:

- 1 ā
*F*; *if**a, a*ā*b*ā*F*;*then b*ā*F*,*if a*ā*b*ā*F*,*then a*ā*c*ā*b*ā*c*ā*F and c*ā*a*ā*c*ā*b*ā*F*.

*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)ā² if a, b* ā *F, then a* ā *b* ā *F (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 x* ā *F and x* ā¤ *y, then y* ā *F*. *From x* ā§ *y* = *x it follows that x* ā *y* = 1. *By Definition 2.4 (i) and (ii), it follows that y* ā *F. Let a, b* ā *F. From Proposition 2.2 (vii), it follows that b* ā¤ 1 ā *b. From Definition 2.4 (iii), it then follows that* (a ā 1) ā (*a* ā *b*) ā *F. Hence, by Definition 2.4 (i) and (ii), a* ā *b* ā *F*.

([2, 14]). *Let F be a filter of an EQ-algebra E. For all a, b, aā² ,bā², c, e*, š ā *F such that a* ā¼ *b and aā²* ā¼ *bā²* ā *F, the following holds*:

*if e*ā*F and e*ā¤ š,*then*š ā*F*;*if e, e*ā¼ š ā*F*,*then*š ā*F*;*a*ā*b*ā*F*, (*a*ā*b*) ā (*b*ā*c*) ā*F, where a*ā*b*:= (*a*ā*b*) ā§ (*b*ā*a*);- (
*a*ā§*aā²*) ā¼ (*b*ā§*bā²*) ā*F*; - (
*a*ā*c*) ā¼ (*b*ā*c*) ā*F and*(*c*ā*a*) ā¼ (*c*ā*b*) ā*F*, - (
*a*ā¼*aā²*) ā¼ (*b*ā¼*bā²*) ā*F*.

As is usually done, given a filter *F* of an *EQ*-algebra *E*, we can define a binary relation on *E* by

From Proposition 2.6, we immediately have the following theorem.

([2, 14]). *Let F be a filter of an EQ-algebra E. The relation* ā”_{F} *is a congruence relation on E*.

([15]). *A poset* (*D*, ā¤) *is called an* upward directed set *if for any x,y* ā *D there exists z* ā *D such that x* ā¤ *z and y* ā¤ *z*.

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

Recall that a set *A* with a family šÆ of its subsets is called a *topological space*, denoted by (*A*, šÆ), if *A*, ā
ā šÆ, the intersection of any finite number of the members of šÆ is in šÆ, and the arbitrary union of members of šÆ is in šÆ. The members of šÆ are called *open sets* of *A*, and the complement of an open set *U*, *A* ā *U*, is a *closed set*. A subfamily {*U*_{Ī±} }_{Ī±āI} of šÆ is called a *base* of šÆ if for each *x* ā *U* ā šÆ there is an Ī± ā *I* such that *x* ā *U*_{Ī±} ā *U* .A subset *P* of *A* is a *neighborhood* of *x* ā *A*, if there exists an open set *U* such that *x* ā *U* ā *P*. Let šÆ_{x} denote the totality of all neighborhoods of *x* in *A*, then subfamily š„_{x} of šÆ_{x} is a *fundamental system of neighborhoods* of *x*, if for each *U*_{x} in šÆ_{x}, there exists a *V*_{x} in *V*_{x} such that *V*_{x} ā *U*_{x}. If every point *x* in *A* has a countable fundamental system of neighborhoods, then we say that the space (*A*, šÆ) satisfies the *first axiom of countability* or is *first-countable*. A topological space (*A*, šÆ) is a *zero-dimensional space* if šÆ has a clopen base. A topological space (*A*, šÆ) is called a *regular space* if for any closed subset *C* of *A* and *x* ā *A* such that *x* ā *C*, then there exist disjoint open sets *U, V* such that *x* ā *U* and *C* ā *V*, or equivalently, for any open subset *U* containing *x*, there exists open subset *V* such that *x* ā *V* ā *VĢ* ā *U* .A topological space (*A*, šÆ) is called a *completely regular space*, if for every *x* ā *X* and every closed set *F* ā *A* such that *x* ā *F* there exists a continuous function *š* : *A* ā [0,1] such that *š*(*x*) = 0 and *š*(*y*) = 1 for *y* ā *F*. Let (*A*, šÆ) and (*B*, š„) be two topological spaces, a mapping *š* of *A* to *B* is *continuous* if *š*^{ā1}(*U*) ā šÆ for any *U* ā š„. The mapping *š* from (*A*, šÆ) to (*B*, š„) is called a *homeomorphism* if š is bijective, and š and š^{ā1} are continuous, or equivalently, if š is bijective, continuous and open (closed). The mapping *š* from (*A*, šÆ) to (*B*, š„) is called a *quotient map* if *š* is surjective, and *V* ā š„ if and only if *š*^{ā1}(*V*) ā šÆ. A topological space (*A*, šÆ) is *compact* if each open cover of *A* is reducible to a finite subcover, and *locally compact* if for every *x* ā *A* there exists a neighborhood *U* of *x* such that *ÅŖ* is a compact subspace of *A*.

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

*T*_{0}: For each*x*,*y*ā*X*and*x*ā*y*, there is at least one of them has a neighborhood excluding the other.*T*_{1}: For each*x*,*y*ā*X*and*x*ā*y*each has neighborhood not containing the other.*T*_{2}: For each*x*,*y*ā*X*and*x*ā*y*both have disjoint neighborhoods*U*,*V*ā Ļ such that*x*ā*U*and*y*ā*V*.

A topological space satisfying *T*_{i} is called a *T*_{i}-space, for any *i* = 0, 1, 2. A *T*_{2}-space is also known as a Hausdorff space.

Let (A, ā) be an algebra of type 2 and šÆ be a topology on *A*. Then (*A*, ā, šÆ) is called a *left (right) topological algebra*, if for all *a* ā *A* the map ā : *A* ā *A* is defined by *x* ā¦ *a* ā *x* (*x* ā¦ *x* ā *a*) is continuous, or equivalently, for any *x* ā *A* and any open subset *V* containing *a* ā *x*(*x* ā *a*) there exists an open subset *W* containing *x* such that *a* ā *W* ā *V* (*W* ā *a* ā *V*). A right and left topological algebra (*A*, ā, šÆ) is called a *semitopological* algebra. Moreover, if the operation ā is continuous, or equivalently, for each *x, y* ā *A* and each open subset *W* containing *x* ā *y*, there exist two open subsets *V*_{1} and *V*_{2} containing *x* and *y* respectively, such that *V*_{1} ā *V*_{2} ā *W*, then (*A*, ā, šÆ) is called a *topological algebra*.

## 3 Uniformity in *EQ*-algebras

From now on, we write *E* instead of the *EQ*-algebra < *E*, ā§, ā, ā¼, 1 > for convenience, unless otherwise stated.

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

*U*ā*V*= {(*x*,*y*) ā*X*Ć*X*: (*x*,*z*) ā*U*, (*y*,*z*) ā*V*, for some*z*ā*X*};*U*^{ā1}= {(*x*,*y*) ā*X*Ć*X*: (*y*,*x*) ā*U*};- Ī = {(
*x*,*x*) ā*X*Ć*X*:*x*ā*X*}.

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

(U1) Ī ā *U for any U* ā š;

(U2) *if U* ā š, *then U*^{ā1} ā š;

(U3) *if U* ā š, *then there exists V* ā š *such that V* ā *V* ā *U*;

(U4) *if U, V* ā š, *then U* ā© *V* ā š;

(U5) *if U* ā š *and U* ā *V* ā *X* Ć *X*, *then V* ā š.

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

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

*Let* ā§ *be an arbitrary family of filters of E which is closed under intersection*. *If U*_{F} = {(*x*, *y*) ā *E* Ć *E* : *x* ā”_{F} *y*} *and* š^{ā} = {*U*_{F} : *F* ā ā§}, *then* š^{ā} *satisfies conditions (U1)-(U4)*.

(U1): Since *F* is a filter of *E*, we have *x* ā”_{F} *x*, for any *x* ā *E*. Hence Ī ā *U*_{F} for all *U*_{F} ā š^{ā}.

(U2): For any *U*_{F} ā š^{ā}, we have

(U3): For any *U*_{F} ā š^{ā}, the transitivity of ā”_{F} implies that *U*_{F} ā *U*_{F} ā *U*_{F}.

(U4): For any *U*_{F}, *U*_{J} ā š^{ā}, we claim that *U*_{F} ā© *U*_{J} = *U*_{Fā©J}. If (*x*, *y*) ā *U*_{F} ā© *U*_{J}, then *x* ā”_{F} *y* and *x* ā”_{J} *y*. Hence *x* ā¼ *y* ā *F* and *x* ā¼ *y* ā *J*. Then *x* ā¼ *y* ā *F* ā© *J* and so (*x*, *y*) ā *U*_{Fā©J}. Conversely, let (*x*, *y*) ā *U*_{Fā©J}. Then *x* ā”_{Fā©J} *y*, hence *x* ā¼ *y* ā *F* ā© *J*, and thus *x* ā¼ *y* ā *F*, *x* ā¼ *y* ā *J*. Therefore *x* ā”_{F} *y* and *x* ā”_{J} *y*, which means that (*x*, *y*) ā *U*_{F} ā© *U*_{J}. So *U*_{F} ā© *U*_{J} = *U*_{Fā©J}. Since *F*, *J* ā ā§, then *F* ā© *J* ā ā§ and so *U*_{F} ā© *U*_{J} ā š^{ā}.āā”

*Let š* = {*U* ā *E* Ć *E* : ā *U*_{F} ā š^{ā} *s.t.* *U*_{F} ā *U* }, *where* š^{ā} comes from Theorem 3.2. Then š satisfies a uniformity on *E* and the pair (*E*, š) is a uniform structure.

By Theorem 3.2, the collection š satisfies the conditions (U1)-(U4). It suffices to show that š satisfies (U5). Let *U* ā š and *U* ā *V* ā *E* Ć *E*. Then there exists *U*_{F} ā *U* ā *V*, which means that *V* ā š.āā”

Let *x* ā *E* and *U* ā š. Define *U*[*x*] := {*y* ā *E* : (*x*, *y*) ā *U*}. Clearly, if *V* ā *U*, then *V*[*x*] ā *U*[*x*].

*Let E be an EQ-algebra. Then*

*is a topology on E, where K comes from Theorem 3.3*.

Clearly, ā
and the set *E* belong to šÆ. It is clear that šÆ is closed under arbitrary union. Finally to show that šÆ is closed under finite intersection, let *G*, *H* ā šÆ and suppose that *x* ā *G* ā© *H*. Then there exist *U*, *V* ā š such that *U*[*x*] ā *G* and *V*[*x*] ā *H*. If *W* = *U* ā© *V*, then *W* ā š. Also *W*[*x*] ā *U*[*x*] ā© *V*[*x*] and so *W*[*x*] ā *G* ā© *H*, hence *G* ā© *H* ā šÆ. Thus šÆ is a topology on *E*.āā”

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

*Let* ā§ *be an arbitrary family of filters of an EQ-algebra E which is closed under intersection. Then the topology šÆ comes from Theorem 3.4 is called a* uniform topology *on E induced by* ā§.

We denote the uniform topology šÆ obtained from an arbitrary family of filters ā§ by šÆ_{Ī}, and if ā§ = {*F*}, we denote it by šÆ_{F}.

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

*We can easily check that* < *E*, ā§, ā, ā¼, 1 > *is an EQ-algebra. Consider the filter F* = {*b*, 1}, *and* ā§ = {*F*}. *Therefore as in Theorem 3.2, we construct* š^{ā} = {*U*_{F}} = {{(*x*, *y*) : *x* ā”_{F} *y*}} = {{(0,0), (*a*, *a*), (*b*, *b*), (*b*, 1), (1, *b*), (1, 1)}}. *We can check that* (*E*, š) *is a uniform space, where* š = {*U* : *U*_{F} ā *U*_{F}}. *Neighborhoods are* *U*_{F} [0] = {0}, *U*_{F}[*a*] = {*a*}, *U*_{F} [*b*] = {*b*, 1}, *U*_{F} [1] = {*b*, 1}. *From above we get that* šÆ_{F} = {ā
, {0}, {*a*}, {*b*, 1}, {0, *a*}, {0, *b*, 1}, {*a*, *b*, 1}, {0, *a*, *b*, 1}}. *Thus* (*E*, š£_{F}) *is a uniform topological space*.

## 4 Topological properties of the space (*E*, šÆ_{Ī})

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

Let *E* be an *EQ*-algebra and *C*, *D* be subsets of *E*. Then we define *C* ā *D* as follows: *C* ā *D* = {*x* ā *y* : *x* ā *C*, *y* ā *D*}, where ā ā {ā§, ā, ā¼}.

*Let š¤ be a topology on* E. *Then* (E, š¤) *is called a* topological *EQ*-algebra(*TEQ*-algebra for short) *if the operations* ā§, ā *and* ā¼ *are continuous with respect to š¤*.

Recall that a topological space (*X*, š¤) is a discrete space if for any *x* ā *X*, {*x*} is an open set.

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

*The pair* (*E*, šÆ_{Ī}) *is a TEQ-algebra*.

By Definition 4.1, it suffices to show that ā is continuous, where ā ā {ā§, ā, ā¼}. Indeed, assume that *x* ā *y* ā *G*, where *x*, *y* ā *E* and *G* is an open subset of *E*. Then there exist *U* ā š, *U*[*x* ā *y*] ā *G*, and a filter F such that *U*_{F} ā š^{ā} and *U*_{F} ā *U*. We claim that the following relation holds:

Let *h*ā*k* ā *U*_{F} [*x*]ā*U*_{F} [*y*]. Then *h* ā *U*_{F} [*x*] and *k* ā *U*_{F} [*y*] we get that *x* ā”_{F} *h* and *y* ā”_{F} *k*. Hence *x* ā *y* ā”_{F} *h*ā*k*. From that we obtain (*x* ā *y*, *h* ā *k*) ā *U*_{F} ā *U*. Hence *h* ā *k* ā *U*_{F} [*x* ā *y*] ā *U*[*x* ā *y*]. Then *h* ā *k* ā *G*. Clearly, *U*_{F} [*x*] and *U*_{F} [*y*] are neighborhoods of *x* and *y*, respectively. Therefore, the operation ā is continuous.āā”

*In Example 3.6, it is easy to check that* (*E*, šÆ_{F}) *is a TEQ-algebra*.

*Let* ā§ *be a family of filters of E which is closed under intersection*. *Any filter in the collection* ā§ *is a clopen subset of E for the topology* šÆ_{Ī}.

Let *F* be a filter of *E* in ā§ and *y* ā *F*^{c}. Then *y* ā *U*_{F} [*y*] and we get *F*^{c} ā āŖ{*U*_{F}[*y*] : *y* ā *F*^{c}}. We claim that for all *y* ā *F*^{c}, *U*_{F} [*y*] ā *F*^{c}. If *z* ā *U*_{F} [*y*], then *z* ā”_{F} *y*. Hence *z* ā¼ *y* ā *F* .If *z* ā *F*, by Lemma 2.6 (i), we get that *y* ā *F*, which is a contradiction. So *z* ā *F*^{c} and we get āŖ{*U*_{F} [*y*] : *y* ā *F*^{c}} ā *F*^{c}. Hence *F*^{c} = āŖ{*U*_{F} [*y*] : *y* ā *F*^{c}}. Since *U*_{F} [*y*] is open for all *y* ā *E*, it follows that *F* is a closed subset of *E*. We show that *F* = āŖ{*U*_{F} [*y*] : *y* ā *F*}. If *y* ā *F*, then *y* ā *U*_{F} [*y*] and we get *F* ā āŖ{*U*_{F} [*y*] : *y* ā *F*}. Let *y* ā *F* .If *z* ā *U*_{F} [*y*], then *z* ā”_{F} *y* and so *y* ā¼ *z* ā *F*. Since *y* ā *F*, by Lemma 2.6 (i), *z* ā *F*, and we get āŖ{*U*_{F} [*y*] : *y* ā *F*} ā *F* .So *F* is also an open subset of *E*.āā”

*Let* ā§ *be a family of filters of E which is closed under intersection. For any x* ā *E and F* ā ā§, *U*_{F} [*x*] *is a clopen subset of E for the topology* šÆ_{Ī}.

First we show that (*U*_{F} [*x*])^{c} is open. If *y* ā (*U*_{F} [*x*])^{c}, then *y* ā¼ *x* ā *F*^{c}. We claim that *U*_{F} [*y*] ā (*U*_{F} [*x*])^{c}. If *z* ā *U*_{F} [*y*], then *z* ā (*U*_{F} [*x*])^{c}, otherwise *z* ā *U*_{F} [*x*], we get that *z* ā¼ *y* ā *F* and *z* ā¼ *x* ā *F*. Since *F* is a filter, we get that (*x* ā¼ *z*) ā (*z* ā¼ *y*) ā *F*. By (*x* ā¼ *z*) ā (*z* ā¼ *y*) < *x* ā¼ *y* and *F* is a filter, it follows that *x* ā¼ *y* ā *F*, which is a contradiction. Hence *U*_{F} [*y*] ā (*U*_{F} [*x*])^{c} for all *y* ā (*U*_{F} [*x*])^{c}, and so *U*_{F} [*x*] is closed. It is clear that *U*_{F} [*x*] is open. So *U*_{F} [*x*] is a clopen subset of *E*.āā”

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

*The space* (*E*, šÆ_{Ī}) is not, in general, a connected space.

It clearly follows from Theorem 4.6.āā”

šÆ_{ā§} = šÆ_{J}, where *J* = ā©{*F* : *F* ā ā§}.

Let š and š^{ā} be as in Theorems 3.2 and 3.3. Now consider ā§_{0} = {*J*}, define (š_{0})^{ā} = {*U*_{J}} and š_{0} = {*U* : *U*_{J} ā *U*}. Let *G* ā šÆ_{Ī}. So for each *x* ā *G*, there is *U* ā š such that *U*[*x*] ā *G*. From *J* ā *F* we get that *U*_{J} ā *U*_{F} for any filter *F* of ā§. Since *U* ā š, there exists *F* ā ā§ such that *U*_{F} ā *U*. Hence *U*_{J} [*x*] ā *U*_{F} [*x*] ā *G*. Since *U*_{J} ā š_{0}, we get that *G* ā šÆ_{J}. So šÆ_{Ī} ā šÆ_{J}. Conversely, let *H* ā šÆ_{J}. Then for any *x* ā *H* there is *U* ā š_{0} such that *U*[*x*] ā *H*. Hence *U*_{J} [*x*] ā *H*. Since ā§ is closed under intersection, so *J* ā ā§. Then we get *U*_{J} ā š and so *H* ā šÆ_{Ī}. Therefore, šÆ_{J} ā šÆ_{Ī}.āā”

*Let F and J be filters of E and F* ā *J*. *Then J is clopen in the topological space* (*E*, šÆ_{F}).

Consider ā§ = {*F*, *J*}. Then by Theorem 4.8, šÆ_{Ī} = šÆ_{F}. Hence by Theorem 4.5, *J* is clopen in the topological space (*E*, šÆ_{F}).āā”

*Let* ā§ *be a family of filters of E which is closed under intersection and J* = ā©{*F* : *F* ā ā§}. *We have the following statements*:

*By Theorem 4.8, we know that*šÆ_{Ī}= šÆ_{J}.*For any U*ā š,*x*ā*E, we can get that**U*_{J}[*x*] ā*U*[*x*].*Hence*šÆ_{Ī}*is equivalent to*{*A*ā*E*: ā*x*ā*A*,*U*_{J}[*x*] ā*A*}.*So A*ā*E is open set if and only if for all x*ā*A*,*U*_{J}[*x*] ā*A if and only if A*= āŖ_{xāA}*U*_{J}[*x*];*For all x*ā*E, by (i), we know that**U*_{J}[*x*]*is the smallest neighborhood of x*;*Let š*_{J}= {*U*_{J}[*x*] :*x*ā*E*}.*By (i) and (ii), it is easy to check that š*šÆ_{J}is abase of_{J};*For all x*ā*E*, {*U*_{J}[*x*]}*is a denumerable fundamental system of neighborhoods of x*.

*If F is a filter of E, then for all x* ā *E*, *U*_{F} [*x*] *is a clopen compact set in the topological space* (*E*, šÆ_{F}).

By Theorem 4.6, it is enough to show that *U*_{F} [*x*] is a compact set. Let *U*_{F} [*x*] ā āŖ_{Ī±āI} *O*_{Ī±}, where each *O*_{Ī±} is an open set of *E*. Since *x* ā *U*_{F} [*x*], there exists Ī± ā *I* such that *x* ā *O*_{Ī±}. Then *U*_{F} [*x*] ā *O*_{Ī±}. Hence *U*_{F} [*x*] is compact. Therefore *U*_{F} [*x*] is a clopen compact set in the topological space (*E*, šÆ_{F}).āā”

*Let* ā§ *be a family of filters of E which is closed under intersection*. *Then* (*E*, šÆ_{Ī}) *is a first-countable, zero-dimensional, disconnected and completely regular space*.

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

*Let* ā§ *be a family of filters of E which is closed under intersection*. *Then* (*E*, šÆ_{Ī}) *is a discrete space if and only if there exists F* ā ā§ *such that U*_{F} [*x*] = {*x*} *for all x* ā *E*.

Let šÆ_{Ī} be a discrete topology on *E*. If for any *F* ā ā§, there exists *x* ā *E* such that *U*_{F} [*x*] ā {*x*}. Let *J* = ā©ā§. Then *J* ā ā§, there exists *x*_{0} ā *E* such that *U*_{J} [*x*_{0}] ā {*x*_{0}}. It follows that there exists *y*_{0} ā *U*_{F} [*x*_{0}] and *x*_{0} ā *y*_{0}. By Remark 4.10 (ii), *U*_{J} [*x*_{0}] is the smallest neighborhood of *x*_{0}. Hence {*x*_{0}} is not an open subset of *E*, which is a contradiction. Conversely, for any *x* ā *E*, there exists *F* ā ā§ such that *U*_{F} [*x*] = {*x*}. Hence {*x*} is an open set of *E*. Therefore, (*E*, šÆ_{Ī}) is a discrete space.āā”

*Let* ā§ *be a family of filters of E which is closed under intersection*, *J* = ā©ā§ *and E be a separated EQ-algebra. Then the following conditions are equivalent*:

- (
*E*, šÆ_{J})*is a discrete space*; *J*= {1}.

- ā (
*ii*): By Theorem 4.13, we have*U*_{J}[1] = {1}. We show that*J*ā*U*_{J}[1]. Let*x*ā*J*. By Proposition 2.2 (vii), we get that*x*ā¤*x*ā¼ 1. Since*J*is a filter and*x*ā*J*, hence*x*ā¼ 1 ā*J*. So*x*ā*U*_{J}[1]. It follows that*J*ā*U*_{J}[1]. Since*U*_{J}[1] = {1} and 1 ā*J*. Therefore,*J*= {1}. - ā (
*i*): Let*J*= {1}. Since*E*is separated, we can get that*U*_{J}[*x*] = {*x*}. It follows that (*E*, šÆ_{J}) is discrete.āā”

*Let* ā§ *be a family of filters of E which is closed under intersection*, *J* = ā©ā§ *and E be a separated EQ-algebra*. *Then* (*E*, šÆ_{J}) *is a Hausdorff space if and only if J* = {1}.

Let (*E*, šÆ_{J}) be a Hausdorff space. First we show that for any *x* ā *E*, *U*_{J} [*x*] = {*x*}. If there exists *x* ā *y* ā *U*_{J} [*x*], then *y* ā *U*_{J} [*x*] ā© *U*_{J} [*y*]. By Remark 4.10 (ii), *U*_{J} [*x*] and *U*_{J} [*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 *y* ā *U*_{J} [*x*] ā© *U*_{J} [*y*] ā *U* ā© *V* ā ā
, 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.āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras*. *A mapping Ļ* : *E*_{1} ā *E*_{2} *is called an EQ-morphism from E*_{1} *to E*_{2} *if*

*for any*ā ā {ā§, ā, ā¼}.

*If, in addition, the mapping Ļ is bijective, then we call Ļ an EQ*-isomorphism.

*Note that Ļ*(1) = 1

*when Ļ is an EQ-morphism*.

*Let Ļ* : *E*_{1} ā *E*_{2} *be an EQ-morphism. Then the following properties hold*:

*if F is a filter of E*_{2},*then the set Ļ*^{ā1}(*F*)*is a filter of E*_{1};*if Ļ is surjective and F is a filter of E*_{1},*then Ļ*(*F*)*is a filter of E*_{2}.

It is easy to prove by definition of filters.āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras and F be a filter of E*_{2}. *If Ļ* : *E*_{1} ā *E*_{2} *is an EQ-isomorphism, then*

For any (*a, b*) ā *U*_{Ļā1(F)} ā *a* ā¼ *b* ā *Ļ*^{ā1}(*F*) ā *Ļ*(*a*) ā¼ *Ļ*(*b*) ā *F* ā (*Ļ*(*a*), *Ļ*(*b*)) ā *U*_{F}.āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras and F be a filter of E*_{2}. *If Ļ* : *E*_{1} ā *E*_{2} *is an EQ-isomorphism, then the following properties hold*:

*for any a*ā*E*_{1},*Ļ*(*U*_{Ļā1(F)}[*a*]) =*U*_{F}[*Ļ*(*a*)];*for any b*ā*E*_{2},*Ļ*^{ā1}(*U*_{F}[*b*]) =*U*_{Ļā1(F)}[*Ļ*^{ā1}(*b*)].

(i) Let *b* ā *Ļ**U*_{Ļā1(F)}[*a*]). Then there exists *c* ā *U*_{Ļā1(F)}[*a*] such that *b* = *Ļ*(*c*). It follows that *a* ā¼ *c* ā *Ļ*^{ā1}(*F*) ā *Ļ*(*a*) ā¼ *Ļ*(*c*) ā *F* ā *Ļ*(*a*) ā¼ *b* ā *F* ā *b* ā *U*_{F} [*Ļ*(*a*)].

Conversely, *b* ā *U*_{F} [*Ļ*(*a*)] ā *Ļ*(*a*) ā¼ *b* ā *F* ā *Ļ*^{ā1}(*Ļ*(*a*) ā¼ *b*) ā *Ļ*^{ā1}(*F*) ā *a* ā¼ *Ļ*^{ā1}(*b*) ā *Ļ*^{ā1}(*F*) ā *Ļ*^{ā1}(*b*) ā *U*_{Ļā1(F)}[*a*] ā *b* ā *Ļ*(*U*_{Ļā1(F)}[*a*]).

(ii) *a* ā *Ļ*^{ā1}(*U*_{F}[*b*]) ā *Ļ*(*a*) ā *U*_{F}[*b*] ā *Ļ*(*a*) ā¼ *b* ā *F* ā *Ļ*^{ā1}(*Ļ*(*a*) ā¼ *b*) ā *Ļ*^{ā1}(*F*) ā *a* ā¼ *Ļ*^{ā1}(*b*) ā *Ļ*^{ā1}(*F*) ā *a* ā *U*_{Ļā1(F)}[*Ļ*^{ā1}(*b*)].āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras and F be a filter of E*_{2}. *If Ļ* : *E*_{1} ā *E*_{2} *is an EQ-isomorphism, then Ļ is a continuous map from* (*E*_{1}, šÆ_{Ļā1(F)}) to (*E*_{2}, šÆ_{F}).

Let *A* ā šÆ_{F}. By Remark 4.10 (i), we can get that *A* = āŖ_{aāA} *U*_{F} [*a*]. It follows that *Ļ*^{ā1}(*A*) = *Ļ*^{ā1}(āŖ_{aāA} *U*_{F} [*a*]) = āŖ_{aāA} *Ļ*^{ā1}(*U*_{F} [*a*]). We claim that if *b* ā *Ļ*^{ā1} (*U*_{F} [*a*]), then *U*_{Ļā1(F)}[*b*] ā *Ļ*^{ā1}(*U*_{F} [*a*]). Indeed, let *c* ā *U*_{Ļā1(F)}[*b*], we get that *c* ā¼ *b* ā *Ļ*^{ā1} (*F*), so *Ļ*(*c*) ā¼ *Ļ*(*b*) ā *F*. Since *Ļ*(*b*) ā *U*_{F} [*a*], we get that *Ļ*(*b*) ā¼ *a* ā *F*. It follows that *Ļ*(*c*) ā¼ *a* ā *F*. Thus we have that *Ļ*(*c*) ā *U*_{F} [*a*]. So *c* ā *Ļ*^{ā1}(*U*_{F} [*a*]). Hence *Ļ*^{ā1}(*U*_{F} [*a*]) = āŖ_{bāĻā1(UF[a])} *U*_{Ļā1(F)} [*b*] ā šÆ_{Ļā1(F)}. Therefore *Ļ*^{ā1}(*A*) = āŖ_{aāA} *Ļ*^{ā1} (*U*_{F} [*a*]) ā šÆ_{Ļā1(F)}. So *Ļ* is a continuous map.āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras and F be a filter of E*_{2}. *If Ļ* : *E*_{1} ā *E*_{2} *is an EQ-isomorphism, then Ļ is a quotient map from* (*E*_{1}, šÆ_{Ļā1(F)}) to (*E*_{2}, šÆ_{F}).

From Theorem 4.20 we get that *Ļ* is a continuous surjective map. It is enough to show that *Ļ* is an open map. Let *A* be an open set of (*E*_{1}, šÆ_{Ļā1(F)}). We claim that *Ļ*(*A*) is an open set of (*E*_{2}, šÆ_{F}). Let *a* ā *Ļ*(*A*). We shall show that *U*_{F} [*a*] ā *Ļ*(*A*). Indeed, for any *b* ā *U*_{F} [*a*], we get that *b* ā¼ *a* ā *F*. By Lemma 4.15, we have *Ļ*^{ā1}(*a*) ā¼ *Ļ*^{ā1}(*b*) ā *Ļ*^{ā1}(*F*). Hence *Ļ*^{ā1}(*b*) ā *U*_{Ļā1(F)}[*Ļ*(*a*)]. Since *a* ā *Ļ*(*A*) and *Ļ* is injective we get that *Ļ*^{ā1}(*a*) ā *A*. By Remark 4.10 (i), it follows that *U*_{Ļā1(F)}[*Ļ*^{ā1}(*a*)] ā *A*. So *Ļ*^{ā1}(*b*) ā *A*, we get that *b* ā *Ļ*(*A*). Therefore, *U*_{F} [*a*] ā *Ļ*(*A*). So *Ļ* is a quotient map.āā”

*Let E*_{1} *and E*_{2} *be EQ-algebras and F be a filter of E*_{2}. *If Ļ* : *E*_{1} ā *E*_{2} *is an EQ-isomorphism, then Ļ is a homeomorphism map from* (*E*_{1}, šÆ_{Ļā1(F)}) to (*E*_{2}, šÆ_{F}).

It clearly follows from Theorem 4.21.āā”

Recall that a uniform space (*X*, š) is totally bounded if for each *U* ā š, there exist *x*_{1},..., *x*_{1} ā *X* such that

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

*the topological space*(*E*, šÆ_{F})*is compact*;*the topological space*(*E*, šÆ_{F})*is totally bounded*;*there exists P*= {*x*_{1}, ... ,*x*_{n}} ā*E such that for all a*ā*E there exists x*_{i}ā*P such that a*ā”_{F}*x*._{i}

- ā (2): The proof is straightforward.
- ā (3): Since (
*E*, šÆ_{F}) is totally bounded, there exist*x*_{1},...,*x*_{n}ā*E*such that . Now let$E={\u0101\x88\u0156}_{i=1}^{n}{U}_{F}[{x}_{i}]$ *a*ā*E*. Then there exists*x*_{i}, such that*a*ā*U*_{F}[*x*_{i}], therefore*a*ā¼*x*_{i}, ā*F*i.e.*a*ā”_{F}*x*_{i}. - ā (1): For any
*a*ā*E*, by hypothesis, there exists*x*_{i}ā*P*such that*a*ā¼*x*_{i}ā*F*. We can get that*a*ā āŖ_{F}[*x*_{i}], hence . Now let$E={\u0101\x88\u0156}_{i=1}^{n}{U}_{F}[{x}_{i}]$ *E*= āŖ_{Ī±āI}*O*_{Ī±}, where each*O*_{Ī±}is an open set of*E*. Then for any*x*_{i}ā*E*there exists Ī±_{i}ā*I*such that*x*_{i}, ā*O*_{Ī±i}. Since*O*_{Ī±i}is an open set,*U*_{F}[*x*_{i}] ā*O*_{Ī±i}, so we have thatTherefore$$E=\underset{i\u0101\x88\x88I}{\u0101\x88\u0156}{U}_{F}[{x}_{i}]\u0101\x8a\x86\underset{i=1}{\overset{n}{\u0101\x88\u0156}}{O}_{{\mathrm{\u012a\pm}}_{i}}.$$ , whence ($E={\u0101\x88\u0156}_{i=1}^{n}{O}_{{\mathrm{\u012a\pm}}_{i}}$ *E*, šÆ_{F}) is compact.

*If F is a filter of E such that F*^{c} *is a finite set, then the topological space* (*E*, šÆ_{F}) *is compact*.

Let *E* = āŖ_{Ī±āI} *O*_{Ī±}, where each *O*_{Ī±} is an open subset of *E*. Let *F*^{c} = {*x*_{1},... ,*x*_{n}}. Then there exist Ī±, Ī±_{1},...,Ī±_{n} ā *I* such that 1 ā *O*_{Ī±}, *x*_{1} ā *O*_{Ī±1},...,*x*_{n} ā *O*_{Ī±n}. Then *U*_{F} [1] ā *O*_{Ī±}, but *U*_{F} [1] = *F*. Hence

*If F is a filter of E, then F is a compact set in the topological space* (*E*, šÆ_{F}).

Let *F* ā āŖ_{Ī±āI} *O*_{Ī±}, where each *O*_{Ī±} is open set of *E*. Since 1 ā *F*, there is Ī± ā I such that 1 ā *O*_{Ī±}. Then *F* = *U*_{F} [1] ā *O*_{Ī±}. Hence F is a compact set in the topological space (*E*, šÆ_{F}).āā”

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

*Let E be an EQ-algebra and* (*D*, ā¤) *be an upward directed set. If for any* Ī± ā *D we have a*_{Ī±} ā *E*, *then we call* {*a*_{Ī±}}_{Ī±āD} *a* net *of E*.

*Let* {*a*_{Ī±}}_{Ī±āD} *be a net of E*. *In the topological space* (*E*, šÆ_{F}), *say that* {*a*_{Ī±}}_{Ī±āD}

- converges to the point
*a*of*E if for any neighborhood U of a, there exists d*_{0}ā*D such that a*_{Ī±}ā*U for any*Ī± ā„*d*_{0}; - Cauchy sequence
*if there exists d*_{0}ā*D such a*_{Ī±}ā”_{F}*a*_{Ī²}*for any*Ī±,*Ī²*ā„*d*_{0}.

A net {*a*_{Ī±}}_{iāD}, which converges to *a* is said to be convergent. For simplicity, we write lim*a*_{Ī±} = *a* and we say that *a* is a limit of {*a*_{Ī±}}_{Ī±āD}.

*Consider the TEQ-algebra* (*E*, šÆ_{F}) *in Example 4.4. Clearly*, (ā, ā¤) *is an upward directed set, where* ā *is a natural number set*. *We define* {*a*_{n}}_{nāā} *as a*_{0} = 0, *a*_{1} = *a*, *a*_{2} = *b*, *a*_{n} = 1, *n* ā„ 3. *It is easy to check that* {*a*_{n}}_{nāā} *is a net of E*. *Let n*_{0} = 3. *For any neighborhood U of 1, if n* ā„ 3, *then* 1 ā *U*. *Therefore*, lim*a*_{n} = *1*.

*Let* {*a*_{Ī±}}_{Ī±āD} *and* {*b*_{Ī±}}_{Ī±āD} *be nets of E and F be a filter of E. Then in the topological space* (*E*, šÆ_{F}) *we have*:

*if*lim*b*_{Ī±}=*b and*lim*a*_{Ī±}=*a, for some a, b*ā*E, then the sequence*{*a*_{Ī±}ā*b*_{Ī±}}_{Ī±āD}*is convergent and*lim*a*_{Ī±}ā*b*_{Ī±}=*a*ā*b, for any operation*ā ā {ā§, ā, ā¼};*any convergent sequence of E is a Cauchy sequence*.

- Let lim
*a*_{Ī±}=*a*, lim*b*_{Ī±}=*b*and ā ā {ā§, ā, ā¼}, for some*a, b*ā*E*. For any neighborhood*W*of*a*ā*b*we get that*U*_{F}[*a*ā*b*] ā*W*. Clearly,*U*_{F}[*a*] and*U*_{F}[*b*] are neighborhoods of*a*and*b*, respectively. By hypothesis, there exist*d*_{1},*d*_{2}ā*D*such that*a*_{Ī±}ā*U*_{F}[*a*] and*b*_{Ī±}ā*U*_{F}[*b*], for any Ī± ā„*d*_{1}and Ī± ā„*d*_{2}. Since*D*is an upward directed set, then there exists*d*_{0}ā*D*such that*d*_{0}ā„*d*_{1}and*d*_{0}ā„*d*_{2}. By Theorem 4.3, we get that*U*_{F}[*a*] ā*U*_{F}[*a*] ā*U*_{F}[*a*ā*b*]. So*a*_{Ī±}ā*U*_{F}[*a*] and*b*_{Ī±}ā*U*_{F}[*b*], for any Ī± ā„*d*_{0}. It follows that*a*ā*b*ā*U*_{F}[*a*] ā*U*_{F}[*a*] ā*U*_{F}[*a*ā*b*] ā*U*, for any Ī± ā„*d*_{0}ā*D*. Therefore, lim*a*_{Ī±}ā*b*_{Ī±}=*a*ā*b*. - Let {
*a*_{Ī±}}_{Ī±āD}be a net of*E*and lim*a*_{Ī±}=*a*. For the neighborhood*U*_{F}[*a*] of*a*, there exists*d*ā*D*such that*a*_{Ī±}ā*D*, for any Ī± ā„*d*. So if Ī±,*Ī²*ā„*d*, then*a*_{Ī±},*a*_{Ī²}ā*U*_{F}[*a*] that is*a*_{Ī±}ā”_{F}*a*and*a*_{Ī²}ā”_{F}*a*. It follows that*a*_{Ī±}ā”_{F}*a*_{Ī²}. Therefore, {*a*_{Ī±}}_{Ī±āD}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 šÆ_{Ī} and proposed the concept of the topological *EQ*-algebra. We then stated and proved special properties of (*E*, šÆ_{Ī}). Especially, we proved that (*E*, šÆ_{Ī}) 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.

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).

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