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Uniform topology on EQ-algebras

  • Jiang Yang , Xiao Long Xin EMAIL logo and Peng Fei He
Published/Copyright: April 14, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

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.

Keywords: Uniform space; Topological EQ-algebra; Filter; Converge sequence
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 x ≤ y if and only if x ∧ y = x;

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

(E3) x ∼ x = 1, (reflexivity axiom);

(E4) ((x ∧ y) ∼ z) ⊗ (t ∼ x) ≤ z ∼ (t ∧ y), (substitution axiom);

(E5) (x ∼ y) ⊗ (z ∼ t) ≤ (x ∼ z) ∼ (y ∼ t), (congruence axiom);

(E6) (x ∧ y ∧ z) ∼ x ≤ (x ∧ y) ∼ x, (monotonicity axiom);

(E7) x ⊗ y ≤ x ∼ y. (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, x → y := (x ∧ y) ∼ x and x̄ := x ∼ 1. Then the following properties hold for all x, y, z ∈ E:

  1. x ⊗ y ≤ x ∧ y ≤ x, y;

  2. z ⊗ (x ∧ y) ≤ (z ⊗ x) ∧ (z ⊗ y);

  3. x ∼ y ≤ x → y;

  4. x → x = 1;

  5. (x ∼ y) ⊗ (y ∼ z) ≤ x ∼ z;

  6. (x → y) ⊗ (y → z) ≤ x → z;

  7. x ≤ x̄. 1̄ = 1;

  8. x ⊗ (x ∼ y) ≤ ȳ;

  9. (x → y) ⊗ (y → x) ≤ x ∼ y ≤ (x → y) ∧ (y → x);

  10. if x ≤ y → z, then x ⊗ y ≤ z̄;

  11. if x ≤ y < z, then z ∼ x ≤ z ∼ y and x ∼ z ≤ x ∼ y.

Definition 2.3

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

  1. separated if for all a, b ∈ E, a ∼ b = 1 implies a=b;

  2. good if for all a ∈ E, 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, c ∈ E we have that:

  1. 1 ∈ F;

  2. if a, a → b ∈ F; then b ∈ F,

  3. if a → b ∈ F, then a ⊗ c → b ⊗ c ∈ F and c ⊗ a → c ⊗ b ∈ F.

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, b ∈ F, then a ⊗ b ∈ 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 ∈ F and x ≤ y, then y ∈ F. From x ∧ y = x it follows that x → y = 1. By Definition 2.4 (i) and (ii), it follows that y ∈ F. Let a, b ∈ F. From Proposition 2.2 (vii), it follows that b ≤ 1 → b. From Definition 2.4 (iii), it then follows that (a ⊗ 1) → (a ⊗ b) ∈ F. Hence, by Definition 2.4 (i) and (ii), a ⊗ b ∈ F.

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 a ∼ b and a′ ∼ b′ ∈ F, the following holds:

  1. if e ∈ F and e ≤ 𝑓, then 𝑓 ∈ F;

  2. if e, e ∼ 𝑓 ∈ F, then 𝑓 ∈ F;

  3. a ↔ b ∈ F, (a → b) ⊗ (b → c) ∈ F, where a ↔ b := (a → b) ∧ (b → a);

  4. (a ∧ a′) ∼ (b ∧ b′) ∈ F;

  5. (a ⊗ c) ∼ (b ⊗ c) ∈ F and (c ⊗ a) ∼ (c ⊗ b) ∈ F,

  6. (a ∼ a′) ∼ (b ∼ b′) ∈ 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 relation ≡F is a congruence relation on E.

Definition 2.8

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

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, 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 neighborhood of x ∈ A, if there exists an open set U such that x ∈ U ⊆ P. 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 Vx ⊆ Ux. 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 x ∈ A such that x ∉ C, then there exist disjoint open sets U, V such that x ∈ U and C ⊆ V, or equivalently, for any open subset U containing x, there exists open subset V such that x ∈ V ⊆ V̄ ⊆ U .A topological space (A, 𝒯) is called a completely regular space, if for every x ∈ X and every closed set F ⊂ A such that x ∉ F there exists a continuous function 𝑓 : A → [0,1] such that 𝑓(x) = 0 and 𝑓(y) = 1 for y ∈ F. 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 x ∈ A 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, y ∈ X and x ≠ y, there is at least one of them has a neighborhood excluding the other.

  2. T1 : For each x, y ∈ X and x ≠ y each has neighborhood not containing the other.

  3. T2 : For each x, y ∈ X and x ≠ y both have disjoint neighborhoods U, V ∈ τ such that x ∈ U and y ∈ V.

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 a ∈ A the map ∗ : A → A is defined by x ↦ a ∗ x (x ↦ x ∗ a) is continuous, or equivalently, for any x ∈ A and any open subset V containing a ∗ x(x ∗ a) there exists an open subset W containing x such that a ∗ W ⊆ V (W ∗ a ⊆ V). A right and left topological algebra (A, ∗, 𝒯) is called a semitopological algebra. Moreover, if the operation ∗ is continuous, or equivalently, for each x, y ∈ A and each open subset W containing x ∗ y, there exist two open subsets V1 and V2 containing x and y respectively, such that V1 ∗ V2 ⊆ W, 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. U ∘ V = {(x, y) ∈ X × X : (x, z) ∈ U, (y, z) ∈ V, for some z ∈ X};

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

  3. Δ = {(x, x) ∈ X × X : x ∈ X}.

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 V ∘ V ⊆ U;

(U4) if U, V ∈ 𝓚, then U ∩ V ∈ 𝓚;

(U5) if U ∈ 𝓚 and U ⊆ V ⊆ X × 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

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

Proof

(U1): Since F is a filter of E, we have x ≡F x, for any x ∈ E. 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 UF ∘ UF ⊆ UF.

(U4): For any UF, UJ ∈ 𝓚∗, we claim that UF ∩ UJ = UF∩J. If (x, y) ∈ UF ∩ UJ, then x ≡F y and x ≡J y. Hence x ∼ y ∈ F and x ∼ y ∈ J. Then x ∼ y ∈ F ∩ J and so (x, y) ∈ UF∩J. Conversely, let (x, y) ∈ UF∩J. Then x ≡F∩J y, hence x ∼ y ∈ F ∩ J, and thus x ∼ y ∈ F, x ∼ y ∈ J. Therefore x ≡F y and x ≡J y, which means that (x, y) ∈ UF ∩ UJ. So UF ∩ UJ = UF∩J. Since F, J ∈ ∧, then F ∩ J ∈ ∧ and so UF ∩ UJ ∈ 𝓚∗. □

Theorem 3.3

Let 𝓚 = {U ⊆ E × E : ∃ UF ∈ 𝓚∗ s.t. UF ⊆ U }, 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 U ⊆ V ⊆ E × E. Then there exists UF ⊆ U ⊆ V, which means that V ∈ 𝓚. □

Let x ∈ E and U ∈ 𝓚. Define U[x] := {y ∈ E : (x, y) ∈ U}. Clearly, if V ⊆ U, 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 x ∈ G ∩ H. Then there exist U, V ∈ 𝓚 such that U[x] ⊆ G and V[x] ⊆ H. If W = U ∩ V, then W ∈ 𝓚. Also W[x] ⊆ U[x] ∩ V[x] and so W[x] ⊆ G ∩ H, hence G ∩ H ∈ 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

Let ∧ be 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) : x ≡F 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 : UF ⊆ UF}. 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 C ∗ D as follows: C ∗ D = {x ∗ y : x ∈ C, y ∈ D}, 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 ∧, ⊗ and ∼ are continuous with respect to 𝓤.

Recall that a topological space (X, 𝓤) is a discrete space if for any x ∈ X, {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 x ∗ y ∈ G, where x, y ∈ E and G is an open subset of E. Then there exist U ∈ 𝓚, U[x ∗ y] ⊆ G, and a filter F such that UF ∈ 𝓚∗ and UF ⊆ U. We claim that the following relation holds:

UF[x]∗UF[y]⊆UF[x∗y]⊆U[x∗y].

Let h∗k ∈ UF [x]∗UF [y]. Then h ∈ UF [x] and k ∈ UF [y] we get that x ≡F h and y ≡F k. Hence x ∗ y ≡F h∗k. From that we obtain (x ∗ y, h ∗ k) ∈ UF ⊆ U. Hence h ∗ k ∈ UF [x ∗ y] ⊆ U[x ∗ y]. Then h ∗ k ∈ G. 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

Let ∧ be a family of filters of E which is closed under intersection. Any filter in the collection ∧ is a clopen subset of E for the topology 𝒯Λ.

Proof

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

Theorem 4.6

Let ∧ be a family of filters of E which is closed under intersection. For any x ∈ E 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 y ∼ x ∈ Fc. We claim that UF [y] ⊆ (UF [x])c. If z ∈ UF [y], then z ∈ (UF [x])c, otherwise z ∈ UF [x], we get that z ∼ y ∈ F and z ∼ x ∈ F. Since F is a filter, we get that (x ∼ z) ⊗ (z ∼ y) ∈ F. By (x ∼ z) ⊗ (z ∼ y) < x ∼ y and F is a filter, it follows that x ∼ y ∈ F, 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 : UJ ⊆ U}. Let G ∈ 𝒯Λ. So for each x ∈ G, there is U ∈ 𝓚 such that U[x] ⊆ G. From J ⊆ F we get that UJ ⊆ UF for any filter F of ∧. Since U ∈ 𝓚, there exists F ∈ ∧ such that UF ⊆ U. Hence UJ [x] ⊆ UF [x] ⊆ G. Since UJ ∈ 𝓚0, we get that G ∈ 𝒯J. So 𝒯Λ ⊆ 𝒯J. Conversely, let H ∈ 𝒯J. Then for any x ∈ H 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 F ⊆ J. 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

Let ∧ be 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, x ∈ E, we can get that UJ [x] ⊆ U[x]. Hence 𝒯Λ is equivalent to {A ⊆ E : ∀x ∈ A, UJ [x] ⊆ A}. So A ⊆ E is open set if and only if for all x ∈ A, UJ [x] ⊆ A if and only if A = ∪x∈A UJ [x];

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

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

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

Lemma 4.11

If F is a filter of E, then for all x ∈ E, 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 x ∈ UF [x], there exists α ∈ I such that x ∈ Oα. 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

Let ∧ be 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 x ∈ E. 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] : x ∈ E}. 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 x ∈ E 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

Let ∧ be 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 x ∈ E.

Proof

Let 𝒯Λ be a discrete topology on E. If for any F ∈ ∧, there exists x ∈ E such that UF [x] ≠ {x}. Let J = ∩∧. Then J ∈ ∧, there exists x0 ∈ E such that UJ [x0] ≠ {x0}. It follows that there exists y0 ∈ UF [x0] and x0 ≠ y0. 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 x ∈ E, 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

Let ∧ be 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 J ⊆ UJ[1]. Let x ∈ J. By Proposition 2.2 (vii), we get that x ≤ x ∼ 1. Since J is a filter and x ∈ J, hence x ∼ 1 ∈ J. So x ∈ UJ [1]. It follows that J ⊆ UJ [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

Let ∧ be 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 x ∈ E, UJ [x] = {x}. If there exists x ≠ y ∈ UJ [x], then y ∈ UJ [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 y ∈ UJ [x] ∩ UJ [y] ⊆ U ∩ V ≠ ∅, 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 : E1 → E2 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 : E1 → E2 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 : E1 → E2 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) ⇔ a ∼ b ∈ φ−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 : E1 → E2 is an EQ-isomorphism, then the following properties hold:

  1. for any a ∈ E1, φ(Uφ−1(F) [a]) = UF [φ(a)];

  2. for any b ∈ E2, φ−1(UF[b]) = Uφ−1(F)[φ−1(b)].

Proof

(i) Let b ∈ φUφ−1(F)[a]). Then there exists c ∈ Uφ−1(F)[a] such that b = φ(c). It follows that a ∼ c ∈ φ−1(F) ⇒ φ(a) ∼ φ(c) ∈ F ⇒ φ(a) ∼ b ∈ F ⇒ b ∈ UF [φ(a)].

Conversely, b ∈ UF [φ(a)] ⇒ φ(a) ∼ b ∈ F ⇒ φ−1(φ(a) ∼ b) ∈ φ−1(F) ⇒ a ∼ φ−1(b) ∈ φ−1(F) ⇒ φ−1(b) ∈ Uφ−1(F)[a] ⇒ b ∈ φ(Uφ−1(F)[a]).

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

Theorem 4.20

Let E1 and E2 be EQ-algebras and F be a filter of E2. If φ : E1 → E2 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 = ∪a∈A UF [a]. It follows that φ−1(A) = φ−1(∪a∈A UF [a]) = ∪a∈A φ−1(UF [a]). We claim that if b ∈ φ−1 (UF [a]), then Uφ−1(F)[b] ⊆ φ−1(UF [a]). Indeed, let c ∈ Uφ−1(F)[b], we get that c ∼ b ∈ φ−1 (F), so φ(c) ∼ φ(b) ∈ F. Since φ(b) ∈ UF [a], we get that φ(b) ∼ a ∈ F. It follows that φ(c) ∼ a ∈ F. Thus we have that φ(c) ∈ UF [a]. So c ∈ φ−1(UF [a]). Hence φ−1(UF [a]) = ∪b∈φ−1(UF[a]) Uφ−1(F) [b] ∈ 𝒯φ−1(F). Therefore φ−1(A) = ∪a∈A 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 : E1 → E2 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 a ∈ φ(A). We shall show that UF [a] ⊆ φ(A). Indeed, for any b ∈ UF [a], we get that b ∼ a ∈ F. 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 a ∈ φ(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 b ∈ φ(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 : E1 → E2 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,..., x1 ∈ X such that X=∪i=1nU[xi] .

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 a ∈ E there exists xi ∈ P such that a ≡F xi.

Proof

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

  2. ⇒ (3): Since (E, 𝒯F) is totally bounded, there exist x1,..., xn ∈ E such that E=∪i=1nUF[xi] . Now let a ∈ E. Then there exists xi, such that a ∈ UF [xi], therefore a ∼ xi, ∈ F i.e. a ≡F xi.

  3. ⇒ (1): For any a ∈ E, by hypothesis, there exists xi ∈ P such that a ∼ xi ∈ F. We can get that a ∈ ∪F [xi], hence E=∪i=1nUF[xi] . Now let E = ∪α∈I Oα, where each Oα is an open set of E. Then for any xi ∈ E there exists αi ∈ I 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=∪i=1nOαi , 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,...,αn ∈ I such that 1 ∈ Oα, x1 ∈ Oα1,...,xn ∈ Oαn. Then UF [1] ⊆ Oα, but UF [1] = F. Hence E=∪i=1nOαi∪Oα . □

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 ⊆ ∪α∈I 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 aα ∈ E, 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 d0 ∈ D such that aα ∈ U for any α ≥ d0;

  2. Cauchy sequence if there exists d0 ∈ D such aα ≡F aβ for any α, β ≥ d0.

A net {aα}i∈D, 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, where ℕ is 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, b ∈ E, then the sequence {aα ∗ bα}α∈D is convergent and limaα ∗ bα = a ∗ b, for any operation ∗ ∈ {∧, ⊗, ∼};

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

Proof

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

  2. Let {aα}α∈D be a net of E and limaα = a. For the neighborhood UF [a] of a, there exists d ∈ D such that aα ∈ D, for any α ≥ d. So if α, β ≥ d, then aα, aβ ∈ UF [a] that is aα ≡F a and aβ ≡F a. It follows that aα ≡F 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).

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Received: 2016-9-17
Accepted: 2017-1-26
Published Online: 2017-4-14

© 2017 Yang et al.

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

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https://doi.org/10.1515/math-2017-0032
Keywords for this article
Uniform space; Topological EQ-algebra; Filter; Converge sequence
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