A note on a method for constructing uniform topologies on algebras associated with logical systems

Abstract

In the recent years, many authors have used a single method for equipping algebraic structures with uniformities which are induced by families of algebraic objects. This paper is devoted to a description of this well-known method in general, and provides insight into those results which are obtained using the method. In fact, we prove that the uniform topology induced by this method coincides with a partition topology generated by an equivalence relation, and illustrate the logic behind the continuity of algebraic operations in these kinds of uniform topologies. Furthermore, the main topological properties of the partition topology induced by a congruence relation are presented. As an application, we explain why many results obtained from this method are trivial. These results have been collected from the works of several mathematicians on more than twenty different algebraic systems over the course of two decades.

Keywords

Algebraic structure Bl-algebra uniform structure partition space partition uniformity

1 Introduction

During the last few decades, several algebraists have shown interest in endowing algebraic systems with topologies using algebraic objects, and studied the disparate properties of such topologies. For example, Luan and Yang [22] (Hoo [16]) applied a proper filter (a system of ideals) of an MV-algebra A to construct a filter topology (a linear topology), which made A into a topological MV-algebra. One of the first steps in the investigation of such systems is, of course, to examine the nature of the topological structure.

There is a trivial method for constructing a uniform structure using an equivalence relation, whose induced uniform topology has well-known properties. For more than two decades, this method was used to equip different algebraic systems with uniform topologies, regardless of the nature of the topological structure in such spaces. In this short paper, we clarify the topological structure, and provide insight into the theory behind many results which are obtained from this method. We collected 29 papers which adopted this well-known method to different algebraic structures by using different kinds of equivalence relations. Furthermore, we note that many results of these papers are based on a single logic.

Haveshki et al. utilized the aforementioned method for BL-algebras in [15], which we recalled in Section 3. Moreover, the uniform structure generated by an equivalence relation is explained in this section. Recently, we verified in [3] that a uniform topology induced by this method on an MV-algebra coincides with a filter topology generated by a proper filter. More generally, we show in the third section that the uniform topology induced by this method coincides with the partition topology, and that the algebraic operations are continuous when the equivalence relation is a congruence. The topological properties of partition topologies are well-known. Therefore, in order to prevent further development of similar theories on different algebraic structures, we show that many results of the papers mentioned in Table 1 depend completely on the shape of open sets in the partition topology, and so are independent of our choice of the algebraic system. The proofs of our results in Section 3 are straightforward and should be well-known. We state some of them just for the sake of completeness.

Table 1
Adoption of the PUM to algebraic systems

Type of equivalence relation and underlying structure Refs.

Filter (BL-algebras) [14 , 21]

Extreme fuzzy filter (BL-algebras) [37]

Filter (MV-algebras) [3]

Riesz ideal (effect algebras) [29]

Weakly algebraic ideal (effect algebras) [23]

Filter (EQ-algebras) [34]

Regular ideal (BCC-algebras) [25]

BCC-ideal (BCC-algebras) [2]

Filter (residuated lattices) [12, 13]

Fuzzy filter (residuated lattices) [11]

Filter (BE-algebras) [24, 27]

Ideal (difference algebra) [4]

Deductive system (Hilbert algebra) [5, 26]

d^*-ideal (d-algebra) [6]

Ideal (subtraction algebras) [1]

Ideal (BCI-algebras) [28, 35]

Filter (fuzzy implication algebras) [30]

Implicative filter (positive implication algebras) [19]

Dual ideal (BCK-algebras) [20]

Hyper K-ideal (hyper K-algebras) [7]

Extreme fuzzy filter (equality algebras) [36]

Very true filter (very true equality algebras) [33]

S-reflexive hyper MV-filter (hyper MV-algebras) [10]

Type of equivalence relation and underlying structure	Refs.
Filter (BL-algebras)	[14 , 21]
Extreme fuzzy filter (BL-algebras)	[37]
Filter (MV-algebras)	[3]
Riesz ideal (effect algebras)	[29]
Weakly algebraic ideal (effect algebras)	[23]
Filter (EQ-algebras)	[34]
Regular ideal (BCC-algebras)	[25]
BCC-ideal (BCC-algebras)	[2]
Filter (residuated lattices)	[12, 13]
Fuzzy filter (residuated lattices)	[11]
Filter (BE-algebras)	[24, 27]
Ideal (difference algebra)	[4]
Deductive system (Hilbert algebra)	[5, 26]
d^*-ideal (d-algebra)	[6]
Ideal (subtraction algebras)	[1]
Ideal (BCI-algebras)	[28, 35]
Filter (fuzzy implication algebras)	[30]
Implicative filter (positive implication algebras)	[19]
Dual ideal (BCK-algebras)	[20]
Hyper K-ideal (hyper K-algebras)	[7]
Extreme fuzzy filter (equality algebras)	[36]
Very true filter (very true equality algebras)	[33]
S-reflexive hyper MV-filter (hyper MV-algebras)	[10]

2 Preliminaries

Throughout this paper, A is a BL-algebra. By this we mean that (A, ∨ , ∧ , ⊙ , → , 0, 1) is an algebra of type (2, 2, 2, 2, 0, 0) such that (A, ∨ , ∧ , 0, 1) is a bounded lattice, (A, ⊙ , 1) is a commutative monoid, and for every a, b and c in A,

a ⊙ b ≤ c if and only if a ≤ b → c,

(a → b) ∨ (b → a) =1, and

a ∧ b = a ⊙ (a → b).

A non-empty subset F of A is called a filter if it is closed under ⊙ and upwards closed with respect to ≤ (that is, x ≤ y and x ∈ F imply y ∈ F).

Let μ be an n-ary operation on an algebraic structure B. An equivalence relation ≡ on B is called a congruence with respect to μ if μ (a₁, a₂, …, a_n) ≡ μ (b₁, b₂, …, b_n) whenever a_i ≡ b_i for every i ∈ {1, 2, …, n}. For every filter F of A, one can define an equivalence relation ≡_F on A as follows: x ≡ _Fy if and only if x → y, y → x ∈ F, for all x, y ∈ A. Then, ≡_F is a congruence with respect to each algebraic operation μ ∈ {∨ , ∧ , ⊙ , →}. The equivalence class of x ∈ A with respect to ≡_F will be denoted by x/F. The quotient set A/≡ _F with its natural BL-algebraic structure is a BL-algebra, denoted by A/F.

Definition 2.1. [14, 31] A filter F of A is called

maximal, if F is proper and it is not contained in any other proper filter of A;

Boolean, if x ∨ x^* ∈ F for every x ∈ A, where x^* = x → 0;

implicative, if y → (y → x) ∈ F implies y → x ∈ F, for all x, y ∈ A;

positive implicative, if (x → y) → x ∈ F implies x ∈ F, for all x, y ∈ A.

Let X be a non-empty set and $P$ be a partition of X. Then, $P$ is a base for a topology $T$ on X which is called the partition topology generated by $P$ . The pair $(X, T)$ is called a partition space. The partition topology is characterized by $T = {\emptyset} \cup {all unions of the members of P} .$

The set △_X = {(x, x) : x ∈ X} is called the diagonal of X. For arbitrary subsets U and V of X × X, define the composition U ∘ V as {(x, z) : (x, y) ∈ V, (y, z) ∈ U forsome y ∈ X}. The inverse of the relation U is U^-1 = {(y, x) ∈ X × X : (x, y) ∈ U}.

Definition 2.2. Let X be a non-empty set, and $U$ be a non-empty family of subsets of X × X such that

each member U of $U$ contains the diagonal of X,

$U \in U$ implies $U^{- 1} \in U$ ,

$U \in U$ implies V ∘ V ⊆ U for some $V \in U$ ,

$U \cap V \in U$ whenever $U, V \in U$ , and

$U \in U$ and U ⊆ V ⊆ X × X imply $V \in U$ .

Then

U

is called a uniform structure (or a uniformity, for short) on X, and the pair

(X, U)

is called a uniform space.

Let $U$ be a uniformity on X. A subfamily $B$ of $U$ is called a base for $U$ if each member of $U$ contains a member of $B$ .

Theorem 2.3. [9] A non-empty family $B$ of subsets of X × X is a base for some uniformity on X if and only if the following statements are true.

Each member of $B$ contains the diagonal of X;

If $B \in B$ , then B^-1 contains a member of $B$ ;

That $B \in B$ implies C ∘ C ⊆ B for some $C \in B$ ;

If $B, C \in B$ , then D ⊆ B ∩ C for some $D \in B$ .

Theorem 2.4. [9] Let $(X, U)$ be a uniform space and let $T_{U} = {G \subseteq X : \forall x \in G \exists U \in U s . t . U [x] \subseteq G}$ , where U [x] = {y ∈ X : (x, y) ∈ U}. Then $T_{U}$ is a topology on X.

The corresponding topology in Theorem 2.4, $T_{U}$ , is called the uniform topology induced by $U$ . A uniform space $(X, U)$ is said to be totally bounded if for every $U \in U$ , there is a finite subset F of X such that X = ⋃ _x∈FU [x]. We refer the reader to [9] for those topological notions which are not explained here.

3 A topological characterization of the partition uniformity method

At the outset, according to [15], we recall the partition uniformity method for a BL-algebra A.

Let Λ be a family of filters of A which is closed under intersection. The family $K^{*} : = {U_{F} : F \in Λ}$ , where U_F : = {(x, y) ∈ A × A : x ≡ _Fy}, is a base for a uniform structure on A. The uniformity generated by $K^{*}$ is $K = {U \subseteq A \times A : U_{F} \subseteq U for some U_{F} \in K^{*}}$ , and the uniform topology induced by $K$ , denoted by T_Λ, is characterized by the fact that G ∈ T_Λ if and only if for each x ∈ G, U [x] ⊆ G for some $U \in K$ . If Λ = {F}, then T_Λ is denoted by T_F. This method of endowing an algebraic structure with a uniformity is called the partition uniformity method (PUM for short). Apparently, the idea of the PUM comes from Iséki’s method for equipping a BCK-algebra with a quasi-uniformity. See [17, 18]. The following appears as Theorem 5.1 in [15].

Theorem 3.1. If Λ is a family of filters of a BL-algebra which is closed under intersection, then T_Λ = T_J, where J = ⋂ {F : F ∈ Λ}.

Let us note that T_{1} = 2^A and T_A = {∅ , A}. We claim that for every family Λ of filters of A which is closed under intersection, the uniform topology T_Λ on A coincides with the partition topology generated by A/J on A, where J = ⋂ {F : F ∈ Λ}. To prove this claim, we need the following proposition.

Proposition 3.2. Let R be an equivalence relation on a non-empty set X. Then,

$B = {R}$ is a base for a uniformity, say $U_{R}$ , on X;

$B_{a} = {a / R}$ is a local base at a ∈ X;

the uniform topology $T_{U_{R}}$ coincides with the partition topology generated by X/R on X.

Proof. The uniformity $U_{R}$ generated by $B$ consists of all subsets U of X × X such that R ⊆ U, and the uniform topology $T_{U_{R}}$ is characterized by $T_{U_{R}} = {G \subseteq X : \forall x \in G \exists U \in U_{R} s . t . U [x] \subseteq G}$ .

For every a ∈ X, the set a/R is a subset of X, and R [x] = a/R for each x ∈ a/R. So $a / R \in T_{U_{R}}$ , for every a ∈ X. Now fix a ∈ X. For any open set U containing a, a/R ⊆ U implies that $B_{a} = {a / R}$ is a local base at a ∈ X, and (2) is proved. To prove (3), it is enough to notice that X/R is a base for $T_{U_{R}}$ if and only if $T_{U_{R}}$ coincides with the partition topology generated by X/R on X.□ The uniform structure $U_{R}$ is called partition uniformity. We now come to our main result.

Corollary 3.3. Let Λ be a family of filters of A which is closed under intersection. Then, T_Λ coincides with the partition topology generated by A/J, where J = ⋂ {F : F ∈ Λ}.

Proof. The intersection of any family of filters of A is still a filter of A, so U_J is an equivalence relation on A. Therefore, by Proposition 3.2 and Theorem 3.1, V ∈ T_Λ if and only if it is a union of a family of equivalence classes {x/J : x ∈ A}. That is, T_Λ coincides with the partition topology generated by A/J, and the proof is complete.□ The PUM was straightforwardly adopted to several different algebraic systems. See Table 1. Many results presented in these papers are similar and independent of the algebraic properties of the underlying structures. In fact, these results depend completely on the shape of open sets in the partition topology. Some topological properties of partition spaces are collected in the following proposition.

Proposition 3.4. Let R be an equivalence relation on X, and $(X, T)$ be a partition space generated by X/R. Then,

X is first-countable, zero-dimensional, completely regular, normal, locally compact and locally connected;

R = △ _X ⇔ X isdiscrete ⇔ X is T₂ ⇔ X is T₁ ⇔ X is T₀;

R = X × X ⇔ X is indiscrete ⇔ X is connected;

X is totally bounded ⇔ Xis countably compact ⇔ Xis compact ⇔ X/Ris finite.

Remark 3.5. It is not hard to see that many results of the aforementioned papers are immediate consequences of Proposition 3.4 and the topological characterization of partition topology. Examples include [15, Theorems 4.8, 5.7, Proposition 3.9, Corollaries 3.6, 4.9], [37, Theorems 4.3, 4.4, 4.10, 4.11, 4.12, 4.14, Corollary 4.4], [29, Proposition 4.6, Theorem 4.7], [23, Theorem 3.5, Propositions 3.2, 3.6, Lemma 3.4], [34, Theorems 4.6, 4.12, 4.13, 4.23, Corollary 4.7], [25, Proposition 4.3, 4.4], [2, Theorems 4.11, 4.12], [12, Corollary 3.15], [13, Theorems 4.8, 4.12, 4.16, Corollaries 4.6, 4.7, 4.10], [11, Theorems 3.7, 3.9, 3.17, 3.22, 3.23, 3.25, Corollaries 3.10, 3.18, 3.21], [24, Theorems 3.5, 3.6], [27, Theorems 4.6, 5.5, 5.8, Proposition 3.8, Corollary 4.7], [4, Theorems 4.6, 4.11, 4.12, Proposition 3.6, Corollary 4.7], [5, Theorem 4.8, Proposition 3.8, Corollaries 3.6, 4.9], [26, Corollary 3.7], [6, Theorems 4.3, 4.9, 4.11, 4.12, Proposition 3.9, Corollaries 3.6, 4.4], [1, Theorems 4.6, 4.11, 4.12, Proposition 3.6, Corollary 4.7], etc.

There are many different kinds of congruence relations on algebraic systems. For example, read about the theory of filters on residuated structures in [8] and [32]. Using the PUM with different types of congruence relations on a fixed algebraic system does not change the nature of the topological structure. Although the shape of open sets in the partition topology may change, the results are still obvious. Let us explain this point further with an example.

Example 3.6. Let F be a maximal and Boolean filter of a BL-algebra A. It is well-known [31, Theorem 2] that for every x ∈ A, x ∈ F or x^* ∈ F. So A/F = {1/F, 0/F}. Therefore, T_F = {∅ , 1/F, 0/F, A}. Clearly, T_F is compact and hence, [15, Theorem 5.8] is obvious. Furthermore, following [14, Proposition 3.19], every maximal and (positive) implicative filter of A is Boolean. So, [14, Theorem 3.23] is also trivial.

Remark 3.7. The uniform topology induced by the PUM in each of the following results coincides with the discrete topology: [30, Theorem 2.3], [2, Theorem 3.4], [19, Theorem 3.3], [1, Theorem 3.4] and [7, Theorem 3.8].

Now consider the following, which is a well-known question in the general theory of topological algebras.

Under what conditions does a non-discrete non-trivial topology exist on an algebraic system A which makes it into a topological algebra?

To provide an almost obvious answer, we need the following proposition.

Proposition 3.8. Let $T$ be a partition topology generated by X/R on X, and μ : A × A × ⋯ × A → A be an n-ary operation. If R is a congruence with respect to μ, then μ is continuous, when A × A × ⋯ × A carries the product topology.

Proof. Consider a = (a₁, a₂, …, a_n) ∈ A × A × ⋯ × A. The only base open set containing μ (a) is μ (a)/R. Also, $a_{i} / R \in T$ for i = 1, 2, …, n. For an arbitrary element μ (x₁, x₂, …, x_n) of the set μ (a₁/R × a₂/R × ⋯ × a_n/R), that x_i R a_i for every i ∈ {1, 2, …, n} implies μ (x₁, x₂, …, x_n) R μ (a). Hence, μ (x₁, x₂, …, x_n) ∈ μ (a)/R .

So, μ (a₁/R × a₂/R × ⋯ × a_n/R) ⊆ μ (a)/R implies the continuity of μ at the point a. That is, μ is continuous.□

A natural way to answer •is to build a non-identity non-trivial congruence relation R on A, and then equip A with the partition topology generated by A/R. From an algebraic point of view, the first part of this way is significant, but the second part is obvious in the sense of topology.

Remark 3.9. Following Proposition 3.8, one observes that the following theorems are trivial: [15, Theorem 4.1], [37, Theorem 4.2], [34, Theorem 4.3], [13, Theorem 4.15], [11, Theorem 3.27], [27, Theorem 4.1], [4, Theorem 4.1], [5, Theorem 4.1], [6, Theorem 4.1] and [1, Theorem 4.1]. We also note that the idea behind Theorem 3.13 of [21] is of this form.

4 Conclusion

For more than two decades, some authors used a well-known method for constructing uniform structures on some algebraic systems, and investigated the topological properties of such systems regardless of the nature of the topological structure in such spaces. We verified that, in fact, they equipped algebraic systems with partition uniformities generated by equivalence relations, and illuminated the triviality behind many results which were obtained from this method.

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