On topological basic algebras

1 Introduction

Two algebraic structures including MV-algebras and orthomodular lattices are employed based on the logic of quantum mechanics. Chajda [10] introduced the basic algebras to generalize orthomodular lattices similarly in which MV-algebras generalize Boolean algebras. A basic algebra (B ; ⊕ , ∼ , 0, 1), is a bounded lattice whose sections are antitone involutions. MV-algebras coincide with associative basic algebras and orthomodular lattices can be identified with basic algebras satisfying y ⊕ (x ∧ y). In addition, lattice effect algebras, which are algebraic structures for evaluating the logical foundations of quantum mechanics can be regarded as a subclass of basic algebras. Historically, Abbot [1] studied implication reduct for Boolean algebra named implication algebra. Further, chajda [11] introduced implication basic algebras, which are common generalization of orthoimplication algebras and implication reducts of MV-algebras.

Most important objects of mathematics represent a combination of algebraic and topological structure. It is well-known that the topological structures of basic algebras can describe some properties of quantum measures, which can help to characterize some algebraic operations of basic algebras.

During the recent decade, some mathematicians studied algebraic logical structures with their topologies and investigated their properties. Borzooei et al. [2–4, 18] defined (semi) topological BL-algebras and proved that other operations of BL-algebra A, ∨ and ∧, are continuous if (A, {⊙ , →} , U) is a topological BL-algebra. They found some conditions under which a semitopological BL-algebra is a T₀, T₁ or T₂-space. In addition, they studied metrizability on (semi) topological BL-algebras, along with their relationship between separation axioms and (semi) topological quotient BL-algebras. Further, Haveshki et al. [13] introduced the topology induced by uniformity on BL-algebras. Hoo [14] proposed topological notions on MV-algebras in order to define topological MV-algebras so that their operations ⊕, ⊖ , ∗ , … are continuous. Then, he considered linear topologies on an MV-algebra A, by family of ideals. furthermore, he defined I-adic topology and completion of A with respect to this topology, where I indicates an ideal of A. Recently, Najafi et al. [17] defined (para, quasi) topological MV-algebras and evaluated their related properties. The results indicated being a topological MV-algebra is regarded as the equivalent conditions for ( A , T ) , where A shows an MV-algebra and T is considered as a topology on A. Additionally, they obtained topologies by which an MV-algebra can be a (para, semi) topological. Luo et al. [16], showed that every weakly algebraic ideal of an effect algebra E induces a uniform topology (weakly algebraic ideal topology) where E is a first-countable, zero-dimensional, disconnected, locally compact and completely regular topological space, and the operation ⊕ of effect algebras is continuous with respect to these topologies.

The presented study aimed to introduce topologies leading to the operations of a basic algebra or implication basic algebra to be semicontinuous or continuous, along with evaluating their properties. To this aim, (semi) topological basic algebras and (semi) topological implication basic algebras are defined and some examples are presented. First, the various operations are defined based on basic algebras. Then, the relationship between semicontinuity and continuity operations is studied and we investigate that under which condition a (semi) topological basic algebra is a T _i - space for i ∈ {0, 1, 2} are found.

In general, the current research of topological logic algebras is mainly based on filters or ideals, which increased the motivation for conducting the present work.

It is known that 0 ≠ I ⊆ M is a congruence kernel of an MV-algebra M if and only if I is an ideal of MV-algebra i.e., if and only if it satisfies the conditions: (P1) a ⊕ b ∈ I for all a, b ∈ I, (P2) if a ∈ I and a ≥ b, then b ∈ I. Krňávek and Kühr [15] called such subsets pre-ideals. But for a basic algebra B , these conditions are not enough for I to be a congruence kernel of B . Chajda et al. [12] defined conditions (I1)-(I3) for ( 0 ∈ ) I ⊆ B . I is called an ideal, if I satisfied (I1), (I2) and (I3) and weak ideal, if it satisfied only (I1) and (I2). For commutative basic algebra B , any weak ideal is an ideal. They defined weak congruence of a basic algebra that means an equivalence relation Θ on B which satisfies the compatibility condition, (a, b) ∈ Θ if and only if (c ø a, c ø b) ∈ Θ (Θ is not a congruence). They proved that, if I is a (weak) ideal, Θ _I is a (weak) congruence which is defined by (x, y) ∈ Θ _I if and only if x ø y ∈ I and y ø x ∈ I and conversely, if Θ is a (weak) congruence, then (weak) congruence class [0]  _Θ is an (weak) ideal. Chajda [8] introduced (weak)filters and (weak)congruences for basic algebras too, and found the same results.

In this paper, we find some relations between (weak)ideals and (weak)filters of basic algebras. Then we attempt to construct some topologies by employing (weak) filters and (weak) ideals that will turn it into a left topological basic algebra which it is a zero dimensional, normal, disconnected, locally compact and completely regular (left) topological space. Also, we present the notion of quotient basic algebras and study the interaction of topological basic algebras and topological quotient basic algebras. Finally, for any implication basic algebra IB of cardinality n and filter F, we find an implication basic algebra IB * of cardinality n + 1 and filter F^* = F ∪ {z^*}, which z * ∉ IB and we prove that there exists at least, one nontrivial regular and normal topological implication basic algebra of cardinality n.

From the algebraic point of view, basic algebras slightly generalize MV-algebras. In addition, MV-algebras coincide with associative basic algebra. Further, we observe that topological MV-algebras, which are MV-algebras with topologies such that all operations are continuous with respect to these topologies, generalize topological basic algebras in a certain sense. Let MV-algebra M be as a basic algebra. Then, I is an ideal of M if and only if I satisfies in (I1) which is called "pre-ideal". We can see that, pre-ideals, weak ideals and ideals of M coincide each other and I is an ideal of M as a basic algebra [12]. The same conditions are available for filters of MV-algebras [8]. More explicitly, if ( M , T ) is a topological MV-algebra, then ( M , T ) is a topological basic algebra, because, all of the operations defined like MV-algebras, x ⊙ y : = ∼ (∼ x ⊕ ∼ y), x ⊖ y : = x ⊙ ∼ y, x ∨ y : = (x ⊖ y) ⊕ y and x ∧ y : = ∼ (∼ x ∨ ∼ y).

2 Preliminaries

In this section, we recollect some definitions and results about topology and basic algebras which will be used in this paper. For more details, we refer to the [4–12].

A basic algebra is an algebra ( B ; ⊕ , ∼ , 0 ) of type (2, 1, 0) that satisfies the identities:

x ⊕ 0 = x,

It is evident that every basic algebra can be redefined in the above signatures. In particular, → allows to reduce the number of defining identities (B1)–(B4) to (BI1) (x → 0) →0 = x, (BI2) (x → y) → y = (y → x) → x, (BI3) (((x → y) → y) → z) → (x → z) =1, where 1 = 0 →0.

A basic algebra B is called commutative if it satisfies the identity x ⊕ y = y ⊕ x, for any x , y ∈ B . Let us note that the binary operation ⊕ is not necessarily commutative nor associative. In any basic algebra ( B ; ⊕ , ∼ , 0 ) , the binary relation ≤ defined by x ≤ y ⇔ ∼ x ⊕ y = 1 is a partial order on B .

The following theorem provides some properties of basic algebras.

Theorem 2.1. Let ( B ; ⊕ , ∼ , 0 ) be a basic algebra. Then for all x , y , z ∈ B , the following properties hold:

x ⊕ 0 =0 ⊕ x = x, ∼ x ⊕ x = x ⊕ ∼ x = 1, x ⊕ 1 =1 ⊕ x = 1,

An algebra ( IB ; → ) of type (2) is called an implication basic algebra if for any x , y , z ∈ IB , it satisfies the following axioms: (IB1) (x → x) → x = x, (IB2) (x → y) → y = (y → x) → x, (IB3) (((x → y) → y) → z) → (x → z) = x → x. In any implication basic algebra ( IB ; → ) , there exists a constant 1 ∈ IB such that x → x = 1 and the binary operation ≤ defined by x ≤ y ⇔ x → y = 1 is a partial order and ( IB ; ≤ ) is a join-semilattice with the greatest element 1, where (x → y) → y = (y → x) → x = x ∨ y.

Lemma 2.2. Let ( IB ; → ) be an implication basic algebra. Then for each x , y , z ∈ IB , the following properties hold:

1 → x = x, x → 1 =1,

Theorem 2.3. Let ( B ; ⊕ , ∼ , 0 ) be a basic algebra. If we define x → y : = ∼ x ⊕ y, then ( B ; → ) is an implication basic algebra. Conversely, every implication basic algebra ( B ; → ) with least element 0, together with operations ∼ and ⊕ defined by ∼x : = x → 0 and x ⊕ y : = (x → 0) → y is a basic algebra.

A nonempty subset F of a basic algebra B (implication basic algebra IB ) containing 1 is called a weak filter of B ( IB ) if for all a , b , c ∈ B ( IB ) , (F1) a ∈ F and a → b ∈ F imply b ∈ F, (F2) a → b ∈ F and b ≤ a imply (b → c) → (a → c) ∈ F. A weak filter F is called a filter if (F3) a → b ∈ F and b → a ∈ F imply (c → a) → (c → b) ∈ F. A nonempty subset I of a basic algebra B containing 0 is called a weak ideal of B if for all a , b , c ∈ B , (I1) b ∈ I and a ø b ∈ I imply a ∈ I, (I2) a ø b ∈ I and b ≤ a imply (c ø b) ø (c ø a) ∈ I. A weak ideal I is called an ideal, if, for any a , b , c ∈ B , (I3) a ø b ∈ I and b ø a ∈ I imply (a ø c) ø (b ø c) ∈ I.

Notations. Let B be a basic algebra. We define mappings I, P, D, M, Q, C from B × B to B by, I (x, y)   : = x  →  y, P (x, y)   : = x  ⊕  y, D (x, y) : = x ⊙ y, M (x, y) : = x ⊖ y, Q (x, y) : = x ⇝ y, C (x, y) : = x ø y.

In the following, we give some preliminary definitions resulting in topology.

It is worth noting that a set A with a family T = { U α } α ∈ I of its subsets is called a topological space, denoted by ( A , T ) , if A , ∅ ∈ T , any finite intersection, and any arbitrary union of members of T is in T . The members of T are called "open sets" of A. The complement of U ∈ T , that is, A \ U is considered as a closed set. A subset P of A is a neighborhood of x ∈ A, if there exists an open set U such that x ∈ U ⊆ P. In addition, a subfamily {U _α } _α∈I of T is a base of T if an α ∈ I is available for each x ∈ U ∈ T such that x ∈ U _α  ⊆ U. Or, equivalently, each U in T is the union of members of {U _α } _α∈I. A subfamily {U _β } _β∈I of T is a subbase for T if the family of finite intersections of members of {U _β } _β∈I forms a base for T . Let (A, ∗) be an algebra of type (2) and T a topology on A. Then the operation ∗ is called left (right) continuous, if the map l a ∗ : A → A ( r a ∗ : A → A ) defined by l a ∗ ( x ) = x ∗ a ( r a ∗ ( x ) = a ∗ x ) is continuous for all a ∈ A. Or, equivalently, for any x in A and any open set U of x * a (a * x), there exists an open set V of x such that V * a ⊆ U (a * V ⊆ U) and ∗ is called semicontinuous, if the map l a ∗ and r a ∗ are continuous for all a ∈ A. Furthermore, ∗ is called continuous, if two open sets (neighborhoods) U and V are available for x and y, respectively, for any x, y in A and any open set (neighborhood) W of x * y such that U * V ⊆ W.

Let ( A , T 1 ) and ( B , T 2 ) be considered as two topological spaces. A mapping f : A → B is called a homeomorphism if f is a bijection and f and f^-1 are continuous. A topological space ( A , T ) is a T₀-space if there exists a neighborhood containing one point but not the other for each pair of distinct points. a T₁-space if each point lies in a neighborhood not containing the other for each pair of distinct points, a T₂-space (Hausdorff space) if there exist two disjoint neighborhoods of x₁ and x₂, respectively, for each pair of distinct points x₁ and x₂. Further, it is considered as a regular space if every neighborhood of a point contains the closure of a neighborhood of that point, a normal space if there exist two disjoint open sets U and V for each disjoint closed sets C and D such that we have C ⊆ U and D ⊆ V, a T₃-space if it is a T₁-space and a regular space, a T₄-space if it is a T₁-space and a normal space or equivalently, if there exists an open set H for each closed set C and each open set O containing C, such that C ⊆ H ⊆ H ¯ ⊆ O , a zero dimensional space if it has clopen base (close and open base).

In the present study, B = ( B ; ⊕ , ∼ , 0 ) and IB = ( IB ; → ) denote a basic algebra and an implication basic algebra, respectively, and T demonstrates a topology on B ( IB ), unless other cases are stated.

3 (Semi) Topological basic algebras

In this section, we introduce the notions of semitopological and topological (implication) basic algebras.

Definition 3.1. Let T be a topology on B ( IB ). Then ( B , T ) ( ( IB , T ) ) is called a

left topological (implication) basic algebra if → is left continuous,

Example 3.2. Let B = { 0 , a , b , 1 } be a lattice whose Hasse diagram is shown in Fig. 1 and the operations ⊕ and ∼ on B be defined as shown in Table 1. Routine calculations show that ( B ; ⊕ , ∼ ) is a basic algebra. It is not difficult to check that T = { { 0 , a } , { b , 1 } , B , ∅ } is a topology on B and ( B , T ) is a topological basic algebra.

5 Induced (semi) topological basic algebras

In this section, we show that in every basic algebra B , any arbitrary family of filters (ideals) induce a topology on B , which is completely regular, normal and locally compact. In Theorem 5.31, it is shown that for any n ≥ 3, there exists a nontrivial topological implication basic algebra, which is normal and regular.

Definition 5.1. [12] Let Θ be an equivalence relation on B .

Θ is called a weak congruence if for all a , b , c ∈ B , (CW) (a, b) ∈ Θ ⇒ (c ø a, c ø b) ∈ Θ.

Theorem 5.2. Let Θ be an equivalence relation on B . Then θ is a weak congruence if and only if it satisfies ( a , b ) ∈ Θ ⇒ ( a → c , b → c ) ∈ Θ ( CC ) Proof. If Θ satisfies (CW) or (CC) and (a, b) ∈ Θ, then (∼ a, ∼ b) ∈ Θ. Now, suppose (CC) is satisfied. For ∼ c ∈ B , we get (∼ a →  ∼ c, ∼ b →  ∼ c) ∈ Θ. This means that (∼ (c ø a) , ∼ (c ø b)) ∈ Θ, which implies that (c ø a, c ø b) ∈ Θ. Conversely assume that (CW) is satisfied. Let (a, b) ∈ Θ, then (∼ a, ∼ b) ∈ Θ. For ∼ c ∈ B , we get (∼ c ø ∼ a, ∼ c ø ∼ b) = (∼ (a → c) , ∼ (b → c)) ∈ Θ, which implies that (a → c, b → c) ∈ Θ. □

Theorem 5.3. [7, 12] (i) Let Θ be a (weak) congruence on B . Then [1]  _Θ and [0]  _Θ are a (weak) filter and an (a weak) ideal of B , respectively.

(ii) Let Θ be a congruence on IB . Then [1]  _Θ is a filter of IB .

(iii) Let F be a (weak) filter of B . Then the relation Θ _F is defined by ( a , b ) ∈ Θ F if and only if a → b ∈ F and b → a ∈ F is a (weak) congruence on B such that [1]  _{Θ F}  = F.

If F is a filter of IB , then the relation Θ _F is a congruence relation on IB such that [1]  _{Θ F}  = F.

(iv) Let I be an (a weak) ideal of B . Then the relation Θ _I is defined by ( a , b ) ∈ Θ I if and only if a ø b ∈ I and b ø a ∈ I , is a (weak) congruence relation on B such that [0]  _{Θ I}  = I.

Lemma 5.4. Let I (F) be an ideal (resp. filter) of B . Then the following conditions hold:

If x ∈ I and y ∈ B , then x ø y ∈ I,

Proof. (i) Let x ∈ I and y ∈ B . Then by (BA₇), (x ø y) ø x = 0 ∈ I, thus by (I1), we get that x ø y ∈ I. (ii) Let x ∈ B and y ∈ F. Then by (BA₈), y → (x → y) =1 ∈ F, thus, by (F1) we get that x → y ∈ F. Also, since y ∈ F, by (F1), x ∨ y = (x → y) → y ∈ F. (iii) Let I be an ideal (filter) of B and x, y ∈ I (resp. F). By (i) and (ii), x ø y ∈ I and y ø x ∈ I. Hence x Θ _I  y. If F is a filter,then by the similar way, we can prove that x Θ _F  y. □

Theorem 5.5. Let I be an (a weak) ideal of B and F =  ∼ I. Then, for all a , b ∈ B , we get that, ( a , b ) ∈ Θ I if and only if ( a , b ) ∈ Θ F .

Proof. Let (a, b) ∈ Θ _I . Then ∼ (∼ b →  ∼ a) = a ø b ∈ I and ∼ (∼ a →  ∼ b) = b ø a ∈ I. Hence ∼a →  ∼ b ∈ F and ∼b →  ∼ a ∈ F. By Proposition 4.4, F is a (weak) filter. Then by (F2), if c = 0, then a → b ∈ F and b → a ∈ F. The converse is obvious. □

Definition 5.6. For an arbitrary element x ∈ B ( IB ) and a nonempty subset U of B ( IB ) , the set U [x] is defined by U [ x ] = { y ∈ B ∣ x ∗ y ∈ U , y ∗ x ∈ U } , for any ∗ ∈ {→ , ø} (resp. ∗ ∈ {→}).

Remark 5.7. In Definition 5.6, if I is a (weak)ideal of B , then I [x] is obviously the equivalence class [ x ] I = { y ∈ B ∣ x Θ I y } . It is satisfied for any (weak) filter of B and IB , too.

Theorem 5.8. Let I be an arbitrary family of ideals of B which is closed under finite intersections. Then T I = { U ⊆ B ∣ ∀ x ∈ U , ∃ I ∈ I , I [ x ] ⊆ U } , is a topology on B with the base β = { I [ x ] ∣ x ∈ B , I ∈ I } = { [ x ] I ∣ x ∈ B , I ∈ I } and ( B , T I ) is a topological basic algebra and T I is called the induced topology by I .

Proof. It is easy to prove that T I is closed under arbitrary unions. T I is a nontrivial topology because for any x ∈ B and I ∈ I , I [ x ] ∈ T I (at least, there is a trivial ideal { 0 } ∈ B ). Note that for any x ∈ B , I [x] is the smallest neighborhood of x and T I is called the induced topology by I .

Suppose that I is a family of ideals of B . Let x , y ∈ B and U ∈ T I such that x ø y ∈ U. Then there exists an ideal I such that I [x ø y] ⊆ U. Since Θ _I is a congruence relation and I [x] is a class of Θ _I , we have I [x] ø I [y] ⊆ I [x ø y] ⊆ U. Hence by Proposition 3.7, ( B , T I ) is a topological basic algebra. Clearly, β is a base for T I .□

Theorem 5.9. Let F be an arbitrary family of filters of B which is closed under finite intersections. Then T F = { U ⊆ B ∣ ∀ x ∈ U , ∃ F ∈ F , F [ x ] ⊆ U } is a topology on B with the base β = { F [ x ] ∣ x ∈ B , F ∈ F } = { [ x ] F ∣ x ∈ B , F ∈ F } and ( B , T F ) is a topological basic algebra and T F is called the induced topology by F .

Proof. The proof is similar to the proof of Theorem 5.8. □

In the rest of paper, if I = { I } and F = { F } , then T F and T I are presented by T F and T I , respectively.

Example 5.10. Consider B = { 0 , a , b , 1 } be the basic algebra ( B ; ⊕ , ∼ ) as in Example 3.2. Routine calculations, show that the set of ideals of B is { { 0 , a } , { 0 , b } , { 0 } , B } .

Now, consider I = { I 0 , I 1 , I 2 } which I₀ = {0}, I₁ = {0, a} and I₂ = {0, b} are ideals of B . Then I₀ [x] = {0}, for all x ∈ B and I 1 [ x ] = { { 0 , a } , x ∈ { 0 , a } { x } , x ∈ { b , 1 } and I 2 [ x ] = { { 0 , b } , x ∈ { 0 , b } { x } , x ∈ { a , 1 }

Thus β = {{0} , {1} , {a} , {b} , {a, 0} , {b, 0}} is a base for T I and ( B , T I ) is a topological basic algebra.

Theorem 5.11. Let I be an arbitrary family of weak ideals of B which is closed under finite intersections. Then ( B , T I ) is a left topological basic algebra which β = { I [ x ] ∣ x ∈ B , I ∈ I } is the base of T I .

Proof. Suppose that I is a family of weak ideals of B . Let a , y ∈ B and U ∈ T I such that a ø y ∈ U. Then there exists I ∈ I such that I [a ø y] ⊆ U. We prove that a ø I [y] ⊆ I [a ø y]. Let b ∈ I [y]. Then b Θ _I y, where Θ _I is a weak congruence. By (CW), a ø b Θ _I  a ø y. Now, a ø b ∈ a ø I [y] ⊆ I [a ø y] ⊆ U. Thus ø is right continuous. Hence, by Corollary 3.9, the operation → is left continuous and ( B , T I ) is a left topological basic algebra. □

Corollary 5.12. Let F be an arbitrary family of weak filters of B which is closed under finite intersections. Then ( B , T ) is a left topological basic algebra which β = { F [ x ] ∣ x ∈ B , F ∈ F } is the base of T F .

Theorem 5.13. If I is an (a weak) ideal of B , then the induced topology by I is equal to induced topology by the (weak) filter F =  ∼ I.

Proof. By Theorem 5.5, the proof is clear. □

Example 5.14. Consider B = { 0 , a , b , c , d , e , f , 1 } be the basic algebra ( B ; ⊕ , ∼ ) as in Example 3.3. Routine calculations, show that I = {0, c} is a weak ideal of B . Then, I 1 [ x ] = { { 0 , c } , x ∈ { 0 , c } { a , e } , x ∈ { a , e } { b , f } , x ∈ { b , f } { d , 1 } , x ∈ { d , 1 }

Thus β = {{0, c} , {a, e} , {b, f} , {d, 1}} is a base for T I and ( B , T I ) is a left topological basic algebra. Easily, we can see that F = {1, d} is a weak filter of B and the induced topology by weak filter F is equal to induced topology by weak ideal I.

Proposition 5.15. If F is a nontrivial filter of B , then ∼F∩ F = ∅.

Proof. Let F be a filter of B and x ∈ ∼ F ∩ F. Then ∼x = (x → 0) ∈ F and x ∈ F. Thus, by (F1), 0 ∈ F and F = B , which is a contradiction. Now, let I be an ideal of B . Then by Proposition 4.4, ∼I is a filter and then ∼I∩ I = F ∩ ∼ F = ∅. □

Corollary 5.16. If I is a nontrivial ideal of B , then ∼I∩ I = ∅.

Proof. Let I be an ideal of B . Then by Proposition 4.4, ∼I = F is a filter of B and so ∼I ∩ I = ∼F∩ F = ∅. □

Definition 5.17. Let I be an ideal of B . I is called supermaximal if B = ∼ I ∪ I .

Remark 5.18. By Proposition 4.4, I is an ideal of B , if and only if ∼I is a filter of B . Then, the dual notion of supermaximal filters can be introduced such as supermaximal ideal.

Theorem 5.19. Let I be an ideal of B . Then the following statements are equivalent:

I is a supermaximal ideal of B ,

Proof. (i) ⇒ (ii) Let I be a supermaximal ideal of B and a ∈ B = I ∪ ∼ I . If a ∈ I, a/I = 0/I = I and if a ∈ ∼ I, a/(∼ I) =1/(∼ I) = (∼ I). So, B / F = { I , ∼ I } is a base for topology T I . Hence T I = { ∅ , I , ∼ I , B } . (ii) ⇒ (iii) It is straightforward. (iii) ⇒ (i) Let I be an ideal of B . Then ∅ ≠ [ 0 ] I = I ∈ T I . By Proposition 4.4 and Corollary 5.16, β = {I, ∼ I} is a base of T I and B = ∼ I ∪ I . Hence, I is a supermaximal ideal of B . □

Example 5.20. Let B = { 0 , a , b , c , d , e , f , g , h , i , j, 1} be the lattice that Hasse diagram is shown in Fig. 4 and the operations ⊕, ∼ and ø on B are defined as shown in Table 4. Routine calculations show that (B ; ⊕ , ∼) is a basic algebra. Moreover, it is not difficult to check that I₁ = {0, a} is a weak ideal and I = {0, b, d, f, h, j} is a supermaximal ideal of B . Now, let I = { I } . I [ x ] = { { 0 , b , d , f , h , j } , x ∈ { 0 , b , d , f , h , j } { a , c , e , g , i , 1 } , x ∈ { a , c , e , g , i , 1 } Then {{0, b, d, f, h, j} , {a, c, e, g, i, 1}} is a base for T I . Hence ( B , T I ) is a topological basic algebra.

OPEN IN VIEWER

Fig. 4

( B ; ⊕ , ∼ ) .

Now, consider the weak ideal I₁ = {0, a}. Then I 1 = { { 0 , a } , x ∈ { 0 , a } { 1 , b } , x ∈ { 1 , b } { c , h } , x ∈ { c , h } { d , g } , x ∈ { d , g } { e , j } , x ∈ { e , j } { f , i } , x ∈ { f , i } Hence {{0, a} , {1, b} , {c, h} , {d, g} , {e, j} , {f, i}} is a base for T I 1 . Therefore, ( B , T I 1 ) is a left topological basic algebra.

Proposition 5.21. Let I be a proper ideal (filter) of B . In the topological space ( B , T I ) , the following properties hold:

For each x ∈ I, I [x] = I.

Proof. (i) The proof is clear.

(ii) Let x ∈ B . Then I [x] is an open set. It is sufficient to prove that (I [x])  ^c is an open set, too. Let y ∈ I [x]  ^c . Then I [x]∩ I [y] = ∅, which yields I [y] ⊆ (I [x])  ^c . Since I [y] is an open set, I [x]  ^c is an open set. Therefore, I [x] (and so I = I [0]) is a clopen subset of B .

(iii) Suppose that O is an arbitrary open covering of I [x]. Then there exists an open set V ∈ O containing x and thus I [x] ⊆ V, because I [x] is the smallest neighborhood of x in this topology. Therefore, I [x] and I [0] = I (I [1] = I) are compact.

(iv) By (ii), I is a proper clopen subset of B , then ( B , T I ) is disconnected and it is zero dimensional, because β = { I [ x ] ∣ x ∈ B } is a clopen base for topology T I . Since for any x ∈ B , I [x] is a compact neighborhood of x, ( B , T I ) is a locally compact space. Now, let x ∈ B and V be a neighborhood of x. Then by (ii), there is a closed neighborhood I [x] of x which I [x] ⊆ V. Therefore ( B , T I ) is regular. Since ( B , T I ) is locally compact, ( B , T I ) is completely regular. Now, let C and O be a closed and open set, respectively, such that C ⊆ O. Since for x ∈ C, I [x] is the smallest neighborhood of x, then I [x] ⊆ O. By (ii), I [x] is a clopen subset of B . Then H = ⋃ _x∈CI [x] is a clopen subset of B . Therefore C ⊆ H ⊆ H ¯ ⊆ O , which implies that ( B , T I ) is a normal space. □

Now, in the following, we present the notion of quotient basic algebras and study the interaction of topological basic algebras and topological quotient basic algebras and find their properties.

Let B be a basic algebra and F be a filter of B . We define ⊕, → and ∼ on B / F by x / F ⊕ y / F = ( x ⊕ y ) / F , x / F → y / F = ( x → y ) / F and ∼ ( x / F ) = ( ∼ x ) / F .

It is easy to check that quotient algebra ( B / F ; ⊕ , ∼ ) is a basic algebra. Let T be a topology on B and π F : B ⟼ B / F be a canonical projection. Suppose T is a topology on B and U is a subset of B / F . Then U is an open subset of B / F if and only if π^-1 (U) is an open subset of T . Now, consider T π F = { U ⊆ B / F ∣ π F - 1 ( U ) ∈ T } .

T π F is the topology on B / F which is called the quotient topology induced by π _F . It is well known that it is the greatest topology on B / F making π _F continuous.

If we consider l a → ( x ) = L a ( x ) = x → a and r a → ( x ) = R a ( x ) = a → x , then it is easy to show that for each a ∈ B , π _F  ∘ L _a  = L_a/F ∘ π _F and π _F  ∘ R _a  = R_a/F ∘ π _F .

In the following, we give some conditions that B / F becomes a topological basic algebra.

Proposition 5.22. Let ( B , T ) be a topological basic algebra and F be a filter of B . If natural projection π F : B ⟼ B / F is an open map, then the quotient basic algebra B / F with quotient topology is a topological basic algebra.

Proof. It is sufficient to prove that → is continuous to quotient topology. Let V be an open neighborhoods of (x → y)/F = x/F → y/F. Then π F - 1 ( V ) is an open subset of B and x → y ∈ π F - 1 ( V ) = U . Since ( B , T ) is a topological basic algebra, there exist open neighborhoods U₁, U₂ of x and y, respectively, such that U 1 → U 2 ∈ π F - 1 ( V ) . Then V₁ = π _F  (U₁) and V₂ = π _F  (U₂) are open sets in B / F and x/F ∈ V₁ and y/F ∈ V₂, because π _F is an open map. Hence V₁ → V₂ ⊆ π _F  (U₁ → U₂) ⊆ U. Therefore → is continuous. □

Proposition 5.23. Let ( B , T F ) be a topological basic algebra induced by F and F be a filter of B . If F ⊆ ∩ { F i ∣ F i ∈ F } , then π F : B ⟼ B / F is an open map.

Proof. Let F be an arbitrary family of filters of B which is closed under finite intersection and F be a filter of B . By Theorem 5.9, β = { F i [ x ] ∣ x ∈ B , F i ∈ F } = { [ x ] F i ∣ x ∈ B , F i ∈ F } is a base for T F . So, it is sufficient to show that for all [x]  _{F i}  ∈ β, π _F  ([x]  _{F i} ) is an open set or π F - 1 π F ( [ x ] F i ) ∈ T F . Let [x]  _{F i}  ∈ β, for x ∈ B and F i ∈ F . Suppose a ∈ π F - 1 π F ( [ x ] F i ) . Then a/F = π _F  (a) ∈ π _F  ([x]  _{F i} ). Hence, there exists b ∈ [x]  _{F i} such that a/F ∈ π _F  (b) = b/F and aΘ _F b. Since aΘ _F b and F ⊆ F _i , we get that aΘ _{F i} b and bΘ _{F i} x. By transitivity condition, aΘ _{F i} x i.e, a ∈ [x]  _{F i} . So π F - 1 π F ( [ x ] F i ) ⊆ [ x ] F i . Clearly, [ x ] F i ⊆ π F - 1 π F ( [ x ] F i ) . Therefore, π F - 1 π F ( [ x ] F i ) = [ x ] F i ∈ T F and π is an open map. □