A topology induced by uniformity on BL-algebras - Semantic Scholar
Math. Log. Quart. 53, No. 2, 162 – 169 (2007) / DOI 10.1002/malq.200610035
A topology induced by uniformity on BL-algebras Masoud Haveshki∗ , Esfandiar Eslami∗∗ , and Arsham Borumand Saeid∗∗∗ Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran Received 27 April 2006, revised 25 December 2006, accepted 2 January 2007 Published online 15 March 2007 Key words Uniformity, BL-algebra, filters, topological BL-algebra. MSC (2000) 03G25, 54E15, 54H99 In this paper, we consider a collection of filters of a BL-algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction BL-algebras have been invented by P. H´ajek [3] in order to provide an algebraic proof of the completeness theorem of “Basic Logic” (BL, for short) arising from the continuous triangular norms, familiar in the fuzzy logic framework. The above notion is generalized to an algebraic system in which the required conditions are fulfilled (Definition 2.1). Filters in BL-algebras are also defined. A filter is a special subset of the BL-algebra that contains elements somehow “close” to 1, which is called the largest element. This leads us to think of a neighborhood system in a BL-algebra, collecting those subsets whose elements are “close” to each other. Some researchers [1, 2, 3] choose to have equivalence or congruence classes of some equivalence or congruence relation with respect to a given filter, respectively. In this paper, we consider a collection of filters and use congruence relation with respect to filters to define a uniformity and turn the BL-algebra into a uniform topological space with the desired subset as the open sets. Towards our goal, we renew some needed algebraic notions in Section 2. Then we consider the uniformity based on congruence relations with respect to a given collection of filters and construct the induced topology by this uniformity in Section 3. In the last sections we study the properties of these topologies such as compactness regarding different filters. Our future research will be investigating the effect of these topologies when we translate their meaning into the related BL-logic.
2 Preliminaries Definition 2.1 [3] A BL-algebra is an algebra (A, ∧, ∨, ∗, →, 0, 1) with four binary operations ∧, ∨, ∗, → and two constants 0, 1 such that (BL1) (A, ∧, ∨, 0, 1) is a bounded lattice; (BL2) (A, ∗, 1) is a commutative monoid; (BL3) ∗ and → form an adjoint pair, i. e., c ≤ a → b if and only if a ∗ c ≤ b for all a, b, c ∈ A; (BL4) a ∧ b = a ∗ (a → b); (BL5) (a → b) ∨ (b → a) = 1. ∗
e-mail: haveshki [email protected] e-mail: [email protected] ∗∗∗ Corresponding author: e-mail: [email protected] ∗∗
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Lemma 2.2 [3] In each BL-algebra A, the following relations hold for all x, y, z ∈ A: (1) x ∗ (x → y) ≤ y. (2) x ≤ (y → (x ∗ y)). (3) x ≤ y if and only if x → y = 1. Definition 2.3 [9] In each BL-algebra A, the order of an element x ∈ A is the smallest positive integer n such that xn = x ∗ · · · ∗ x = 0, denoted by ord(x) = n, and if no such n exist, then ord(x) = ∞. Definition 2.4 [3] A filter of a BL-algebra A is a nonempty subset F of A such that for all a, b ∈ A, we have (1) a, b ∈ F implies a ∗ b ∈ F ; (2) a ∈ F and a ≤ b imply b ∈ F . Definition 2.5 [7] A proper filter M of a BL-algebra A is called maximal (or ultrafilter) if it is not properly contained in any other proper filter of A. Proposition 2.6 [7] Any proper filter of a BL-algebra A can be extended to a maximal filter. Definition 2.7 [3] Let A be a BL-algebra and F a filter of A. F is a prime filter if and only if for all x, y in A, x → y ∈ F or y → x ∈ F . Definition 2.8 [9] A BL-algebra A is called local if it has a unique maximal filter. Proposition 2.9 [9] A BL-algebra A is local if and only if for all x ∈ A, ord(x) < ∞ or ord(x ∗ ) < ∞, where x∗ = x → 0. Definition 2.10 An element x of a BL-algebra A is a zero divisor if there is an element y ∈ A, y = 0, such that x ∗ y = 0. Definition 2.11 [2] Let X ⊆ A. The filter of A generated by X will be denoted by X . We have ∅ = {1} and
X = {a ∈ A | x1 ∗ x2 ∗ · · · ∗ xn ≤ a for some n ∈ N and some x1 , x2 , . . . , xn ∈ X}. Definition 2.12 [3] A congruence relation on a BL-algebra A is an equivalence relation R on A. Moreover, if xRy and uRv, then we have (Cg1) (x ∗ u)R(y ∗ v), (Cg2) (x → u)R(y → v) and (u → x)R(v → y), (Cg3) (x ∧ u)R(y ∧ v) and (x ∨ u)R(y ∨ v). Theorem 2.13 [3] Let F be a filter of a BL-algebra A. Define x ≡F y
if and only if x → y ∈ F and y → x ∈ F .
Then ≡F is a congruence relation on A. Definition 2.14 [8] Let A be a BL-algebra. A filter F is called Boolean if for all x ∈ A we have x ∨ x ∗ ∈ F . Theorem 2.15 [8] Let A be a BL-algebra and F a subset of A. Then the following conditions are equivalent: (1) F is a maximal and Boolean filter. (2) F is a prime and Boolean filter. (3) F is a proper filter, and x ∈ F or x∗ ∈ F for all x ∈ A.
3 Uniformity in a BL-algebra From now on (A, ∧, ∨, ∗, →, 0, 1) is a BL-algebra. Let X be a nonempty set and U, V be any subsets of X × X. Define U ◦ V = {(x, y) ∈ X × X | (z, y) ∈ U and (x, z) ∈ V for some z ∈ X}, U −1 = {(x, y) ∈ X × X | (y, x) ∈ U }, ∆ = {(x, x) ∈ X × X | x ∈ X}.
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Definition 3.1 [5] By a uniformity on X we shall mean a nonempty collection K of subsets of X × X which satisfies the following conditions: (U1) ∆ ⊆ 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 called a uniform structure (uniform space). Theorem 3.2 Let Λ be an arbitrary family of filters of A which is closed under intersection. If UF = {(x, y) ∈ A × A | x ≡F y} and
K∗ = {UF | F ∈ Λ},
then K∗ satisfies the conditions (U1) – (U4). P r o o f. (U1): Since F is a filter of A, we have x ≡F x for any x ∈ A, hence ∆ ⊆ UF for all UF ∈ K∗ . (U2): For any UF ∈ K∗ , we have (x, y) ∈ (UF )−1 ⇔ (y, x) ∈ UF ⇔ y ≡F x ⇔ x ≡F y ⇔ (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 , y → x ∈ F , x → y ∈ J, and y → x ∈ J. Then x ≡F ∩J y and hence (x, y) ∈ UF ∩J . Conversely, let (x, y) ∈ UF ∩J . Then x ≡F ∩J y, hence x → y ∈ F ∩ J and y → x ∈ F ∩ J, and thus x → y ∈ F , y → x ∈ F , x → y ∈ J, and y → x ∈ J. Therefore x ≡F y and x ≡J y, which means that (x, y) ∈ UF ∩ UJ . So UF ∩ UJ = UF ∩J . Since F, J ∈ Λ, then F ∩ J ∈ Λ, UF ∩ UJ ∈ K∗ . Theorem 3.3 Let K = {U ⊆ A × A | UF ⊆ U for some UF ∈ K∗ }. Then K satisfies a uniformity on A and the pair (A, K) is a uniform structure. P r o o f. 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 ⊆ A × A. Then there exists UF ⊆ U ⊆ V , which means that V ∈ K. This proves the theorem. Let x ∈ A and U ∈ K. Define U [x] := {y ∈ A | (x, y) ∈ U }. Theorem 3.4 Given a BL-algebra A, T = {G ⊆ A | (∀x ∈ G)(∃U ∈ K)(U [x] ⊆ G)} is a topology on A. P r o o f. Clearly, ∅ and the set A belong to T . Also from the very nature of that definition, 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, and so G ∩ H ∈ T . Thus T is a topology on A. Note that for any x in A, U [x] is an open neighborhood of x. Lemma 3.5 Let F be a filter of A. If F = {1}, then UF = U{1} . P r o o f. Since F = {1}, there is z ∈ F such that z = 1. By (BL3), z ≤ (1 → (z ∗ 1)). Since F is a filter, z = 1 → z ∈ F . By Lemma 2.2, z → 1 = 1 ∈ F . Hence 1 ∈ UF [z], and then (z, 1) ∈ UF . On the other hand, since z = 1, (z, 1) ∈ / U{1} . c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Corollary 3.6 Let F be a filter of A. Then F = {1} ∈ Λ if and only if T is a discrete topology. P r o o f. Let F = {1} ∈ Λ. Then UF ∈ K∗ . Hence UF [x] ∈ T for all x ∈ A. On the other hand, UF [x] = {y | y ≡F x} = {y | x → y = 1, y → x = 1} = {x}. Hence T is a discrete topology on A. Conversely, let T be a discrete topology on A. Then {x} is open for all x ∈ A. For arbitrary x ∈ A, there exists a filter F ∈ Λ such that UF [x] ⊆ {x}. Since x ∈ UF [x], UF [x] = {x}. In particular, there is a filter F in Λ such that UF [1] = {1} and, moreover, UF [1] = F . Then we have F = {1}. Definition 3.7 Let (A, K) be a uniform structure. Then the topology T is called the uniform topology on A induced by K. Example 3.8 Let B = {0, a, b, c, 1}. Define ∗ and → as follows: ∗ 1 0 a b c 1 1 0 a b c 0 0 0 0 0 0 a a 0 a a a b b 0 a b a c c 0 a a c ,
→ 1 0 a b c
1 1 1 1 1 1
0 a b c 0 a b c 1 1 1 1 0 1 1 1 0 c 1 c 0 b b 1 .
Easily we can check that (B, ∗, →, 0, 1) is a BL-algebra, whose lattice (B, ∧, ∨, 0, 1) is given by the partial order 1 c
@ @
@
@ @
b
@a 0.
Consider the filter F = {b, 1}, and let Λ = {F }. Therefore as in Theorem 3.2, we construct K∗ = {{UF }} = {{(x, y) | x ≡F y}} = {{(1, 1), (0, 0), (a, a), (b, b), (c, c), (1, b), (b, 1), (a, c), (c, a)}}. We can check that (B, K) is a uniform space, where K = {U | UF ⊆ U }. Open neighborhoods are UF [0] = {0},
UF [a] = {a, c},
UF [b] = {b, 1},
UF [c] = {a, c},
UF [1] = {1, b}.
From above we get that T = {{0}, {a, c}, {b, 1}, {0, a, c}, {0, b, 1}, {a, b, c, 1}, {0, a, b, c, 1}, ∅}. Thus (B, T ) is a uniform topological space. Proposition 3.9 The topological space (A, T ) is completely regular. P r o o f. See [5, Theorem 14.2.9].
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4 Topological property of the space (A, T ) Let A be a BL-algebra and C, D subsets of A. Then we define C ∗ D and C → D, respectively, as follows: C ∗ D = {x ∗ y | x ∈ C, y ∈ D},
C → D = {x → y | x ∈ C, y ∈ D}.
Let A be a BL-algebra and T a topology defined on the set A. Then we say that the pair (A, T ) is a topological BL-algebra if the BL-algebraic operations ∗ and → are continuous with respect to T . The continuity of the BL-algebraic operations ∗ and → is equivalent to having the following properties satisfied: (a) Let O be an open set and a, b ∈ A such that a ∗ b ∈ O. Then there are open sets O1 and O2 such that a ∈ O1 , b ∈ O2 , and O1 ∗ O2 ⊆ O. (b) Let O be an open set and a, b ∈ A such that a → b ∈ O. Then there are O1 and O2 such that a ∈ O1 , b ∈ O2 , and O1 → O2 ⊆ O. Theorem 4.1 The pair (A, T ) is a topological BL-algebra. P r o o f. Let us first prove (a). Indeed, assume that x ∗ y ∈ G, where x, y ∈ A and G is an open subset of A. Then there exist U ∈ K, U [x ∗ y] ⊆ G, and a filter F such that UF ⊆ U . We claim that the following relation holds: UF [x] ∗ UF [y] ⊆ U [x ∗ y]. Indeed, for h ∈ UF [x] and k ∈ UF [y] we get 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. Thus the condition (a) is verified. To verify (b) with a similar argument as above, we get UF [x] → UF [y] ⊆ UF [x → y]. Theorem 4.2 [5] Let X be a set and S ⊂ P(X × X) be a family such that for every U ∈ S the following conditions hold: (1) ∆ ⊆ U . (2) U −1 contains a member of S. (3) There exists V ∈ S such that V ◦ V ⊆ U . Then there exists a unique uniformity U for which S is a subbase. Theorem 4.3 If we let B = {UF | F is a filter of A}, then B is a subbase for a uniformity of A, we denote this topology by S. P r o o f. Since ≡F is an equivalence relation, it is clear that B satisfies the axioms of Theorem 4.2. We say that the topology σ is finer than τ if τ ⊆ σ as subsets of the power set. Then we have: Corollary 4.4 The topology S is finer than T . Proposition 4.5 If we let M = {UM | M is a maximal filter of A}, then M is a subbase for a uniformity of A, we denote this topology by Max. Corollary 4.6 The topology T is finer than Max. Theorem 4.7 Any filter in the collection Λ is a clopen subset of A.
P r o o f. Let F be a filter of A in Λ and y ∈ F c . Then y ∈ UF [y] and we get F c ⊆ {UF [y] | y ∈ F c }. We claim that for all y ∈ F c , UF [y] ⊆ F c . If z ∈ U F [y], then z ≡F y. Hence z → y ∈ cF . Ifz ∈ F , then y ∈ cF , which is a contradiction. So z ∈ F c and we get {UF [y] | y ∈ F c } ⊆ F c . Hence F = {UF [y] | y ∈ F }, and since UF [y] is open for all y∈ A, F is a closed subset. 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 , z ∈ F , and we get {UF [y] | y ∈ F } ⊆ F . So F is also an open subset of A.
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Theorem 4.8 For any x ∈ A and F ∈ Λ, UF [x] is a clopen subset of A. P r o o f. We show that (UF [x])c is open. If y ∈ (UF [x])c , then x → y ∈ F c or y → x ∈ F c . Let y → x ∈ F c . Hence by Theorem 4.1 and Theorem 4.7, (UF [y] → UF [x]) ⊆ UF [y → x] ⊆ F c . We claim that UF [y] ⊆ (UF [x])c . If z ∈ UF [y], then z → x ∈ (UF [z] → UF [x]) = UF [z → x] ⊆ UF [y → x] ⊆ F c . So z → x ∈ F c , we get z ∈ (UF [x])c . 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 A. A topological space X is connected if and only if has only X and ∅ as clopen subsets. Therefore we have: Corollary 4.9 The space (A, T ) is not, in general, a connected space.
5 Some connection between uniform topology and filters We denote the uniform topology obtained by an arbitrary family Λ by TΛ , and if Λ = {F }, we denote it by TF . Theorem 5.1 TΛ = TJ , where J = {F | F ∈ Λ}. P r o o f. Let K and K∗ be as in Theorems 3.2 and 3.3. Now consider Λ0 = {J}, define (K0 )∗ = {UJ } and K0 = {U | UJ ⊆ U }. Let G ∈ TΛ . So for all x ∈ G, there is U ∈ K such that U [x] ⊆ G. From J ⊆ F we get that UJ ⊆ UF for all filters F of A. Since U ∈ K, there exists F ∈ Λ such that UF ⊆ U . Hence UJ [x] ⊆ UF [x] ⊆ G. Since UJ ∈ K0 , G ∈ TJ . So TΛ ⊆ TJ . Conversely, let H ∈ TJ , then for all x ∈ H there is U ∈ K0 such that U [x] ⊆ H. So UJ [x] ⊆ H, and since Λ is closed under intersection, J ∈ Λ. Then we get UJ ∈ K and so H ∈ TΛ . Thus TJ ⊆ TΛ . Corollary 5.2 Let F and J be filters of A and F ⊆ J. Then J is clopen in the topological space (A, T F ). P r o o f. Consider Λ = {F, J}. Then by Theorem 5.1, TΛ = TF , and therefore J is clopen in the topological space (A, TF ). Theorem 5.3 Let F and J be filters of A. Then TF ⊆ TJ if and only if J ⊆ F . P r o o f. Let J ⊆ F . Consider Λ1 = {F }, K1 ∗ = {UF }, K1 = {U | UF ⊆ U }, Λ2 = {J}, K2 ∗ = {UJ }, K2 = {U | UJ ⊆ U }.
and
Let G ∈ TF . Then for all x ∈ G, there exists U ∈ K1 such that U [x] ⊆ G. Since J ⊆ F , we have UJ ⊆ UF , and since UF [x] ⊆ G, we get UJ [x] ⊆ G. UJ ∈ K2 and so G ∈ TJ . Conversely, let TF ⊆ TJ . In contrary let a ∈ J \ F . Since F ∈ TF , by hypothesis we get that F ∈ TJ . Then for all x ∈ F , there exists U ∈ K2 such that U [x] ⊆ F , and so J = UJ [1] ⊆ F . Corollary 5.4 Let F and J be filters of A. Then F = J if and only if TF = TJ . Corollary 5.5 The following conditions are equivalent: (1) There exists a proper filter M of a BL-algebra A such that TM ⊆ TF for all filters F of A. (2) A is local. (3) For all x ∈ A, ord(x) < ∞ or ord(x∗ ) < ∞. www.mlq-journal.org
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P r o o f. (1) ⇔ (2): Let TM ⊆ TF for all filters F of A. Then by Theorem 5.3, F ⊆ M for all filters F of A. Hence M is a unique maximal filter of A. Conversely, let A be local. Then there exists a unique maximal filter M of A. By Proposition 2.6, F ⊆ M for all filters F of A. Thus by Theorem 5.3, TM ⊆ TF . (2) ⇔ (3): This is clear by Proposition 2.9. Theorem 5.6 Let M = A \ ZD(A), where ZD(A) = {x ∈ A | x is zero divisor}, then the following conditions are equivalent: (1) If TF ⊂ TM , then F = A. (2) M is a maximal filter. (3) ord(x) < ∞ for all x ∈ ZD(A). P r o o f. (1) ⇔ (2): This is clear by Theorem 5.3. (2) ⇔ (3): Let M be a maximal filter and x ∈ ZD(A), so x ∈ M c . Since M is a maximal filter, we have that M ∪ {x} = A. Hence x1 ∗ x2 ∗ · · · ∗ xn ≤ 0 for some xi ∈ M ∪ {x}. Then 0 = x1 ∗ x2 ∗ · · · ∗ xn = f1 ∗ f2 ∗ · · · ∗ fm ∗ xt for some t > 0, and fi ∈ M for all 1 ≤ i ≤ n. We show that t = 0. In contrary suppose that t = 0, then f1 ∗ f2 ∗ · · · ∗ fm = 0 ∈ M, which is a contradiction. So xt = 0 and thus ord(x) = t < ∞. Conversely, let every zero divisor of A be of finite order. If M ⊂ F ⊆ A, then there exists a ∈ F \ M , thus a is a zero divisor, i. e., there exists n such that an = 0. Therefore F = A. Recall that a uniform space (X, K) is totally bounded if for each U ∈ K, there exist x 1 , . . . , xn ∈ X such n that X = i=1 U [xi ], and X is compact if any open cover of X has a finite subcover. Theorem 5.7 Let F be filter of A. Then the following conditions are equivalent: (1) The topological space (A, TF ) is compact. (2) The topological space (A, TF ) is totally bounded. (3) There exists P = {x1 , x2 , . . . , xn } ⊆ A such that for all a ∈ A there exists xi ∈ P such that a ≡F xi . P r o o f. (1) ⇒ (2): This is clear by [5, Theorem 14.3.8]. (2) ⇒ (3): Let UF ∈ K. Since (A, TF ) is totally bounded, there exist x1 , x2 , . . . , xn ∈ F such that n A = i=1 UF [xi ]. Now let a ∈ A. Then there exists xi such that a ∈ UF [xi ], therefore a → xi ∈ F and xi → a ∈ F . (3) ⇒ (1): For any a ∈ Aby hypothesis there exists xi ∈ P such that a → xi ∈ F and xi → a ∈ F . Theren fore a ∈ UF [xi ], hence A = i=1 UF [xi ]. Now let A = α∈I Oα , where each Oα is an open set of A. Then for any xi ∈ A there exists αi ∈ I such that xi ∈ Oαi . Since Oαi is an open set, UF [xi ] ⊆ Oαi , so n n A = i=1 UF [xi ] ⊆ i=1 Oαi . n Therefore A = i=1 Oαi which means that (A, TF ) is compact. Theorem 5.8 Let F be a maximal and Boolean filter of A. Then the topological space (A, T F ) is compact. P r o o f. Let A = α∈I Oα , where each Oα is an open set of A. Then there exist α, β ∈ I such that 1 ∈ Oα and 0 ∈ Oβ , thus UF [1] ⊆ Oα and UF [0] ⊆ Oβ .
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On the other hand, we have UF [1] = {x | x → 1 ∈ F and 1 → x ∈ F } = {x | x ∈ F }, UF [0] = {x | x → 0 ∈ F and 0 → x ∈ F } = {x | x∗ ∈ F }. Since F is a maximal and Boolean filter, by Theorem 2.15, wehave x ∈ F or x∗ ∈ F for all x ∈ A. Thus for all x ∈ A we have x ∈ UF [1] or x ∈ UF [0], therefore A = Oα Oβ , which means that (A, TF ) is compact. Remark 5.9 In general, the converse of the theorem above is not valid. To show this, consider the topological space of Example 3.8. Since B is finite, the topological space (B, TF ) is compact for the filter F = {1, b}, but F is not a maximal filter of B. Theorem 5.10 If F is a filter of A such that F c is a finite set, then the topological space (A, TF ) is compact. P r o o f. Let A = α∈I Oα , where each Oα is an open subset of A. Let F c = {x1 , x2 , . . . , xn }. Then there exist α, α1 , α2 , . . . , αn ∈ I such that 1 ∈ Oα , x1 ∈ Oα1 , . . . , xn ∈ Oαn . Then UF [1] ⊆ Oα , but UF [1] = F . Hence A = Oα1 ∪ Oα2 ∪ · · · ∪ Oαn . Theorem 5.11 If F is a filter of A, then F is a compact set in the topological space (A, T F ). P r o o f. Let F ⊆ α∈I Oα , where each Oα is an open set of A. 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 (A, TF ). Theorem 5.12 If F is a filter of A, then for all x ∈ A, UF [x] is a compact set in the topological space (A, TF ). P r o o f. Let UF [x] ⊆ α∈I Oα , where each Oα is an open set of A. Since x ∈ UF [x], there exists α ∈ I such that x ∈ Oα . Then UF [x] ⊆ Oα . Hence UF [x] is compact. Acknowledgements comments.
The authors would like to express their sincere thanks to the referees for their valuable suggestions and
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A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo BL-algebra: part I. Mult. Val. Logic 8 (5 – 6), 673 – 714 (2002). A. Di Nola and L. Leustean, Compact representations of BL-algebra. Arch. Math. Logic 42, 737 – 761 (2003). P. H´ajek, Metamathematics of Fuzzy Logic (Kluwer Academic Publishers, 1988). I. M. James, Introduction to Uniform Spaces (Cambridge Untversity Press, 1990). K. D. Joshi, Introduction to General Topology (New Age International Publisher, 1997). W. Page, Topological Uniform Structures (Dover Publication, Inc., 1988). E. Turunen, BL-algebras of basic fuzzy logic. Mathware and Soft Computing 6, 49 – 61 (1999). E. Turunen, Boolean deductive systems of BL-algebras. Arch. Math. Logic 40, 467 – 473 (2001). E. Turunen and S. Sessa, Local BL-algebra. International J. Multiple-Valued Logic 6, 229 – 249 (2001). S. Willard, General Topology (Adison-Wesley Publishing Company, 1970).
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A topology induced by uniformity on BL-algebras Masoud Haveshki∗ , Esfandiar Eslami∗∗ , and Arsham Borumand Saeid∗∗∗ Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran Received 27 April 2006, revised 25 December 2006, accepted 2 January 2007 Published online 15 March 2007 Key words Uniformity, BL-algebra, filters, topological BL-algebra. MSC (2000) 03G25, 54E15, 54H99 In this paper, we consider a collection of filters of a BL-algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction BL-algebras have been invented by P. H´ajek [3] in order to provide an algebraic proof of the completeness theorem of “Basic Logic” (BL, for short) arising from the continuous triangular norms, familiar in the fuzzy logic framework. The above notion is generalized to an algebraic system in which the required conditions are fulfilled (Definition 2.1). Filters in BL-algebras are also defined. A filter is a special subset of the BL-algebra that contains elements somehow “close” to 1, which is called the largest element. This leads us to think of a neighborhood system in a BL-algebra, collecting those subsets whose elements are “close” to each other. Some researchers [1, 2, 3] choose to have equivalence or congruence classes of some equivalence or congruence relation with respect to a given filter, respectively. In this paper, we consider a collection of filters and use congruence relation with respect to filters to define a uniformity and turn the BL-algebra into a uniform topological space with the desired subset as the open sets. Towards our goal, we renew some needed algebraic notions in Section 2. Then we consider the uniformity based on congruence relations with respect to a given collection of filters and construct the induced topology by this uniformity in Section 3. In the last sections we study the properties of these topologies such as compactness regarding different filters. Our future research will be investigating the effect of these topologies when we translate their meaning into the related BL-logic.
2 Preliminaries Definition 2.1 [3] A BL-algebra is an algebra (A, ∧, ∨, ∗, →, 0, 1) with four binary operations ∧, ∨, ∗, → and two constants 0, 1 such that (BL1) (A, ∧, ∨, 0, 1) is a bounded lattice; (BL2) (A, ∗, 1) is a commutative monoid; (BL3) ∗ and → form an adjoint pair, i. e., c ≤ a → b if and only if a ∗ c ≤ b for all a, b, c ∈ A; (BL4) a ∧ b = a ∗ (a → b); (BL5) (a → b) ∨ (b → a) = 1. ∗
e-mail: haveshki [email protected] e-mail: [email protected] ∗∗∗ Corresponding author: e-mail: [email protected] ∗∗
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Lemma 2.2 [3] In each BL-algebra A, the following relations hold for all x, y, z ∈ A: (1) x ∗ (x → y) ≤ y. (2) x ≤ (y → (x ∗ y)). (3) x ≤ y if and only if x → y = 1. Definition 2.3 [9] In each BL-algebra A, the order of an element x ∈ A is the smallest positive integer n such that xn = x ∗ · · · ∗ x = 0, denoted by ord(x) = n, and if no such n exist, then ord(x) = ∞. Definition 2.4 [3] A filter of a BL-algebra A is a nonempty subset F of A such that for all a, b ∈ A, we have (1) a, b ∈ F implies a ∗ b ∈ F ; (2) a ∈ F and a ≤ b imply b ∈ F . Definition 2.5 [7] A proper filter M of a BL-algebra A is called maximal (or ultrafilter) if it is not properly contained in any other proper filter of A. Proposition 2.6 [7] Any proper filter of a BL-algebra A can be extended to a maximal filter. Definition 2.7 [3] Let A be a BL-algebra and F a filter of A. F is a prime filter if and only if for all x, y in A, x → y ∈ F or y → x ∈ F . Definition 2.8 [9] A BL-algebra A is called local if it has a unique maximal filter. Proposition 2.9 [9] A BL-algebra A is local if and only if for all x ∈ A, ord(x) < ∞ or ord(x ∗ ) < ∞, where x∗ = x → 0. Definition 2.10 An element x of a BL-algebra A is a zero divisor if there is an element y ∈ A, y = 0, such that x ∗ y = 0. Definition 2.11 [2] Let X ⊆ A. The filter of A generated by X will be denoted by X . We have ∅ = {1} and
X = {a ∈ A | x1 ∗ x2 ∗ · · · ∗ xn ≤ a for some n ∈ N and some x1 , x2 , . . . , xn ∈ X}. Definition 2.12 [3] A congruence relation on a BL-algebra A is an equivalence relation R on A. Moreover, if xRy and uRv, then we have (Cg1) (x ∗ u)R(y ∗ v), (Cg2) (x → u)R(y → v) and (u → x)R(v → y), (Cg3) (x ∧ u)R(y ∧ v) and (x ∨ u)R(y ∨ v). Theorem 2.13 [3] Let F be a filter of a BL-algebra A. Define x ≡F y
if and only if x → y ∈ F and y → x ∈ F .
Then ≡F is a congruence relation on A. Definition 2.14 [8] Let A be a BL-algebra. A filter F is called Boolean if for all x ∈ A we have x ∨ x ∗ ∈ F . Theorem 2.15 [8] Let A be a BL-algebra and F a subset of A. Then the following conditions are equivalent: (1) F is a maximal and Boolean filter. (2) F is a prime and Boolean filter. (3) F is a proper filter, and x ∈ F or x∗ ∈ F for all x ∈ A.
3 Uniformity in a BL-algebra From now on (A, ∧, ∨, ∗, →, 0, 1) is a BL-algebra. Let X be a nonempty set and U, V be any subsets of X × X. Define U ◦ V = {(x, y) ∈ X × X | (z, y) ∈ U and (x, z) ∈ V for some z ∈ X}, U −1 = {(x, y) ∈ X × X | (y, x) ∈ U }, ∆ = {(x, x) ∈ X × X | x ∈ X}.
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Definition 3.1 [5] By a uniformity on X we shall mean a nonempty collection K of subsets of X × X which satisfies the following conditions: (U1) ∆ ⊆ 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 called a uniform structure (uniform space). Theorem 3.2 Let Λ be an arbitrary family of filters of A which is closed under intersection. If UF = {(x, y) ∈ A × A | x ≡F y} and
K∗ = {UF | F ∈ Λ},
then K∗ satisfies the conditions (U1) – (U4). P r o o f. (U1): Since F is a filter of A, we have x ≡F x for any x ∈ A, hence ∆ ⊆ UF for all UF ∈ K∗ . (U2): For any UF ∈ K∗ , we have (x, y) ∈ (UF )−1 ⇔ (y, x) ∈ UF ⇔ y ≡F x ⇔ x ≡F y ⇔ (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 , y → x ∈ F , x → y ∈ J, and y → x ∈ J. Then x ≡F ∩J y and hence (x, y) ∈ UF ∩J . Conversely, let (x, y) ∈ UF ∩J . Then x ≡F ∩J y, hence x → y ∈ F ∩ J and y → x ∈ F ∩ J, and thus x → y ∈ F , y → x ∈ F , x → y ∈ J, and y → x ∈ J. Therefore x ≡F y and x ≡J y, which means that (x, y) ∈ UF ∩ UJ . So UF ∩ UJ = UF ∩J . Since F, J ∈ Λ, then F ∩ J ∈ Λ, UF ∩ UJ ∈ K∗ . Theorem 3.3 Let K = {U ⊆ A × A | UF ⊆ U for some UF ∈ K∗ }. Then K satisfies a uniformity on A and the pair (A, K) is a uniform structure. P r o o f. 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 ⊆ A × A. Then there exists UF ⊆ U ⊆ V , which means that V ∈ K. This proves the theorem. Let x ∈ A and U ∈ K. Define U [x] := {y ∈ A | (x, y) ∈ U }. Theorem 3.4 Given a BL-algebra A, T = {G ⊆ A | (∀x ∈ G)(∃U ∈ K)(U [x] ⊆ G)} is a topology on A. P r o o f. Clearly, ∅ and the set A belong to T . Also from the very nature of that definition, 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, and so G ∩ H ∈ T . Thus T is a topology on A. Note that for any x in A, U [x] is an open neighborhood of x. Lemma 3.5 Let F be a filter of A. If F = {1}, then UF = U{1} . P r o o f. Since F = {1}, there is z ∈ F such that z = 1. By (BL3), z ≤ (1 → (z ∗ 1)). Since F is a filter, z = 1 → z ∈ F . By Lemma 2.2, z → 1 = 1 ∈ F . Hence 1 ∈ UF [z], and then (z, 1) ∈ UF . On the other hand, since z = 1, (z, 1) ∈ / U{1} . c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Corollary 3.6 Let F be a filter of A. Then F = {1} ∈ Λ if and only if T is a discrete topology. P r o o f. Let F = {1} ∈ Λ. Then UF ∈ K∗ . Hence UF [x] ∈ T for all x ∈ A. On the other hand, UF [x] = {y | y ≡F x} = {y | x → y = 1, y → x = 1} = {x}. Hence T is a discrete topology on A. Conversely, let T be a discrete topology on A. Then {x} is open for all x ∈ A. For arbitrary x ∈ A, there exists a filter F ∈ Λ such that UF [x] ⊆ {x}. Since x ∈ UF [x], UF [x] = {x}. In particular, there is a filter F in Λ such that UF [1] = {1} and, moreover, UF [1] = F . Then we have F = {1}. Definition 3.7 Let (A, K) be a uniform structure. Then the topology T is called the uniform topology on A induced by K. Example 3.8 Let B = {0, a, b, c, 1}. Define ∗ and → as follows: ∗ 1 0 a b c 1 1 0 a b c 0 0 0 0 0 0 a a 0 a a a b b 0 a b a c c 0 a a c ,
→ 1 0 a b c
1 1 1 1 1 1
0 a b c 0 a b c 1 1 1 1 0 1 1 1 0 c 1 c 0 b b 1 .
Easily we can check that (B, ∗, →, 0, 1) is a BL-algebra, whose lattice (B, ∧, ∨, 0, 1) is given by the partial order 1 c
@ @
@
@ @
b
@a 0.
Consider the filter F = {b, 1}, and let Λ = {F }. Therefore as in Theorem 3.2, we construct K∗ = {{UF }} = {{(x, y) | x ≡F y}} = {{(1, 1), (0, 0), (a, a), (b, b), (c, c), (1, b), (b, 1), (a, c), (c, a)}}. We can check that (B, K) is a uniform space, where K = {U | UF ⊆ U }. Open neighborhoods are UF [0] = {0},
UF [a] = {a, c},
UF [b] = {b, 1},
UF [c] = {a, c},
UF [1] = {1, b}.
From above we get that T = {{0}, {a, c}, {b, 1}, {0, a, c}, {0, b, 1}, {a, b, c, 1}, {0, a, b, c, 1}, ∅}. Thus (B, T ) is a uniform topological space. Proposition 3.9 The topological space (A, T ) is completely regular. P r o o f. See [5, Theorem 14.2.9].
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4 Topological property of the space (A, T ) Let A be a BL-algebra and C, D subsets of A. Then we define C ∗ D and C → D, respectively, as follows: C ∗ D = {x ∗ y | x ∈ C, y ∈ D},
C → D = {x → y | x ∈ C, y ∈ D}.
Let A be a BL-algebra and T a topology defined on the set A. Then we say that the pair (A, T ) is a topological BL-algebra if the BL-algebraic operations ∗ and → are continuous with respect to T . The continuity of the BL-algebraic operations ∗ and → is equivalent to having the following properties satisfied: (a) Let O be an open set and a, b ∈ A such that a ∗ b ∈ O. Then there are open sets O1 and O2 such that a ∈ O1 , b ∈ O2 , and O1 ∗ O2 ⊆ O. (b) Let O be an open set and a, b ∈ A such that a → b ∈ O. Then there are O1 and O2 such that a ∈ O1 , b ∈ O2 , and O1 → O2 ⊆ O. Theorem 4.1 The pair (A, T ) is a topological BL-algebra. P r o o f. Let us first prove (a). Indeed, assume that x ∗ y ∈ G, where x, y ∈ A and G is an open subset of A. Then there exist U ∈ K, U [x ∗ y] ⊆ G, and a filter F such that UF ⊆ U . We claim that the following relation holds: UF [x] ∗ UF [y] ⊆ U [x ∗ y]. Indeed, for h ∈ UF [x] and k ∈ UF [y] we get 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. Thus the condition (a) is verified. To verify (b) with a similar argument as above, we get UF [x] → UF [y] ⊆ UF [x → y]. Theorem 4.2 [5] Let X be a set and S ⊂ P(X × X) be a family such that for every U ∈ S the following conditions hold: (1) ∆ ⊆ U . (2) U −1 contains a member of S. (3) There exists V ∈ S such that V ◦ V ⊆ U . Then there exists a unique uniformity U for which S is a subbase. Theorem 4.3 If we let B = {UF | F is a filter of A}, then B is a subbase for a uniformity of A, we denote this topology by S. P r o o f. Since ≡F is an equivalence relation, it is clear that B satisfies the axioms of Theorem 4.2. We say that the topology σ is finer than τ if τ ⊆ σ as subsets of the power set. Then we have: Corollary 4.4 The topology S is finer than T . Proposition 4.5 If we let M = {UM | M is a maximal filter of A}, then M is a subbase for a uniformity of A, we denote this topology by Max. Corollary 4.6 The topology T is finer than Max. Theorem 4.7 Any filter in the collection Λ is a clopen subset of A.
P r o o f. Let F be a filter of A in Λ and y ∈ F c . Then y ∈ UF [y] and we get F c ⊆ {UF [y] | y ∈ F c }. We claim that for all y ∈ F c , UF [y] ⊆ F c . If z ∈ U F [y], then z ≡F y. Hence z → y ∈ cF . Ifz ∈ F , then y ∈ cF , which is a contradiction. So z ∈ F c and we get {UF [y] | y ∈ F c } ⊆ F c . Hence F = {UF [y] | y ∈ F }, and since UF [y] is open for all y∈ A, F is a closed subset. 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 , z ∈ F , and we get {UF [y] | y ∈ F } ⊆ F . So F is also an open subset of A.
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Theorem 4.8 For any x ∈ A and F ∈ Λ, UF [x] is a clopen subset of A. P r o o f. We show that (UF [x])c is open. If y ∈ (UF [x])c , then x → y ∈ F c or y → x ∈ F c . Let y → x ∈ F c . Hence by Theorem 4.1 and Theorem 4.7, (UF [y] → UF [x]) ⊆ UF [y → x] ⊆ F c . We claim that UF [y] ⊆ (UF [x])c . If z ∈ UF [y], then z → x ∈ (UF [z] → UF [x]) = UF [z → x] ⊆ UF [y → x] ⊆ F c . So z → x ∈ F c , we get z ∈ (UF [x])c . 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 A. A topological space X is connected if and only if has only X and ∅ as clopen subsets. Therefore we have: Corollary 4.9 The space (A, T ) is not, in general, a connected space.
5 Some connection between uniform topology and filters We denote the uniform topology obtained by an arbitrary family Λ by TΛ , and if Λ = {F }, we denote it by TF . Theorem 5.1 TΛ = TJ , where J = {F | F ∈ Λ}. P r o o f. Let K and K∗ be as in Theorems 3.2 and 3.3. Now consider Λ0 = {J}, define (K0 )∗ = {UJ } and K0 = {U | UJ ⊆ U }. Let G ∈ TΛ . So for all x ∈ G, there is U ∈ K such that U [x] ⊆ G. From J ⊆ F we get that UJ ⊆ UF for all filters F of A. Since U ∈ K, there exists F ∈ Λ such that UF ⊆ U . Hence UJ [x] ⊆ UF [x] ⊆ G. Since UJ ∈ K0 , G ∈ TJ . So TΛ ⊆ TJ . Conversely, let H ∈ TJ , then for all x ∈ H there is U ∈ K0 such that U [x] ⊆ H. So UJ [x] ⊆ H, and since Λ is closed under intersection, J ∈ Λ. Then we get UJ ∈ K and so H ∈ TΛ . Thus TJ ⊆ TΛ . Corollary 5.2 Let F and J be filters of A and F ⊆ J. Then J is clopen in the topological space (A, T F ). P r o o f. Consider Λ = {F, J}. Then by Theorem 5.1, TΛ = TF , and therefore J is clopen in the topological space (A, TF ). Theorem 5.3 Let F and J be filters of A. Then TF ⊆ TJ if and only if J ⊆ F . P r o o f. Let J ⊆ F . Consider Λ1 = {F }, K1 ∗ = {UF }, K1 = {U | UF ⊆ U }, Λ2 = {J}, K2 ∗ = {UJ }, K2 = {U | UJ ⊆ U }.
and
Let G ∈ TF . Then for all x ∈ G, there exists U ∈ K1 such that U [x] ⊆ G. Since J ⊆ F , we have UJ ⊆ UF , and since UF [x] ⊆ G, we get UJ [x] ⊆ G. UJ ∈ K2 and so G ∈ TJ . Conversely, let TF ⊆ TJ . In contrary let a ∈ J \ F . Since F ∈ TF , by hypothesis we get that F ∈ TJ . Then for all x ∈ F , there exists U ∈ K2 such that U [x] ⊆ F , and so J = UJ [1] ⊆ F . Corollary 5.4 Let F and J be filters of A. Then F = J if and only if TF = TJ . Corollary 5.5 The following conditions are equivalent: (1) There exists a proper filter M of a BL-algebra A such that TM ⊆ TF for all filters F of A. (2) A is local. (3) For all x ∈ A, ord(x) < ∞ or ord(x∗ ) < ∞. www.mlq-journal.org
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P r o o f. (1) ⇔ (2): Let TM ⊆ TF for all filters F of A. Then by Theorem 5.3, F ⊆ M for all filters F of A. Hence M is a unique maximal filter of A. Conversely, let A be local. Then there exists a unique maximal filter M of A. By Proposition 2.6, F ⊆ M for all filters F of A. Thus by Theorem 5.3, TM ⊆ TF . (2) ⇔ (3): This is clear by Proposition 2.9. Theorem 5.6 Let M = A \ ZD(A), where ZD(A) = {x ∈ A | x is zero divisor}, then the following conditions are equivalent: (1) If TF ⊂ TM , then F = A. (2) M is a maximal filter. (3) ord(x) < ∞ for all x ∈ ZD(A). P r o o f. (1) ⇔ (2): This is clear by Theorem 5.3. (2) ⇔ (3): Let M be a maximal filter and x ∈ ZD(A), so x ∈ M c . Since M is a maximal filter, we have that M ∪ {x} = A. Hence x1 ∗ x2 ∗ · · · ∗ xn ≤ 0 for some xi ∈ M ∪ {x}. Then 0 = x1 ∗ x2 ∗ · · · ∗ xn = f1 ∗ f2 ∗ · · · ∗ fm ∗ xt for some t > 0, and fi ∈ M for all 1 ≤ i ≤ n. We show that t = 0. In contrary suppose that t = 0, then f1 ∗ f2 ∗ · · · ∗ fm = 0 ∈ M, which is a contradiction. So xt = 0 and thus ord(x) = t < ∞. Conversely, let every zero divisor of A be of finite order. If M ⊂ F ⊆ A, then there exists a ∈ F \ M , thus a is a zero divisor, i. e., there exists n such that an = 0. Therefore F = A. Recall that a uniform space (X, K) is totally bounded if for each U ∈ K, there exist x 1 , . . . , xn ∈ X such n that X = i=1 U [xi ], and X is compact if any open cover of X has a finite subcover. Theorem 5.7 Let F be filter of A. Then the following conditions are equivalent: (1) The topological space (A, TF ) is compact. (2) The topological space (A, TF ) is totally bounded. (3) There exists P = {x1 , x2 , . . . , xn } ⊆ A such that for all a ∈ A there exists xi ∈ P such that a ≡F xi . P r o o f. (1) ⇒ (2): This is clear by [5, Theorem 14.3.8]. (2) ⇒ (3): Let UF ∈ K. Since (A, TF ) is totally bounded, there exist x1 , x2 , . . . , xn ∈ F such that n A = i=1 UF [xi ]. Now let a ∈ A. Then there exists xi such that a ∈ UF [xi ], therefore a → xi ∈ F and xi → a ∈ F . (3) ⇒ (1): For any a ∈ Aby hypothesis there exists xi ∈ P such that a → xi ∈ F and xi → a ∈ F . Theren fore a ∈ UF [xi ], hence A = i=1 UF [xi ]. Now let A = α∈I Oα , where each Oα is an open set of A. Then for any xi ∈ A there exists αi ∈ I such that xi ∈ Oαi . Since Oαi is an open set, UF [xi ] ⊆ Oαi , so n n A = i=1 UF [xi ] ⊆ i=1 Oαi . n Therefore A = i=1 Oαi which means that (A, TF ) is compact. Theorem 5.8 Let F be a maximal and Boolean filter of A. Then the topological space (A, T F ) is compact. P r o o f. Let A = α∈I Oα , where each Oα is an open set of A. Then there exist α, β ∈ I such that 1 ∈ Oα and 0 ∈ Oβ , thus UF [1] ⊆ Oα and UF [0] ⊆ Oβ .
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On the other hand, we have UF [1] = {x | x → 1 ∈ F and 1 → x ∈ F } = {x | x ∈ F }, UF [0] = {x | x → 0 ∈ F and 0 → x ∈ F } = {x | x∗ ∈ F }. Since F is a maximal and Boolean filter, by Theorem 2.15, wehave x ∈ F or x∗ ∈ F for all x ∈ A. Thus for all x ∈ A we have x ∈ UF [1] or x ∈ UF [0], therefore A = Oα Oβ , which means that (A, TF ) is compact. Remark 5.9 In general, the converse of the theorem above is not valid. To show this, consider the topological space of Example 3.8. Since B is finite, the topological space (B, TF ) is compact for the filter F = {1, b}, but F is not a maximal filter of B. Theorem 5.10 If F is a filter of A such that F c is a finite set, then the topological space (A, TF ) is compact. P r o o f. Let A = α∈I Oα , where each Oα is an open subset of A. Let F c = {x1 , x2 , . . . , xn }. Then there exist α, α1 , α2 , . . . , αn ∈ I such that 1 ∈ Oα , x1 ∈ Oα1 , . . . , xn ∈ Oαn . Then UF [1] ⊆ Oα , but UF [1] = F . Hence A = Oα1 ∪ Oα2 ∪ · · · ∪ Oαn . Theorem 5.11 If F is a filter of A, then F is a compact set in the topological space (A, T F ). P r o o f. Let F ⊆ α∈I Oα , where each Oα is an open set of A. 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 (A, TF ). Theorem 5.12 If F is a filter of A, then for all x ∈ A, UF [x] is a compact set in the topological space (A, TF ). P r o o f. Let UF [x] ⊆ α∈I Oα , where each Oα is an open set of A. Since x ∈ UF [x], there exists α ∈ I such that x ∈ Oα . Then UF [x] ⊆ Oα . Hence UF [x] is compact. Acknowledgements comments.
The authors would like to express their sincere thanks to the referees for their valuable suggestions and
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