Filter topologies on MV-algebras
Cuicui Luan & Yichuan Yang
Soft Computing A Fusion of Foundations, Methodologies and Applications ISSN 1432-7643 Soft Comput DOI 10.1007/s00500-017-2574-y
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Author's personal copy Soft Comput DOI 10.1007/s00500-017-2574-y
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Filter topologies on M V -algebras Cuicui Luan1 · Yichuan Yang1
© Springer-Verlag Berlin Heidelberg 2017
Abstract We show that every M V -algebra A can be naturally equipped with a filter topology such that A is a topological M V -algebra, which improves the main results on C-topological lattice-ordered groups in both Yang (Sci China Ser A 52(11):2397–2403, 2009) and Gusi´c (Proc Am Math Soc 126(9):2593–2597, 1998). Furthermore, for a 2divisible M V -algebra A, we show that the filter topological space is Hausdorff if and only if A is semisimple. Keywords M V -algebra · Filter topology · Hausdorff space · Semisimple · Dense
1 Introduction By Mundici categorical equivalence between unital latticeordered abelian groups and M V -algebras, it is interesting to construct the topology on M V -algebras and investigate related properties (Luan 2010; Lapenta 2012) use the idea of Gusi´c (1998), Yang (2009). This paper is a preliminary study of such a topology, some questions that may be considered are posed at the end of the paper. We hope an interested reader will take up the task of developing this idea.
Let A, ⊕, , ¬, 0 be an M V -algebra, a nonempty sublattice F of A \ {0} is said to be a filter if c ∈ F whenever c ≥ a ∈ F. It is easily seen that there are many filters in a nontrivial M V -algebra. Define the open F-ball of radius r ∈ F, with the center x0 ∈ A to be x0 ,r = {x ∈ A|r d(x, x0 ) ∈ F}, where d(x, x0 )) = (x ¬x0 ) ⊕ (x0 ¬x) is the distance between x and x0 , and x y := x ¬y. Then, open F-balls constitute a base of the filter topology on A, which makes A into a topological M V -algebra (Theorem 1) (Luan 2010), which improves the main results on C-topological latticeordered groups in both Yang (2009) and Gusi´c (1998) where the restriction of 2-divisibility is assumed. Note that this result was also obtained independently by Lapenta (2012). An element a of an M V -algebra A is 2-divisible, if there is a b ∈ A such that b = a b. If such a b exists, it is uniquely determined by a and will also be denoted by b = a/2. If each element of A is 2-divisible, we say that A is 2-divisible. A filter F of A is 2-divisible if x ∈ F implies x/2 ∈ F. If A is a 2-divisible M V -algebra, and a ∈ A \ {0}. Put Aa = {x ∈ A|x ≥ a} and denote Aa,n = 1/2n−1 Aa for all n ∈ N, then F = n∈N Aa,n is the minimal set of 2-divisible filter containing a. We prove that a 2-divisible M V -algebra is semisimple if and only if it is Hausdorff with respect to any 2-divisible filter, if and only if every 2-divisible filter is dense in A (Theorem 2).
Communicated by Y. Yang. The authors acknowledge the support of NSFC, Grant 11271040.
B
Yichuan Yang
[email protected] Cuicui Luan
[email protected]
1
School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
2 Basics Recall that an M V -algebra is an algebra A, ⊕, ¬, 0 with a binary operation ⊕, a unary operation ¬ and a constant 0 satisfying the following axioms: MV1) x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z, MV2) x ⊕ y = y ⊕ x,
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MV3) MV4) MV5) MV6)
x ⊕ 0 = x, ¬¬x = x, x ⊕ ¬0 = ¬0, ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x.
Proof By a ≤ b ⊕c and Lemma 1 (iv), there exists an m ∈ A so that b⊕c = a⊕m. By lemma 3, (d a)m = d (a⊕m). Then by (2) and (1), we have ¬(d (b ⊕ c)) ⊕ (d a)
Define the constant 1 and the binary operations and as follows: 1 := ¬0; x y := ¬(¬x ⊕¬y); x y := x ¬y. Then, x ⊕ ¬x = 1
(1)
= ¬((d a) m) ⊕ (d a) = ¬(d a) ⊕ m ⊕ (d a) = 1, whence d a ≥ d (b ⊕ c) follows from Lemma 1. This completes the proof.
According to and MV4), we can easily get ¬(x y) = ¬x ⊕ y.
= ¬(d (a ⊕ m)) ⊕ (d a)
(2)
For any x and y of A, we write x ≤ y iff ¬x ⊕ y = 1. We call ≤ the natural order of A, which endows A with a bounded distributive lattice structure with the join x ∨ y and x ∧ y are given by x ∨ y := ¬(¬x ⊕ y) ⊕ y and x ∧ y := ¬(¬x ∨ ¬y) The following two results are well known. Lemma 1 Let A be an M V -algebra and x, y ∈ A. Then, the following conditions are equivalent.
Lemma 5 Let A be an M V -algebra, if a ≤ b, then for every c ∈ A, a c ≤ b c. Proof ¬b ≤ ¬a follows from a ≤ b and Lemma 1 (vi). By Lemma 2 (i) and (2), for every c ∈ A, one gets ¬b⊕c ≤ ¬
a ⊕ c, and ¬(b c) ≤ ¬(a c). Thus, a c ≤ b c. Let A be an M V -algebra, Chang’s distance function d : A × A −→ A is defined by d(x, y) = (x ¬y) ⊕ (y ¬x)
i) ii) iii) iv) v) vi)
¬x ⊕ y = 1, x ¬y = x y = 0, y = x ⊕ (y x), there is an element z ∈ A such that x ⊕ z = y, x ≤ y, ¬y ≤ ¬x.
We list some properties of distance function on M V -algebras as follows.
Lemma 2 In an M V -algebra A, the natural order ≤ has the following properties: i) If x ≤ y, then for each z ∈ A, x ⊕ z ≤ y ⊕ z and x z ≤ y z, ii) If x y ≤ z, then x ≤ ¬y ⊕ z. Lemma 3 Let A be an M V -algebra and a, b, c ∈ A, then a (b ⊕ c) = (a b) c. Proof By MV1), MV4) and (2), one can get (a b) c = ¬((¬a ⊕ b) ⊕ c) = ¬(¬a ⊕ (b ⊕ c)) = a (b ⊕ c).
Lemma 4 Let A be an M V -algebra. If a, b, c ∈ A, a ≤ b ⊕ c, then for each d ∈ A, d a ≥ d (b ⊕ c).
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d(x, y) = 0 ⇔ x = y;
(4)
d(x, y) = d(y, x);
(5)
d(x, z) ≤ d(x, y) ⊕ d(y, z);
(6)
d(x, y) = d(¬x, ¬y);
(7)
d(x ⊕ s, y ⊕ t) ≤ d(x, y) ⊕ d(s, t).
(8)
3 Filter topologies on M V -algebras Definition 1 Let A be an M V -algebra, a nonempty subset F of A \ {0} is said to be a filter if a, b ∈ F implies a ∧ b ∈ F, and c ∈ F whenever c ≥ a ∈ F. Note that for an element a ∈ A \ {0}, the upper set ↑ a := {x ∈ A | x ≥ a} is the minimal filter which contains a. Definition 2 Let F be an M V -algebra and let F be a filter of A. The open F-ball of radius r ∈ F, with the center x0 ∈ A, is denoted by x0 ,r = {x ∈ A|r d(x, x0 ) ∈ F}.
= ¬(¬a ⊕ b) c
This completes the proof.
(3)
Proposition 1 Let A be an M V -algebra and let F be a filter of A. Then open F-balls constitute a base of a topology on A, we call this topology a filter topology. Proof Since r 0 = ¬(¬r ⊕ 0) = r for all r ∈ F, and by (4) it follows that x0 ∈ x0 ,r , thus, the open F-balls form an open cover of the space A.
Author's personal copy Filter topologies on M V -algebras
Let z 0 ∈ x0 ,r ∩ y0 ,R be arbitrary. By the definition of F-balls, we can assume that r d(z 0 , x0 ) = c1 , R d(z 0 , y0 ) = c2 for some c1 , c2 ∈ F. Then, z 0 ,ε ⊂ x0 ,r ∩ y0 ,R for ε = c1 ∧ c2 . Namely, for x ∈ z 0 ,ε , we have ε d(x, z 0 ) = c3 for some c3 ∈ F. By (6), Lemmata 3, 4 and 5, we get r d(x, x0 ) ≥ r (d(z 0 , x0 ) ⊕ d(z 0 , x)) = (r d(z 0 , x0 )) d(z 0 , x) = c1 d(z 0 , x) ≥ ε d(z 0 , x) = c3 ∈ F it follows that x ∈ x0 ,r . Similar Thus, r d(x, x0 ) ∈ F and arguments implies that x ∈ y0 ,R . This completes the proof.
Recall that a topological M V -algebra is an M V -algebra and a topological space such that both the operations ⊕ and ¬ are continuous. Theorem 1 Every filter topology on an M V -algebra A makes A into a topological M V -algebra. Proof Firstly, we prove that the mapping ⊕ : A × A → A, (x, y) → x ⊕ y is continuous. For any ε ∈ F, let U be an open F-ball of radius ε around x0 ⊕ y0 , and let V be an open F-ball of radius ε around x0 , that is V = {x ∈ A|εd(x, x0 ) ∈ F}. Take x ∈ V , assume that εd(x, x0 ) = cx ∈ F. Let W be an open F-ball of radius cx around y0 . Then, V × W is an open neighborhood around (x0 , y0 ). In fact, pick (x, y) = z ∈ V × W , then ε d(x ⊕ y, x0 ⊕ y0 ) ≥ ε (d(x, x0 ) ⊕ d(y, y0 )) = ¬(¬ε ⊕ (d(x, x0 ) ⊕ d(y, y0 ))) = ¬(¬ε ⊕ d(x, x0 )) d(y, y0 ) = (ε d(x, x0 )) d(y, y0 ) = cx d(y, y0 ) ∈ F by (8), (2), MV4), MV1), and Lemmata 4 and 5. Since (x, y) can be chosen arbitrary, we have ⊕(V × W ) ⊂ U , so the continuity of ⊕ is proved. Next, we will prove that ¬ : A → A, x → ¬x is also a continuous mapping. In fact, for every ε ∈ F, x0 ∈ A, let U0 be an open F-ball of radius ε around ¬x0 , and let V0 be an open F-ball of radius ε around x0 . Take x ∈ V0 , then we have ε d(x, x0 ) ∈ F, then by d(x, x0 ) = d(¬x, ¬x0 ) we have ε d(¬x, ¬x0 ) ∈ C, that is, ¬x ∈ U0 . Since x can be chosen arbitrarily, we have ¬(V0 ) ⊂ U0 . So the continuity of ¬ is verified.
Finally, A forms a topological space based on it’s open F-balls by Proposition 1, therefore, A is a topological M V algebra.
Remark The order structure of M V -algebra can completely be determined by the properties of operations ⊕ and ¬, while lattice-ordered groups’ order structure can not be determined by group operation. Followed by the corresponded result in C-topological lattice-ordered groups and Muncici’s equivalence, the first version of Theorem 1 is appeared in Yang (2006) where an assumption of 2-divisibility is required, and this improved version has been showed independently by Luan (2010) and by Lapenta (2012). Consequently, we have following result that improves a main result on C-topological lattice-ordered groups in both Yang (2009) and Gusi´c (1998) where the restriction of 2-divisibility is also assumed. Corollary 1 Every filter topology on an abelian l-group G makes G into a topological l-group.
4 2-divisible filter topologies on M V -algebras An element a of an M V -algebra A is 2-divisible, if there is a b ∈ A such that b = a b. If such a b exists, it is uniquely determined by a and will also be denoted by b = a/2. It is clear that the minimal element of an M V -algebra is 2divisible. If each element of A is 2-divisible, we say that A is 2-divisible. Definition 3 Let A be an M V -algebra, a filter F of A is 2-divisible if x ∈ F implies x/2 ∈ F. Example The real interval [0, 1] and the generalized real interval [0, ∞] can be equipped with infinitely many 2divisible filter topologies. Assume that A is a 2-divisible M V -algebra, and a ∈ A \ a} and denote Aa,n = 1/2n−1 Aa {0}. Put Aa = {x ∈ A|x ≥ for all n ∈ N, then F = n∈N Aa,n is the minimal set of 2-divisible filter containing a. Definition 4 Let A be an M V -algebra, and let F be a filter. A decreasing sequence {xn }n∈N in A converges to x if x = n {xn } exists, denoted by lim(xn ) = x. The sequence {xn }n∈N in A is said to converge in order to x if there exists a natural number m and a decreasing sequence pn with lim( pn ) = 0 such that d(xn , x) ≤ pn holds for all n ≥ m. The sequence {xn }n∈N in A F-converges to x, if for any ε ∈ F, ∃m ∈ N, d(xn , x) < ε for all n ≥ m, denoted by lim F (xn ) = x. Proposition 2 Let A be an M V -algebra, and let F be a 2divisible filter. If a sequence {xn }n∈N in A F-converges to x, then for any element x0 in A, the sequence {d(xn , x0 )}n∈N F-converges to d(x, x0 ).
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Proof ∀x0 ∈ A, by (6), we have d(xn , x0 ) ≤ d(xn , x) ⊕ d(x, x0 ), and by Lemma 1, MV1) and MV4), we have ¬(¬(¬d(xn , x0 ) ⊕ d(x, x0 ))) ⊕ d(xn , x) = 1, Lemma 1 yields ¬(¬d(xn , x0 ) ⊕ d(x, x0 )) ≤ d(xn , x). Similarly,
Proof Because it is trivial for p = 0, we assume that p > 0. Then, it suffices to prove that any lower bound l of the sequence { p/2n }n∈N satisfies l = 0. In fact, let l be such a lower bound, then 0 ≤ s := l ∨ 0 is still a lower bound of the sequence, so 0 ≤ 2n s ≤ p for all n ∈ N and thus 0 ≤ ns ≤ p for all n ∈ N. It follows that s = 0 since A is semisimple, and thus, l = 0. Conversely, let v ∈ A and assume that nv ≤ p ∈ A for all n ∈ N. Then, v ≤ p/2n for all n ∈ N. Since n ( p/2n ) = 0 implies that v = 0, i.e., A is semisimple. This completes the proof.
For F-convergence, we have
¬(¬d(x, x0 ) ⊕ d(xn , x0 )) ≤ d(xn , x),
Proposition 5 Let A be an archimedean 2-divisible M V algebra. Then, the F-limit for any F-convergent sequence whence {xn }n∈N in A is unique, and A is the F-closure of F with respect to F = n∈N Aa,n for every nonzero element a in A. d(d(xn , x0 ), d(x, x0 )) Proof Assume that F = n∈N Aa,n = n∈N {a/2n−1 ⊕ = ¬(¬d(xn , x0 ) ⊕ d(x, x0 )) ⊕ ¬(¬d(x, x0 ) ⊕ d(xn , x0 )) p| p ∈ A}. Especially, it follows that 0 is a F-limit of the sequence {a/2n−1 }n∈N by Proposition 4. Suppose that x is ≤ d(xn , x) ⊕ d(xn , x) another F-limit of the sequence {a/2n−1 }n∈N , then we get d(x, 0) = x < ε for all ε ∈ F, and so x < a/2n−1 for follows from the definition of Chang’s distance function. all n ∈ N, which implies x = 0 since A is semisimple, Hence, if {xn }n∈N F-converges to x, we have {d(xn , x0 )}n∈N and thus, the F-limit of the sequence {a/2n−1 }n∈N is unique. F-converges to d(x, x0 ) by the 2-divisibility of F. This comAnalogously, we may also prove that the F-limit for any pletes the proof.
F-convergent sequence {xn }n∈N in A is unique using the semisimple property of A. Furthermore, for all p ∈ A, it is The F-closure of a subset S of an M V -algebra A with clear that limn (a/2n−1 ⊕ p) = p and a/2n−1 ⊕ p ∈ F for all a filter F is defined to be the set Cl(S) = {lim F (xn ) ∈ n ∈ N. Thus, we have shown that A is a subset of the closure A|{xn } ⊂ S}. If Cl(S) = A, then S is said to be dense in A. of F. On the other hand, the closure of F is a subset of A by It is straightforward to verify that F-closure has following the definition of F-closure. The proof is completed.
properties. Proposition 3 Let A be an M V -algebra, and F a 2-divisible filter, S, T ⊂ A. Then, (i) (ii) (iii) (iv)
∅ = Cl(∅), S ⊂ Cl(S), S ⊂ T ⇒ Cl(S) ⊂ Cl(T ), Cl(Cl(S)) = Cl(S).
Note that the four properties in Proposition 3 are equivalent to one equation (Yang 2006): S ∪ Cl(Cl(S) ∪ Cl(T )) = Cl(S ∪ T ) \ Cl(∅). Recall that an M V -algebra A is said to be semisimple if for any 0 < a < b ∈ A, ∃n ∈ N such that na < b is not valid. We obtain an order convergence characterization for semisimple 2-divisible M V -algebras. Proposition 4 Let A be a 2-divisible M V -algebra. Then, A is semisimple if and only if every sequence { p/2n }n∈N of positive elements in A order-converges to 0.
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Now, we get a characterization of semi-simplicity, density and Hausdorff property. Theorem 2 Let A be a 2-divisible M V -algebra. Then, the following conditions are equivalent: 1) A is semisimple. 2) Every 2-divisible filter is dense in A. 3) A is Hausdorff with respect to any 2-divisible filter topology. Proof 1)⇒2)⇒3): It follows directly from Propositions 4 and 5. 3)⇒1): Assume that A is not semisimple. Then, there exist two strictly positive elements a and b in A with nb < a for all n ∈ N. We check that b and 2b are not equal. In fact, it is clear that 0 < b ≤ 2b < 1. Suppose that b = 2b = b ⊕ b, then Lemma 1 (ii) implies that 2bb = 0. On the other hand, b ⊕ b b = ¬(¬(b ⊕ b) ⊕ b) ≥ ¬(¬(b ⊕ b) ⊕ b ⊕ b) = ¬0 = 1 by Lemmata 1 and 2. A contradiction. Finally, pick F = n∈N Aa,n , then the two different elements b and 2b can not be separated in the F-topology, and so A is not a Hausdorff space.
Author's personal copy Filter topologies on M V -algebras
Remark 2 Just as pointed out in Yang (2006, 2009), a filter topology on an abelian lattice-ordered group with a strong unit need not be Hausdorf, a filter topology on an M V algebra need not be T2 , either. Moreover, it is obvious that the assumption of 2-divisible in this paper can be replaced by n-divisible for any natural number n ≥ 2.
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Some questions Here, we list some questions that may be considered at a later time. Question 1. Can the restriction of 2-divisibility be canceled in Theorem 2? Question 2. For a distributive lattice D, it is well known that the assumption of the existence of a compatible topology on D that is both compact and Hausdorff is pretty strong. What about the compactness of filter topology? Question 3. How do homomorphisms of MV-algebras behave with respect to the filter topologies?
Gusi´c I (1998) A topology on lattice-ordered groups. Proc Am Math Soc 126(9):2593–2597 Lapenta S (2012) A topology on MV-algebras, preprint Luan C (2010) C-topology on MV-algebras, (Chinese), Master Thesis, Beihang University Yang Y (2006) l-groups and Bézout domains, Dissertation, University of Stuttgart Yang Y (2006) A characterization of closures and interiors. Int Math Forum 1(1–4):139–140 Yang Y (2009) The C-topology on lattice-ordered groups. Sci China Ser A 52(11):2397–2403
Conflict of interest The authors declare that they have no conflict of interest.
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