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Topological FLew -algebras
Accepted Manuscript Topological FLew -algebras Jean B. Nganou, Serge F.T. Tebu
PII: DOI: Reference:
S1570-8683(15)00051-8 http://dx.doi.org/10.1016/j.jal.2015.04.004 JAL 379
To appear in:
Journal of Applied Logic
Received date: 22 October 2014 Accepted date: 24 April 2015
Please cite this article in press as: J.B. Nganou, S.F.T. Tebu, Topological FLew -algebras, Journal of Applied Logic (2015), http://dx.doi.org/10.1016/j.jal.2015.04.004
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TOPOLOGICAL FLew -ALGEBRAS JEAN B. NGANOU, SERGE F. T. TEBU Abstract. The main goal of this article is to introduce topological FLew algebras and study their main properties. We also treat completions of FLew -algebras with respect to inductive family of filters. This work generalizes similar works on MV-algebras [10] and on FLew -algebras equipped with uniform topologies [9]. Key words: complete, MV-algebra, inductive family, Heyting algebra, topological FLew -algebra, profinite FLew -algebra, Stone FLew -algebra.
1. Introduction Universal topological algebras, (which are algebras with topologies with respect to which all operations are continuous) have been introduced and studied thoroughly (see for e.g [4, 12, 13, 19]). Some of the notable classes of topological algebras that have been the objects of more detailed studies include groups [17], lattices [16], othomodular lattices [5], MV-algebras [10]. A residuated lattice is a lattice equipped with a monotone monoidal operation (with a unit e) and a pair of binary operations /, \ satisfying x y ≤ z if and only if x ≤ z/y if and only if y ≤ x\z FL-algebras, which are residuated lattices expanded by a constant ¯0, form the algebraic semantics for the so-called (intuitionistic) substructural logics. These provide a unifying framework for several kinds of logics (Girard’s linear logic, the Lambek calculus, the L ukasiewicz’s many-valued logics, H´ajek fuzzy logics etc). A very important subclass of FL-algebras is that of FLew -algebras, which are FL-algebras extended by the exchange and weakening rules. An FLew -algebra (also known as bounded integral commutative residuated lattice) can be defined [7, 11] as a bounded lattice (L, ∨, ∧, 0, 1) with two additional binary operations , → satisfying: (i) (L, , 1) is a commutative monoid, and (ii) for all x, y, z ∈ L, x y ≤ z if and only if x ≤ y → z. A nonempty subset F of a FLew -algebra L is called a filter of L if it satisfies: (F1) x y ∈ F for 2010 Mathematics Subject Classification. 06D35, 06E15, 06D50. April 30, 2015. 1
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all x, y ∈ F , and (F2) For all x, y ∈ L, if x ≤ y and x ∈ F , then y ∈ F . While, filters of an FLew -algebra are studied for their connection to its congruences, they primarily come from logic itself since they correspond to theories of the logic. Indeed, filters of an FLew -algebra L are just subsets of L that are closed under all deductions. Equivalently, in algebraic logic, filters are sets of designated elements that provide matricial models for the logic. Among the many important subclasses of FLew -algebras, there is that of MV-algebras, which constitute the algebraic counterpart of L ukasiewicz many valued logic. In [10], Hoo introduced topological MV-algebras, and studied their main properties. A close analysis of Hoo’s work reveals that the essential ingredients are the existence of an adjoint pair of operations and the fact that ideals of MV-algebras correspond to its congruences. This prompted us to consider the same study in a more general context where similar ingredients are available, namely FLew -algebras. The main goal of the present work is to generalize Hoo’s work to FLew -algebras. This yields the notion of topological FLew -algebra, which is an FLew -algebra together with topology with respect to which all the operations are continuous. Topological FLew -algebrass were already investigated in [9], but mainly for the uniform topology. The paper is organized as follows. In section 2, we study the general properties of topological FLew -algebras. In section 3, we study a special class of topological FLew -algebras, namely those arising from decreasing families of filters indexed by directed sets. We establish conditions under which the resulting space has certain topological properties, and also study completions of topological FLew -algebras. In addition, we show that such FLew -algebras are Stone if and only of they are compact and Hausdorff. Homomorphisms of FLew -algebras have the usual meaning. A subset F of an FLew -algebras L is called a deductive system of L if: (ds1) 1 ∈ F and (ds2) For every x, y ∈ L, if x, x → y ∈ F , then y ∈ F. It is known that the notions of filters and deductive systems coincide (see e.g.,[8]). We shall use solely the filter terminology in the entire article. Filters of FLew -algebras induce congruences, indeed the following result is wellknown. Proposition 1.1. [18] Let L be an FLew -algebra and let F be a filter of L. The relation x ≡F y if and only if x → y, y → x ∈ F is a congruence on L. If F is a filter of L, the quotient FLew -algebra induced by the congruence ≡F shall be denoted by L/F . In addition the class of an element a ∈ L with respect to ≡F is often denoted by [a]F ; and πF denotes the natural projection L → L/F .
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For every subset X ⊆ L, the smallest filter of L containing X (i.e., the intersection of all filters F of L such that X ⊆ F ) is called the filter generated by X and will be denoted by X . If U and V are subsets of L and ∈ {∨, ∧, →, }, then U V := {x y : x ∈ U, y ∈ V } Remark 1.2. While some aspects of the paper could be treated in the more general context of FL-algebras, or even arbitrary residuated lattices, we restrict our study to bounded integral commutative residuated lattices (FLew algebras). Our choice is mainly motivated by the fact that the definitions and notations in the FLew -algebras context are greatly simplified, and that once established, one could easily adapt the relevant results to that more general context. The main properties of FLew -algebras are summarized in the next result. Proposition 1.3. [6, 18] The following hold in all FLew -algebras L = (L, ∧, ∨, , → , 0, 1) and for all x, y, z ∈ L. x ≤ y if and only if x → y = 1 x (y ∨ z) = (x y) ∨ (x z), x (y ∧ z) = (x y) ∧ (x z)
(i) (ii)
x → (y ∧ z) = (x → y) ∧ (x → z), (x ∨ y) → z = (x → z) ∧ (y → z)
(iii)
(y → x) y ≤ x, x (z → y) ≤ z → x y
(iv)
z → (y → x) = (z y) → x, (y → z) (x → y) ≤ z → x x → y ≤ (y → z) → (x → z), y → x ≤ y z → x z y ≤ x → y, z → (x ∧ y) = (z → x) ∧ (z → y) If x ≤ y, then y → z ≤ x → z, z → x ≤ z → y, x z ≤ y z (x → y) x ≤ x ∧ y, x y ≤ x ∧ y (y → z) (x → y) ≤ z → x
(v) (vi) (vii) (viii) (ix) (x)
For more details on FLew -algebras, the reader may refer to [1, 7, 11] 2. Topological FLew -algebras: general properties Let (L, ∧, ∨, , →, 0, 1) be an FLew -algebras and τ be a topology on L. We say that the pair (L, τ ) is a topological FLew -algebra (TFLew -algebra, for short) if the operations ∧, ∨, →, are continuous with respect to τ . We shall simply say that L is a TFLew -algebra without explicit mention of the topology τ when the topology is clear from the context. We start by presenting two important examples of TFLew -algebras.
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Example 2.1. Consider L = [0, 1], x ∨ y = max(x, y), x ∧ y = min(x, y), x y = max(0, x + y − 1), x → y = min(1 − x + y, 1) Then L is a TFLew -algebra under the natural topology on [0, 1]. More generally, every topological MV-algebra as treated in [10] is a TFLew -algebra. Example 2.2. For every integer n ≥ 2, let Ln be the unique linear Heyting algebra with n elements. Then, each Ln is a TFLew -algebra under the discrete topology. Now, consider L := n≥2 Ln endowed with the product topology. Then L is a TFLew -algebra, in fact a Stone (compact, Hausdorff and zerodimensional) TFLew -algebra. The following fact, which was observed in [9] is very useful for proving continuity of the operations of a TFLew -algebra. Proposition 2.3. [9] Let (L, ∧, ∨, , →, 0, 1) be a FLew -algebra and τ be a topology on L. An operation ∈ {∨, ∧, →, } is continuous if and only if for every open set O of (L, τ ) and x, y ∈ L such that x y ∈ O, there exists open sets O1 , O2 such that x ∈ O1 , y ∈ O2 and O1 O2 ⊆ O. Proposition 2.4. Let L be a TFLew -algebra, and F a filter of L such that πF is open. Then, the quotient FLew -algebra L/F is a TFLew -algebra under the quotient topology. Proof. It is sufficient to prove that ([x]F , [y]F ) → [x]F [y]F := [x y]F is continuous where ∈ {∧, ∨, , →}. Let x, y ∈ L and let W be a neighborhood of [x y]F . Then πF−1 (W ) is open in L and x y ∈ πF−1 (W ). Since L is a TFLew -algebra, there exists neighborhoods U0 of x and V0 of y such that U0 V0 ⊂ πF−1 (W ). Put U := πF (U0 ) and V := πF (V0 ). Then U and V are open in L/F since πF is open. Clearly [x]F ∈ U , [y]F ∈ V and it is straightforward that U V ⊆ πF (U0 V0 ) ⊆ W . Thus, from Proposition 2.3 that ([x]F , [y]F ) → [x]F [y]F := [xy]F is continuous for each ∈ {∧, ∨, , →}. The following observation justifies why the content of this section generalizes some of the main results in [10]. Remark 2.5. Let (A, ⊕, ¬, 0) be an MV-algebra viewed as an FLew -algebra. Let F be a filter of the FLew -algebra A, then ¬F is an ideal of the MV-algebra A. It is straightforward to see that when πF is open, the quotient TFLew algebras A/F and A/¬F are algebraically and topologically isomorphic. Proposition 2.6. For every TFLew -algebra L and F an open filter of L, the quotient topology on L/F is discrete.
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Proof. Note that for every x ∈ L, πF−1 ([x]F ) = [x]F . Therefore, it is enough to prove that [x]F is open for every x ∈ L. Let x ∈ L, and define f, g : L → L by f (a) = a → x and g(a) = x → a. Thus, f and g are continuous and since F is open, then f −1 (F ), g −1 (F ) are open. Now, it is also clear that [x]F = f −1 (F ) ∩ g −1 (F ), which is open. Proposition 2.7. Let L be a TFLew -algebra in which {1} is closed (resp. open). Then {x} is closed (resp. open) for all x ∈ L. Proof. Let f : L2 → L be the continuous map given by f (a, b) = a → b and let g : L3 → L2 be given by g(a, b, c) = (a → b, b → c). Then g is continuous. If {1} is closed, so is {(1, 1)} ⊆ L2 . Let x ∈ L and let h : L → L2 be given by h(x) = g(a, x, a) = (a → x, x → a). Then h is the restriction of g to {a} × L × {a} and hence is continuous. Now h−1 (1, 1) = {x ∈L: x → a = 1, a → x = 1} = {a}. Thus {a} is also closed. A similar argument is used when {1} is open. The following result was obtained in [9], but only when L is endowed by the uniform topology induced by a family of filters closed under the intersection. Theorem 2.8. Let L be a TFLew -algebra and F a filter of L, then, 1. If L/F is Hausdorff, then F is closed and the converse holds if πF is open. 2. L/F is discrete if and only if F is open. Proof. 1. The necessity holds for every filter as proved in [9, Thm. 3.9]. While the proof of the sufficiency given in [9, Thm. 3.9] was intended only for uniform topologies, the only extra hypothesis used was πF is open, which we have. 2. Note if L/F is discrete, then {F } is open in L/F . But, since πF−1 ({F }) = F , then F is open. The converse was established in Proposition 2.6. Theorem 2.9. Let (L, τ ) be a topological FLew -algebra. The connected component of 1 is a closed filter of L. Proof. Let C be the connected component of 1, and let x ∈ C. On one hand, since fx1 : L → L : x → x y is a continuous map, fx1 (C) = x C is connected and it contains x. Hence C ∩ (x C) = ∅. This means that C ∪ (x C) is connected. Since it contains 1, we have C ∪(xC) ⊆ C. Therefore xC ⊆ C. Thus C C ⊆ C. On the other hand let x, x → y ∈ C and fy2 : L → L : x −→ x ∨ y is a continuous map. Then (x → y) x ∈ C. Since (x → y) x ≤ y we have y = [(x → y) x] ∨ y ∈ y ∨ C and fy2 (C) = y ∨ C is connected and contains 1. This means that y ∨ C ⊆ C; hence y ∈ C. This proves that C is a filter.
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Since C is connected, then the closure C of C is connected and is maximal, C = C i.e., C is closed. Proposition 2.10. If F is an open (resp. closed) filter, then for each x ∈ L, [x]F is open (resp. closed). Proof. The proof when F is open was given in the proof of Proposition 2.6. The case F closed is handled similarly. Theorem 2.11. Every open filter is closed. Proof. Suppose that a filter F is open. Then by Proposition 2.10, each con gruence class [x]F is open. But F = L \ {[x]F : x ∈ / F }, which is closed. We would like to point out that a closed filter of a TFLew -algebra needs not be open. For instance, the trivial filter of the TFLew -algebra of Example 2.1 is closed, but not open. Theorem 2.12. If 1 is an isolated point of a filter F , then F is discrete. Proof. Let U is an open neighborhood of 1 such that {1} = U ∩ F and let x ∈ F . Since x → x = 1 ∈ U , there exist neighborhoods V1 and V2 of x such that V1 → V2 ⊂ U . Setting V = V1 ∩ V2 we have V ∩ F → V ∩ F ⊂ (V → V ) ∩ F ⊂ U ∩ F = {1}. Let y ∈ V ∩ F . Then y → x ∈ V ∩ F → V ∩ F = {1}, that is y ≤ x. Similarly x → y ∈ V ∩ F → V ∩ F = {1}, that is x ≤ y. Thus x = y, proving that V ∩ F = {x}. This means that x is also an isolated point of F and hence F is discrete. Theorem 2.13. Suppose that the only closed filters of L are {1} and L. Then the topology is one of the following types: (1) Hausdorff and totally disconnected. (2) Hausdorff and connected, (3) The trivial topology. Proof. Let C be the connected component of 1. By theorem 2.9, C is a closed filter and hence either C = {1} or C = L. If C = {1}, then {1} = {1} and hence L is Hausdorff. For each x ∈ L, let Cx be the connected component containing x. Then Cx → x and x → Cx are connected and containing 1. Hence Cx → x ⊆ C = {1} and x → Cx ⊆ C = {1}; that is Cx = {x}. This means that L is Hausdorff and totaly disconnected. On the other hand, if C = L, then L is connected. We have again {1} = {1} or {1} = L. In the first case L is Hausdorff and connected. In the second case, let U be any nonempty open subset of L. Let x ∈ {1}. Then U ∩ {1} = ∅. Thus 1 ∈ U , that is, every nonempty open subset U of L
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must contain 1. Further, if x ∈ / U for some x ∈ L, then since x → x = 1 ∈ U , we can find open neighborhoods V and W of x such that V → W ⊆ U. But 1 ∈ W , means that x = x → 1 ∈ V → W ⊆ U, a contradiction. Hence U = L, that is, every open set is either empty or is L. Thus the topology is trivial. We recall that an FLew -algebra L is called locally finite if for every x ∈ L \ {0}, there exists a natural number n such that xn = 0. It follows that such an FLew -algebra is simple, i.e, has only two filters: itself and {1}. The following result is therefore an immediate consequence of Theorem 2.13. Corollary 2.14. The topology on a locally finite FLew -algebra is one of the following types: (1) Hausdorff and totally disconnected, (2) Hausdorff and connected, (3) the trivial topology. 3. TFLew -algebras: completions In this section, we construct our main class of examples of TFLew -algebras and use it to treat completions of FLew -algebras. Our approach follows that found in [10] about MV-algebras and their ideals. We start by motivating the study of these concepts. Completions (in particular complete TFLew -algebras) treated here are natural examples of profinite TFLew -algebras. As observed by the authors of [2], profinite algebras play an important role in the study of various logical systems. One fundamental question for a logical system is that of decidability, in other words, does there exist an algorithm which decides for a given formula ϕ (in the language L), whether or not ϕ is a theorem in L. It is well-known that if L is a finitely axiomatizable and has the finite model property (FMP), then L is decidable. But, for languages L with nice algebraic semantics, whether L has the FMP or not reduces to whether or not the variety VL of algebraic model of L is generated by finite VL -algebras. Let L be any FLew -algebra, and let F := {Fi : i ∈ I} be a decreasing family of filters indexed by a directed poset (I, ≤). More explicitly, the following is true: (i) For every i, j ∈ I, there exists k ∈ I such that i ≤ k and j ≤ k and, (ii) For every i, j ∈ J, i ≤ j implies Fj ⊆ Fi . We shall call F an inductive family of filters indexed by (I, ≤). For each a ∈ L and each i ∈ I, for simplicity and clarity, we denote by U (a, i) the congruence class of a modulo Fi , that is U (a, i) = [a]Fi = {x ∈ L : x ≡Fi a} = {x ∈ L : a → x, x → a ∈ Fi }.
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Taking these congruences classes as the system of neighborhoods turns L into a TFLew -algebra. More precisely, Proposition 3.1. Let L be an FLew -algebra, F be an inductive family of filters indexed by (I, ≤) and TF = {G ⊆ L : ∀a ∈ G, ∃i ∈ I, U (a, i) ⊆ G} Then (L, TF ) is a TFLew -algebra. Proof. It is clear that ∅ and the set L belong to TF . Also from the definition, it is clear that TF is closed under arbitrary unions. Finally to show that TF is closed under finite intersection, let G, H ∈ TF and suppose a ∈ G ∩ H. Then there exist i, j ∈ I such that U (a, i) ⊂ G and U (a, j) ⊂ H. Since I is a directed set, there exists k ∈ I such that i ≤ k, j ≤ k. Then U (a, k) ⊂ U (a, i)∩U (a, j); that is U (a, k) ⊂ G ∩ H. Therefore G ∩ H ∈ TF . Thus TF is a topology on L. Next, we prove the continuity of ∧. Let O be an open set of L and x, y ∈ L such that x ∧ y ∈ O. Then there exists i ∈ I, U (x ∧ y, i) ⊆ O and one can show that U (x, i) ∧ U (y, i) ⊆ U (x ∧ y, i). Indeed, let z ∈ U (x, i) ∧ U (y, i), then z = a ∧ b with a ∈ U (x, i) and b ∈ U (y, i). Since a ≡Fi x, b ≡Fi y and ≡Fi is a congruence on L, we have a ∧ b ≡Fi x ∧ y. That is z ≡Fi x ∧ y, hence z ∈ U (x ∧ y, i) ⊆ O. Therefore, ∧ is continuous by Proposition 2.3. The proofs of the continuity of ∨, and → are similar to that of ∧. Thus, (L, TF ) is a TFLew -algebra as claimed. Remark 3.2. We will often denote the TFLew -algebra (L, TF ) simply by LF and refer to TF as the linear topology on L induced by F. Clearly each U (a, i) is open in LF . In addition, observe that for a fixed i ∈ I and a ∈ L, we have U (a, i) = L \ {U (x, i) : x ≡Fi a} and hence each U (a, i) is both open and closed (i.e., clopen) in LF . Moreove, since Fi = U (1, i) for each i ∈ I, then each Fi is clopen. We want to explain more directly how this construction generalizes the one found in [10] for MV-algebras. Indeed, suppose that (A, ⊕, ¬, 0) is an MV-algebra viewed as an FLew -algebra, and let F := {Fi : i ∈ I} be a decreasing family of filters indexed by a directed poset (I, ≤). Consider ¬F := {¬Fi : i ∈ I}. Then ¬F := {¬Fi : i ∈ I} is a decreasing family of ideals of A indexed by a directed poset (I, ≤). The topological MV-algebra on A induced by ¬F as treated by [10] is algebraically and topologically isomorphic to the TFLew -algebra induced by F. For more explicit examples, consider the following.
TOPOLOGICAL FLew -ALGEBRAS
Example 3.3. Let L := Z≥0 be the set of nonnegative integers.
L, ∨, ∧, →, , 0, 1 is an FLew -algebra (indeed a BL-algebra), where
9
Then
x ∨ y = gcd(x, y), x ∧ y = lcm(x, y), x → y = lcm(x, y)/x, x y = xy Fix an integer n ≥ 1 and let D(n) denotes the set of positive divisors of n. Then (D(n), ≤) is a directed set, where d ≤ d if d |d. For each d ∈ D(n), let Fd = d be the principal filter of L generated by d. Then F := (Fd )d∈D(n) is an inductive family of filters. The following result provides a necessary and sufficient condition of the separation axioms for TF . Theorem 3.4. Let L be an FLew -algebra and F be a decreasing family of filters of L indexed by a directed poset (I, ≤). Then the following assertions are equivalent. (i) (L, TF ) is a T1 -space, (ii) i∈I Fi = {1}, (iii) (L, TF ) is a T2 -space Proof. (i) ⇒ (ii) Assume that (L, TF ) is a T1 -space. By contradiction suppose there is x ∈ i∈I Fi with x = 1. Then since TF is a T1 -space, there exists an open set V containing 1 such that x ∈ / V . Therefore, there exists i ∈ I such that Fi = U (1, i) ⊆ V . Hence, x ∈ V which is clearly a contradiction. (ii) ⇒ (iii) Assume that i∈I Fi = {1}. Let x, y ∈ L such that x = y, then x → y = 1 or y → x = 1. Suppose that x → y = 1, then by assumption there exists i ∈ I such that x → y ∈ / Fi . We claim that U (x, i) ∩ U (y, i) = ∅. Indeed, if on the contrary we assume that U (x, i) ∩ U (y, i) = ∅, then there exists a ∈ U (x, i) ∩ U (y, i). Thus x → a, a → y ∈ Fi , hence (x → a) (a → y) ∈ Fi . But since (x → a) (a → y) ≤ x → y, then x → y ∈ Fi , which is a contradiction. One argues the case y → x = 1 similarly. (iii) ⇒ (i) T2 -spaces are T1 -spaces, always. We consider the following standard construction in the context of TFLew algebras. A similar construction for MV-algebras can be found in [10]. Start with the TFLew -algebra LF , where F is a decreasing family of filters of L indexed by a directed poset (I, ≤). Recall that for every i ∈ I, Fi is open in L , thus each L/Fi is discrete. Next, we consider the product topology on F i∈I L/Fi . In addition, note that for every i ≤ j in (I, ≤), there is a natural homomorphism ϕij : L/Fj → L/Fi . It is indeed a simple verification that (L/Fi , ϕij ) is an inverse/projective system, whose limit shall be denoted by := lim L/Fi . L is given the subspace topology of L/Fi , and is called L i∈I ←−
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the completion of L. In addition, there is a natural continuous homomorphism defined by ψ(x) = ([x]F ) and ψ(L) is dense in L. ψ:L→L i i∈I The following are obvious. is injective if and Proposition 3.5. With the preceding notations, ψ : L → L only if i∈I Fi = {1}. is injective if and only if LF is In other words, from Theorem 3.4, ψ : L → L is an isomorphism. Hausdorff. We say that L is complete (w.r.t F) if ψ : L → L → L/Fi denotes the natural projection and Pi := p−1 (1). Then Let pi : L i indexed by the same directed F ∗ := (Pi )i∈I is a decreasing family of filters of L set (I, ≤). Remark 3.6. If I and J are directed sets, then the families of filters F = {Fi : i ∈ I} and G = {Gj : j ∈ J} induce the same topology on L if and only if for each U (x, i) there exists j ∈ J such that U (x, j) ⊆ U (x, i) and for each U (x, j) there exists i ∈ I such that U (x, i) ⊆ U (x, j). In this case the natural homomorphism lim L/Fi −→ lim L/Fj is an isomorphism. Thus the ←− ←− depends only on the topology of L. completion L is the linear topology induced by F ∗ . Proposition 3.7. 1. The topology on L is complete and Hausdorff. 2. L let U (([xi ]F )i∈I , j) = Proof. 1. For each j ∈ I and each ([xi ]Fi )i∈I ∈ L, i {([yi ]Fi )i∈I ∈ L : ([yi ]Fi )i∈I ≡Pj ([xi ]Fi )i∈I }. Then if i = j and Uj = {[xj ]F }. U (([xi ]Fi )i∈I , j) = i∈I Ui , where Ui = pi (L) j Thus U (([xi ]Fi )i∈I , j) is open. 2. This follows easily from the definition of completeness. For the rest of the paper, L denotes a TFLew -algebra and the topology on L is the linear topology induced by an inductive family F of filters. Proposition 3.8. Every compact and Hausdorff TFLew -algebra is complete. Proof. Suppose that L is compact and Hausdorff, then by Proposition 3.4, i∈I Fi = {1}. Therefore, by Proposition 3.5, L is a subspace of L. In addition, for each i, since L = ∪a∈L U (a, i), and L is compact, then there exist a1i , . . . , ani ∈ L such that L = ∪nj=1 U (aji , i). Therefore, L/Fi = {[a1i ]F , . . . , [ani ]F } and L/Fi is finite for each i ∈ I. Thus, i∈I L/Fi is compact and since L is a then compact subspace, L is closed in i∈I L/Fi . But, since L is dense in L, r r = L = L∩L = L ∩ L, where L denotes the relative closure of L in L. L Hence L ⊆ L, which implies L = L and L is complete.
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An important result due to Strauss [16] asserts that a topological Boolean algebra is Stone if and only if it is compact and Hausdorff. We can establish the same result for the TFLew -algebras discussed here. Proposition 3.9. Let L be a TFLew -algebra. Then L is Stone if and only if it is compact and Hausdorff. Proof. Recall that an FLew -algebra is Stone if is compact, Hausdorff and zerodimensional (i,e., has a base of clopen sets). We only need to prove that every compact Hausdorff TFLew -algebra is Stone. Suppose that L is a compact and ˆ But, L ˆ is clearly Hausdorff TFLew -algebra. Then, by Proposition 3.8, L ∼ = L. a Stone TFLew -algebra, as any profinite FLew -algebra is Stone. Therefore, L is a Stone TFLew -algebra. Proposition 3.10. Each U (x, i) is convex and hence L is locally convex. Proof. Let a, b ∈ U (x, i) and z ∈ L such that a ≤ z ≤ b. We can show that z ∈ U (x, i). Note that a ≤ z ≤ b implies a → z, z → b ∈ Fi . Since a, b ∈ U (x, i), we have a → x, x → a, x → b, b → x ∈ Fi . But (a → z) (x → a) ≤ x → z, so x → z ∈ Fi . Similarly (b → x) (z → b) ≤ z → x, so z → x ∈ Fi . Thus x ≡Fi z, therefore z ∈ U (x, i) The following result is motivated by similar considerations for general topological lattices. Proposition 3.11. If L is a topological FLew -algebra and if U is an open subset of L then U ∧ L, U ∨ L, U L, and U → L are all open. Proof. It is well known for general lattices that U ∧ L, U ∨ L are open when U is [5, Fact 3]. To see that U L is open, let a ∈ U and x ∈ L, then there exists i ∈ I such that U (a, i) ⊆ U . Hence, U (a x, i) = U (a, i) U (x, i) ⊆ U (a, i) L ⊆ U L. Thus, U L is open. The proof that U → L is open is similar. We follow the idea of [10] to introduce the following concepts. Definition 3.12. A sequence {xn : n ∈ N} of elements of L is Cauchy if for each i ∈ I there exists a positive integer ni such that for all integers m, n > ni we have xn ≡Fi xm . Definition 3.13. x ∈ L is a limit of a sequence {xn : n ∈ N} of elements of L if for each i ∈ I there exists a positive integer ni such that for all integers n > ni we have xn ≡Fi x. If x is the unique limit of {xn : n ∈ N}, we shall write x = limn→∞ xn .
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The following can be proved using the standard argument. Remark 3.14. When L is Hausdorff, the limit of any sequence {xn : n ∈ N} is unique, if it exists. We provide a complete proof of the next result to illustrate the technique. Theorem 3.15. If L is complete then every Cauchy sequence in L has a unique limit. Proof. If L is complete, then it is Hausdorff and hence by Remark 3.14 limits are unique. Let {xn : n ∈ N} be Cauchy. For each i ∈ I there exists a positive integer ni such that for all integers m, n > ni we have xm ≡Fi xn . Let ni0 = min{n ∈ N : xm ≡Fi0 xn }. Now suppose that i0 < j ∈ I. For j we have a corresponding positive integer nj . Since Fj ⊆ Fi0 , we have xm ≡Fi xn for all positive integers m, n > nj . If i0 < j, we have ni0 < nj . Now, for each i ∈ I, let bi = xni + 1. Then we have a family (bi )i∈I of elements of L such that if i < j we have bi ≡Fi bj . Since L is complete, it follows that we can find x ∈ L with bi ≡Fi x for each ∈ I, that is xni0 +1 ≡Fi0 x. Thus for n > ni0 we have xn ≡Fi0 xni0 +1 ≡Fi0 x When X is a subset of a TFLew -algebra L, X will denote the closure of X in L. Proposition 3.16. Let F be a filter of L. 1. If Fi ⊆ F for some i ∈ I, then F is closed. 2. If F ⊆ {Fi : i ∈ I}, then F ⊆ {Fi : i ∈ I}. Hence {Fi : i ∈ I} is closed. Proof. (1) Suppose x ∈ F = i∈I a∈F U (a, i). Then x ∈ U (a, i) for some a ∈ F , that is, a → x ∈ Fi ⊆ F . Hence a, a → x ∈ F , therefore x ∈ F . (2) Let y ∈ F . Then for each i ∈ I, we have y ∈ U (a, i) for some a ∈ F, that is, a → y ∈ Fi . Since x ∈ F ⊆ Fi , we have that y ∈ Fi . Hence y ∈ i∈I Fi . Taking the case J = i∈I Fi we have that i∈I Fi is closed. The following result is easily checked. Proposition 3.17. Suppose that {Gj : j ∈ J} is another topology. Then the following are equivalent: (1) For each j ∈ J there exists i ∈ I with Fi ⊆ Gj , (2) The {Fi : i ∈ I} topology is finer than the {Gj : j ∈ J} topology, that is, each basic open set U (a, j) in the {Gj , j ∈ J} topology is open in the {Fi , i ∈ I} topology.
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Recall that L is finitely cogenerated if for every family {Gj : j ∈ J} of filters of L such that j∈J Gj = {1}, there exists a finite subset K of J such that j∈K Gj = {1}. Recall [3], that a filter F of an FLew -algebra L is called prime if, x ∨ y ∈ F implies x ∈ F or y ∈ F ; for all x, y ∈ L. Proposition 3.18. If L is finitely cogenerated and Hausdorff, then every prime filter of L is closed. Proof. Suppose that F is a prime filter of L. Since j∈J Gj = {1}, there exists a finite subset K of J such that j∈K Gj = {1} ⊆ F . Since F is prime, by [3], lemma 3(ii) it follows that Gi ⊆ F for some i ∈ K. Hence F is closed by proposition 3.16(1). 4. Final remarks We have introduced and treated the basic properties of TFLew -algebras. A natural topic that is closely related to the completions as treated in section 3 is that of profiniteness. We hope to investigate profinite topological FLew algebras in the same spirit as profinite topological orthomodular lattices [5], and profinite MV-algebras [14]. Ackowledgments: The authors are grateful to the referees whose comments helped improve both the content and the presentation of the paper. References [1] L. Bˇehounek, P. Cintula, P. H´ ajek, Handbook of mathematical fuzzy logic 1, Studies in Logic 38(2011). [2] G. Bezhanishvili, P. J. Morandi, Profinite Heyting algebras and profinite completions of Heyting algebras, Georgian J. Math. bf 16 (2009), No. 1, 29-47. [3] C. Bu¸sneag, D. Piciu, The stable topology for residuated lattices, Soft Comput 16 (2012)1639-1655. [4] T. H. Choe, Zero-dimensional compact associative distributive universal algebras, Proc. Amer. Math. Soc. 42(1974) 607-613. [5] T. H. Choe, R. J. Greechie, Profinite orthomodular lattices, Proceedings of AMS 118(1993). [6] N. Galatos, Variety of residuated lattices, PhD thesis (2003). [7] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at substructural logics, bf 151 Studies in Logic and the foundations of Mathematics, Elsevier, Amsterdam, 2007. [8] B. Van Gasse, G. Deschrijver, C. Cornelis, and E. E. Kerre, Filters of residuated lattices and triangle algebras, Inf. Sciences 18016 (2010)3006-3020.
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[9] S. Ghorbani, A. Hasankhani, Some Properties of Quotient Topology on Residuated Lattices, PU. M. A. 21 (2010)15-26. [10] C. S. Hoo, Topological MV-algebras, Topology Appl. 81(1997)103-121. [11] P. Jipsen, C. Tsinakis, A Survey of Residuated Lattices, in: J. Martinez (ed) Ordered Algebraic Structures, Kluwer, (2002)19-56. [12] P. T. Johnstone, Stone spaces, Cambridge Studies in Adv. Math. 3, Cambridge University Press, Cambridge (1982). [13] L. D. Nel, Universal topological algebra needs closed topological categories, Topology Appl. 12(1981)321-330. [14] J. B. Nganou, Profinite MV-algebras and multisets, Order (2015) DOI 10.1007/s11083-014-9345-5. [15] H. Ono, Substructural Logics and Residuated Lattices- an Introduction, V. F. Hendricks and J. Malinowski (eds.), Trends in Logic 20 (2003)177–212. [16] D. P. Strauss, Topological lattices, Proc. London Math. Soc. 3 18 (1968)217-230. [17] H. Taqdir, Introduction to Topological Groups, Philadelphia: R.E. Krieger Pub. Co(1981). [18] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag,1999. [19] O. Wyler, On the categories of general topology and topological algebras, Arch. Math. 22(1971)7-17. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: [email protected] Department of Mathematics, University of Yaounde I, Cameroon E-mail address: [email protected]
PII: DOI: Reference:
S1570-8683(15)00051-8 http://dx.doi.org/10.1016/j.jal.2015.04.004 JAL 379
To appear in:
Journal of Applied Logic
Received date: 22 October 2014 Accepted date: 24 April 2015
Please cite this article in press as: J.B. Nganou, S.F.T. Tebu, Topological FLew -algebras, Journal of Applied Logic (2015), http://dx.doi.org/10.1016/j.jal.2015.04.004
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TOPOLOGICAL FLew -ALGEBRAS JEAN B. NGANOU, SERGE F. T. TEBU Abstract. The main goal of this article is to introduce topological FLew algebras and study their main properties. We also treat completions of FLew -algebras with respect to inductive family of filters. This work generalizes similar works on MV-algebras [10] and on FLew -algebras equipped with uniform topologies [9]. Key words: complete, MV-algebra, inductive family, Heyting algebra, topological FLew -algebra, profinite FLew -algebra, Stone FLew -algebra.
1. Introduction Universal topological algebras, (which are algebras with topologies with respect to which all operations are continuous) have been introduced and studied thoroughly (see for e.g [4, 12, 13, 19]). Some of the notable classes of topological algebras that have been the objects of more detailed studies include groups [17], lattices [16], othomodular lattices [5], MV-algebras [10]. A residuated lattice is a lattice equipped with a monotone monoidal operation (with a unit e) and a pair of binary operations /, \ satisfying x y ≤ z if and only if x ≤ z/y if and only if y ≤ x\z FL-algebras, which are residuated lattices expanded by a constant ¯0, form the algebraic semantics for the so-called (intuitionistic) substructural logics. These provide a unifying framework for several kinds of logics (Girard’s linear logic, the Lambek calculus, the L ukasiewicz’s many-valued logics, H´ajek fuzzy logics etc). A very important subclass of FL-algebras is that of FLew -algebras, which are FL-algebras extended by the exchange and weakening rules. An FLew -algebra (also known as bounded integral commutative residuated lattice) can be defined [7, 11] as a bounded lattice (L, ∨, ∧, 0, 1) with two additional binary operations , → satisfying: (i) (L, , 1) is a commutative monoid, and (ii) for all x, y, z ∈ L, x y ≤ z if and only if x ≤ y → z. A nonempty subset F of a FLew -algebra L is called a filter of L if it satisfies: (F1) x y ∈ F for 2010 Mathematics Subject Classification. 06D35, 06E15, 06D50. April 30, 2015. 1
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all x, y ∈ F , and (F2) For all x, y ∈ L, if x ≤ y and x ∈ F , then y ∈ F . While, filters of an FLew -algebra are studied for their connection to its congruences, they primarily come from logic itself since they correspond to theories of the logic. Indeed, filters of an FLew -algebra L are just subsets of L that are closed under all deductions. Equivalently, in algebraic logic, filters are sets of designated elements that provide matricial models for the logic. Among the many important subclasses of FLew -algebras, there is that of MV-algebras, which constitute the algebraic counterpart of L ukasiewicz many valued logic. In [10], Hoo introduced topological MV-algebras, and studied their main properties. A close analysis of Hoo’s work reveals that the essential ingredients are the existence of an adjoint pair of operations and the fact that ideals of MV-algebras correspond to its congruences. This prompted us to consider the same study in a more general context where similar ingredients are available, namely FLew -algebras. The main goal of the present work is to generalize Hoo’s work to FLew -algebras. This yields the notion of topological FLew -algebra, which is an FLew -algebra together with topology with respect to which all the operations are continuous. Topological FLew -algebrass were already investigated in [9], but mainly for the uniform topology. The paper is organized as follows. In section 2, we study the general properties of topological FLew -algebras. In section 3, we study a special class of topological FLew -algebras, namely those arising from decreasing families of filters indexed by directed sets. We establish conditions under which the resulting space has certain topological properties, and also study completions of topological FLew -algebras. In addition, we show that such FLew -algebras are Stone if and only of they are compact and Hausdorff. Homomorphisms of FLew -algebras have the usual meaning. A subset F of an FLew -algebras L is called a deductive system of L if: (ds1) 1 ∈ F and (ds2) For every x, y ∈ L, if x, x → y ∈ F , then y ∈ F. It is known that the notions of filters and deductive systems coincide (see e.g.,[8]). We shall use solely the filter terminology in the entire article. Filters of FLew -algebras induce congruences, indeed the following result is wellknown. Proposition 1.1. [18] Let L be an FLew -algebra and let F be a filter of L. The relation x ≡F y if and only if x → y, y → x ∈ F is a congruence on L. If F is a filter of L, the quotient FLew -algebra induced by the congruence ≡F shall be denoted by L/F . In addition the class of an element a ∈ L with respect to ≡F is often denoted by [a]F ; and πF denotes the natural projection L → L/F .
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For every subset X ⊆ L, the smallest filter of L containing X (i.e., the intersection of all filters F of L such that X ⊆ F ) is called the filter generated by X and will be denoted by X . If U and V are subsets of L and ∈ {∨, ∧, →, }, then U V := {x y : x ∈ U, y ∈ V } Remark 1.2. While some aspects of the paper could be treated in the more general context of FL-algebras, or even arbitrary residuated lattices, we restrict our study to bounded integral commutative residuated lattices (FLew algebras). Our choice is mainly motivated by the fact that the definitions and notations in the FLew -algebras context are greatly simplified, and that once established, one could easily adapt the relevant results to that more general context. The main properties of FLew -algebras are summarized in the next result. Proposition 1.3. [6, 18] The following hold in all FLew -algebras L = (L, ∧, ∨, , → , 0, 1) and for all x, y, z ∈ L. x ≤ y if and only if x → y = 1 x (y ∨ z) = (x y) ∨ (x z), x (y ∧ z) = (x y) ∧ (x z)
(i) (ii)
x → (y ∧ z) = (x → y) ∧ (x → z), (x ∨ y) → z = (x → z) ∧ (y → z)
(iii)
(y → x) y ≤ x, x (z → y) ≤ z → x y
(iv)
z → (y → x) = (z y) → x, (y → z) (x → y) ≤ z → x x → y ≤ (y → z) → (x → z), y → x ≤ y z → x z y ≤ x → y, z → (x ∧ y) = (z → x) ∧ (z → y) If x ≤ y, then y → z ≤ x → z, z → x ≤ z → y, x z ≤ y z (x → y) x ≤ x ∧ y, x y ≤ x ∧ y (y → z) (x → y) ≤ z → x
(v) (vi) (vii) (viii) (ix) (x)
For more details on FLew -algebras, the reader may refer to [1, 7, 11] 2. Topological FLew -algebras: general properties Let (L, ∧, ∨, , →, 0, 1) be an FLew -algebras and τ be a topology on L. We say that the pair (L, τ ) is a topological FLew -algebra (TFLew -algebra, for short) if the operations ∧, ∨, →, are continuous with respect to τ . We shall simply say that L is a TFLew -algebra without explicit mention of the topology τ when the topology is clear from the context. We start by presenting two important examples of TFLew -algebras.
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Example 2.1. Consider L = [0, 1], x ∨ y = max(x, y), x ∧ y = min(x, y), x y = max(0, x + y − 1), x → y = min(1 − x + y, 1) Then L is a TFLew -algebra under the natural topology on [0, 1]. More generally, every topological MV-algebra as treated in [10] is a TFLew -algebra. Example 2.2. For every integer n ≥ 2, let Ln be the unique linear Heyting algebra with n elements. Then, each Ln is a TFLew -algebra under the discrete topology. Now, consider L := n≥2 Ln endowed with the product topology. Then L is a TFLew -algebra, in fact a Stone (compact, Hausdorff and zerodimensional) TFLew -algebra. The following fact, which was observed in [9] is very useful for proving continuity of the operations of a TFLew -algebra. Proposition 2.3. [9] Let (L, ∧, ∨, , →, 0, 1) be a FLew -algebra and τ be a topology on L. An operation ∈ {∨, ∧, →, } is continuous if and only if for every open set O of (L, τ ) and x, y ∈ L such that x y ∈ O, there exists open sets O1 , O2 such that x ∈ O1 , y ∈ O2 and O1 O2 ⊆ O. Proposition 2.4. Let L be a TFLew -algebra, and F a filter of L such that πF is open. Then, the quotient FLew -algebra L/F is a TFLew -algebra under the quotient topology. Proof. It is sufficient to prove that ([x]F , [y]F ) → [x]F [y]F := [x y]F is continuous where ∈ {∧, ∨, , →}. Let x, y ∈ L and let W be a neighborhood of [x y]F . Then πF−1 (W ) is open in L and x y ∈ πF−1 (W ). Since L is a TFLew -algebra, there exists neighborhoods U0 of x and V0 of y such that U0 V0 ⊂ πF−1 (W ). Put U := πF (U0 ) and V := πF (V0 ). Then U and V are open in L/F since πF is open. Clearly [x]F ∈ U , [y]F ∈ V and it is straightforward that U V ⊆ πF (U0 V0 ) ⊆ W . Thus, from Proposition 2.3 that ([x]F , [y]F ) → [x]F [y]F := [xy]F is continuous for each ∈ {∧, ∨, , →}. The following observation justifies why the content of this section generalizes some of the main results in [10]. Remark 2.5. Let (A, ⊕, ¬, 0) be an MV-algebra viewed as an FLew -algebra. Let F be a filter of the FLew -algebra A, then ¬F is an ideal of the MV-algebra A. It is straightforward to see that when πF is open, the quotient TFLew algebras A/F and A/¬F are algebraically and topologically isomorphic. Proposition 2.6. For every TFLew -algebra L and F an open filter of L, the quotient topology on L/F is discrete.
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Proof. Note that for every x ∈ L, πF−1 ([x]F ) = [x]F . Therefore, it is enough to prove that [x]F is open for every x ∈ L. Let x ∈ L, and define f, g : L → L by f (a) = a → x and g(a) = x → a. Thus, f and g are continuous and since F is open, then f −1 (F ), g −1 (F ) are open. Now, it is also clear that [x]F = f −1 (F ) ∩ g −1 (F ), which is open. Proposition 2.7. Let L be a TFLew -algebra in which {1} is closed (resp. open). Then {x} is closed (resp. open) for all x ∈ L. Proof. Let f : L2 → L be the continuous map given by f (a, b) = a → b and let g : L3 → L2 be given by g(a, b, c) = (a → b, b → c). Then g is continuous. If {1} is closed, so is {(1, 1)} ⊆ L2 . Let x ∈ L and let h : L → L2 be given by h(x) = g(a, x, a) = (a → x, x → a). Then h is the restriction of g to {a} × L × {a} and hence is continuous. Now h−1 (1, 1) = {x ∈L: x → a = 1, a → x = 1} = {a}. Thus {a} is also closed. A similar argument is used when {1} is open. The following result was obtained in [9], but only when L is endowed by the uniform topology induced by a family of filters closed under the intersection. Theorem 2.8. Let L be a TFLew -algebra and F a filter of L, then, 1. If L/F is Hausdorff, then F is closed and the converse holds if πF is open. 2. L/F is discrete if and only if F is open. Proof. 1. The necessity holds for every filter as proved in [9, Thm. 3.9]. While the proof of the sufficiency given in [9, Thm. 3.9] was intended only for uniform topologies, the only extra hypothesis used was πF is open, which we have. 2. Note if L/F is discrete, then {F } is open in L/F . But, since πF−1 ({F }) = F , then F is open. The converse was established in Proposition 2.6. Theorem 2.9. Let (L, τ ) be a topological FLew -algebra. The connected component of 1 is a closed filter of L. Proof. Let C be the connected component of 1, and let x ∈ C. On one hand, since fx1 : L → L : x → x y is a continuous map, fx1 (C) = x C is connected and it contains x. Hence C ∩ (x C) = ∅. This means that C ∪ (x C) is connected. Since it contains 1, we have C ∪(xC) ⊆ C. Therefore xC ⊆ C. Thus C C ⊆ C. On the other hand let x, x → y ∈ C and fy2 : L → L : x −→ x ∨ y is a continuous map. Then (x → y) x ∈ C. Since (x → y) x ≤ y we have y = [(x → y) x] ∨ y ∈ y ∨ C and fy2 (C) = y ∨ C is connected and contains 1. This means that y ∨ C ⊆ C; hence y ∈ C. This proves that C is a filter.
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Since C is connected, then the closure C of C is connected and is maximal, C = C i.e., C is closed. Proposition 2.10. If F is an open (resp. closed) filter, then for each x ∈ L, [x]F is open (resp. closed). Proof. The proof when F is open was given in the proof of Proposition 2.6. The case F closed is handled similarly. Theorem 2.11. Every open filter is closed. Proof. Suppose that a filter F is open. Then by Proposition 2.10, each con gruence class [x]F is open. But F = L \ {[x]F : x ∈ / F }, which is closed. We would like to point out that a closed filter of a TFLew -algebra needs not be open. For instance, the trivial filter of the TFLew -algebra of Example 2.1 is closed, but not open. Theorem 2.12. If 1 is an isolated point of a filter F , then F is discrete. Proof. Let U is an open neighborhood of 1 such that {1} = U ∩ F and let x ∈ F . Since x → x = 1 ∈ U , there exist neighborhoods V1 and V2 of x such that V1 → V2 ⊂ U . Setting V = V1 ∩ V2 we have V ∩ F → V ∩ F ⊂ (V → V ) ∩ F ⊂ U ∩ F = {1}. Let y ∈ V ∩ F . Then y → x ∈ V ∩ F → V ∩ F = {1}, that is y ≤ x. Similarly x → y ∈ V ∩ F → V ∩ F = {1}, that is x ≤ y. Thus x = y, proving that V ∩ F = {x}. This means that x is also an isolated point of F and hence F is discrete. Theorem 2.13. Suppose that the only closed filters of L are {1} and L. Then the topology is one of the following types: (1) Hausdorff and totally disconnected. (2) Hausdorff and connected, (3) The trivial topology. Proof. Let C be the connected component of 1. By theorem 2.9, C is a closed filter and hence either C = {1} or C = L. If C = {1}, then {1} = {1} and hence L is Hausdorff. For each x ∈ L, let Cx be the connected component containing x. Then Cx → x and x → Cx are connected and containing 1. Hence Cx → x ⊆ C = {1} and x → Cx ⊆ C = {1}; that is Cx = {x}. This means that L is Hausdorff and totaly disconnected. On the other hand, if C = L, then L is connected. We have again {1} = {1} or {1} = L. In the first case L is Hausdorff and connected. In the second case, let U be any nonempty open subset of L. Let x ∈ {1}. Then U ∩ {1} = ∅. Thus 1 ∈ U , that is, every nonempty open subset U of L
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must contain 1. Further, if x ∈ / U for some x ∈ L, then since x → x = 1 ∈ U , we can find open neighborhoods V and W of x such that V → W ⊆ U. But 1 ∈ W , means that x = x → 1 ∈ V → W ⊆ U, a contradiction. Hence U = L, that is, every open set is either empty or is L. Thus the topology is trivial. We recall that an FLew -algebra L is called locally finite if for every x ∈ L \ {0}, there exists a natural number n such that xn = 0. It follows that such an FLew -algebra is simple, i.e, has only two filters: itself and {1}. The following result is therefore an immediate consequence of Theorem 2.13. Corollary 2.14. The topology on a locally finite FLew -algebra is one of the following types: (1) Hausdorff and totally disconnected, (2) Hausdorff and connected, (3) the trivial topology. 3. TFLew -algebras: completions In this section, we construct our main class of examples of TFLew -algebras and use it to treat completions of FLew -algebras. Our approach follows that found in [10] about MV-algebras and their ideals. We start by motivating the study of these concepts. Completions (in particular complete TFLew -algebras) treated here are natural examples of profinite TFLew -algebras. As observed by the authors of [2], profinite algebras play an important role in the study of various logical systems. One fundamental question for a logical system is that of decidability, in other words, does there exist an algorithm which decides for a given formula ϕ (in the language L), whether or not ϕ is a theorem in L. It is well-known that if L is a finitely axiomatizable and has the finite model property (FMP), then L is decidable. But, for languages L with nice algebraic semantics, whether L has the FMP or not reduces to whether or not the variety VL of algebraic model of L is generated by finite VL -algebras. Let L be any FLew -algebra, and let F := {Fi : i ∈ I} be a decreasing family of filters indexed by a directed poset (I, ≤). More explicitly, the following is true: (i) For every i, j ∈ I, there exists k ∈ I such that i ≤ k and j ≤ k and, (ii) For every i, j ∈ J, i ≤ j implies Fj ⊆ Fi . We shall call F an inductive family of filters indexed by (I, ≤). For each a ∈ L and each i ∈ I, for simplicity and clarity, we denote by U (a, i) the congruence class of a modulo Fi , that is U (a, i) = [a]Fi = {x ∈ L : x ≡Fi a} = {x ∈ L : a → x, x → a ∈ Fi }.
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Taking these congruences classes as the system of neighborhoods turns L into a TFLew -algebra. More precisely, Proposition 3.1. Let L be an FLew -algebra, F be an inductive family of filters indexed by (I, ≤) and TF = {G ⊆ L : ∀a ∈ G, ∃i ∈ I, U (a, i) ⊆ G} Then (L, TF ) is a TFLew -algebra. Proof. It is clear that ∅ and the set L belong to TF . Also from the definition, it is clear that TF is closed under arbitrary unions. Finally to show that TF is closed under finite intersection, let G, H ∈ TF and suppose a ∈ G ∩ H. Then there exist i, j ∈ I such that U (a, i) ⊂ G and U (a, j) ⊂ H. Since I is a directed set, there exists k ∈ I such that i ≤ k, j ≤ k. Then U (a, k) ⊂ U (a, i)∩U (a, j); that is U (a, k) ⊂ G ∩ H. Therefore G ∩ H ∈ TF . Thus TF is a topology on L. Next, we prove the continuity of ∧. Let O be an open set of L and x, y ∈ L such that x ∧ y ∈ O. Then there exists i ∈ I, U (x ∧ y, i) ⊆ O and one can show that U (x, i) ∧ U (y, i) ⊆ U (x ∧ y, i). Indeed, let z ∈ U (x, i) ∧ U (y, i), then z = a ∧ b with a ∈ U (x, i) and b ∈ U (y, i). Since a ≡Fi x, b ≡Fi y and ≡Fi is a congruence on L, we have a ∧ b ≡Fi x ∧ y. That is z ≡Fi x ∧ y, hence z ∈ U (x ∧ y, i) ⊆ O. Therefore, ∧ is continuous by Proposition 2.3. The proofs of the continuity of ∨, and → are similar to that of ∧. Thus, (L, TF ) is a TFLew -algebra as claimed. Remark 3.2. We will often denote the TFLew -algebra (L, TF ) simply by LF and refer to TF as the linear topology on L induced by F. Clearly each U (a, i) is open in LF . In addition, observe that for a fixed i ∈ I and a ∈ L, we have U (a, i) = L \ {U (x, i) : x ≡Fi a} and hence each U (a, i) is both open and closed (i.e., clopen) in LF . Moreove, since Fi = U (1, i) for each i ∈ I, then each Fi is clopen. We want to explain more directly how this construction generalizes the one found in [10] for MV-algebras. Indeed, suppose that (A, ⊕, ¬, 0) is an MV-algebra viewed as an FLew -algebra, and let F := {Fi : i ∈ I} be a decreasing family of filters indexed by a directed poset (I, ≤). Consider ¬F := {¬Fi : i ∈ I}. Then ¬F := {¬Fi : i ∈ I} is a decreasing family of ideals of A indexed by a directed poset (I, ≤). The topological MV-algebra on A induced by ¬F as treated by [10] is algebraically and topologically isomorphic to the TFLew -algebra induced by F. For more explicit examples, consider the following.
TOPOLOGICAL FLew -ALGEBRAS
Example 3.3. Let L := Z≥0 be the set of nonnegative integers.
L, ∨, ∧, →, , 0, 1 is an FLew -algebra (indeed a BL-algebra), where
9
Then
x ∨ y = gcd(x, y), x ∧ y = lcm(x, y), x → y = lcm(x, y)/x, x y = xy Fix an integer n ≥ 1 and let D(n) denotes the set of positive divisors of n. Then (D(n), ≤) is a directed set, where d ≤ d if d |d. For each d ∈ D(n), let Fd = d be the principal filter of L generated by d. Then F := (Fd )d∈D(n) is an inductive family of filters. The following result provides a necessary and sufficient condition of the separation axioms for TF . Theorem 3.4. Let L be an FLew -algebra and F be a decreasing family of filters of L indexed by a directed poset (I, ≤). Then the following assertions are equivalent. (i) (L, TF ) is a T1 -space, (ii) i∈I Fi = {1}, (iii) (L, TF ) is a T2 -space Proof. (i) ⇒ (ii) Assume that (L, TF ) is a T1 -space. By contradiction suppose there is x ∈ i∈I Fi with x = 1. Then since TF is a T1 -space, there exists an open set V containing 1 such that x ∈ / V . Therefore, there exists i ∈ I such that Fi = U (1, i) ⊆ V . Hence, x ∈ V which is clearly a contradiction. (ii) ⇒ (iii) Assume that i∈I Fi = {1}. Let x, y ∈ L such that x = y, then x → y = 1 or y → x = 1. Suppose that x → y = 1, then by assumption there exists i ∈ I such that x → y ∈ / Fi . We claim that U (x, i) ∩ U (y, i) = ∅. Indeed, if on the contrary we assume that U (x, i) ∩ U (y, i) = ∅, then there exists a ∈ U (x, i) ∩ U (y, i). Thus x → a, a → y ∈ Fi , hence (x → a) (a → y) ∈ Fi . But since (x → a) (a → y) ≤ x → y, then x → y ∈ Fi , which is a contradiction. One argues the case y → x = 1 similarly. (iii) ⇒ (i) T2 -spaces are T1 -spaces, always. We consider the following standard construction in the context of TFLew algebras. A similar construction for MV-algebras can be found in [10]. Start with the TFLew -algebra LF , where F is a decreasing family of filters of L indexed by a directed poset (I, ≤). Recall that for every i ∈ I, Fi is open in L , thus each L/Fi is discrete. Next, we consider the product topology on F i∈I L/Fi . In addition, note that for every i ≤ j in (I, ≤), there is a natural homomorphism ϕij : L/Fj → L/Fi . It is indeed a simple verification that (L/Fi , ϕij ) is an inverse/projective system, whose limit shall be denoted by := lim L/Fi . L is given the subspace topology of L/Fi , and is called L i∈I ←−
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the completion of L. In addition, there is a natural continuous homomorphism defined by ψ(x) = ([x]F ) and ψ(L) is dense in L. ψ:L→L i i∈I The following are obvious. is injective if and Proposition 3.5. With the preceding notations, ψ : L → L only if i∈I Fi = {1}. is injective if and only if LF is In other words, from Theorem 3.4, ψ : L → L is an isomorphism. Hausdorff. We say that L is complete (w.r.t F) if ψ : L → L → L/Fi denotes the natural projection and Pi := p−1 (1). Then Let pi : L i indexed by the same directed F ∗ := (Pi )i∈I is a decreasing family of filters of L set (I, ≤). Remark 3.6. If I and J are directed sets, then the families of filters F = {Fi : i ∈ I} and G = {Gj : j ∈ J} induce the same topology on L if and only if for each U (x, i) there exists j ∈ J such that U (x, j) ⊆ U (x, i) and for each U (x, j) there exists i ∈ I such that U (x, i) ⊆ U (x, j). In this case the natural homomorphism lim L/Fi −→ lim L/Fj is an isomorphism. Thus the ←− ←− depends only on the topology of L. completion L is the linear topology induced by F ∗ . Proposition 3.7. 1. The topology on L is complete and Hausdorff. 2. L let U (([xi ]F )i∈I , j) = Proof. 1. For each j ∈ I and each ([xi ]Fi )i∈I ∈ L, i {([yi ]Fi )i∈I ∈ L : ([yi ]Fi )i∈I ≡Pj ([xi ]Fi )i∈I }. Then if i = j and Uj = {[xj ]F }. U (([xi ]Fi )i∈I , j) = i∈I Ui , where Ui = pi (L) j Thus U (([xi ]Fi )i∈I , j) is open. 2. This follows easily from the definition of completeness. For the rest of the paper, L denotes a TFLew -algebra and the topology on L is the linear topology induced by an inductive family F of filters. Proposition 3.8. Every compact and Hausdorff TFLew -algebra is complete. Proof. Suppose that L is compact and Hausdorff, then by Proposition 3.4, i∈I Fi = {1}. Therefore, by Proposition 3.5, L is a subspace of L. In addition, for each i, since L = ∪a∈L U (a, i), and L is compact, then there exist a1i , . . . , ani ∈ L such that L = ∪nj=1 U (aji , i). Therefore, L/Fi = {[a1i ]F , . . . , [ani ]F } and L/Fi is finite for each i ∈ I. Thus, i∈I L/Fi is compact and since L is a then compact subspace, L is closed in i∈I L/Fi . But, since L is dense in L, r r = L = L∩L = L ∩ L, where L denotes the relative closure of L in L. L Hence L ⊆ L, which implies L = L and L is complete.
TOPOLOGICAL FLew -ALGEBRAS
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An important result due to Strauss [16] asserts that a topological Boolean algebra is Stone if and only if it is compact and Hausdorff. We can establish the same result for the TFLew -algebras discussed here. Proposition 3.9. Let L be a TFLew -algebra. Then L is Stone if and only if it is compact and Hausdorff. Proof. Recall that an FLew -algebra is Stone if is compact, Hausdorff and zerodimensional (i,e., has a base of clopen sets). We only need to prove that every compact Hausdorff TFLew -algebra is Stone. Suppose that L is a compact and ˆ But, L ˆ is clearly Hausdorff TFLew -algebra. Then, by Proposition 3.8, L ∼ = L. a Stone TFLew -algebra, as any profinite FLew -algebra is Stone. Therefore, L is a Stone TFLew -algebra. Proposition 3.10. Each U (x, i) is convex and hence L is locally convex. Proof. Let a, b ∈ U (x, i) and z ∈ L such that a ≤ z ≤ b. We can show that z ∈ U (x, i). Note that a ≤ z ≤ b implies a → z, z → b ∈ Fi . Since a, b ∈ U (x, i), we have a → x, x → a, x → b, b → x ∈ Fi . But (a → z) (x → a) ≤ x → z, so x → z ∈ Fi . Similarly (b → x) (z → b) ≤ z → x, so z → x ∈ Fi . Thus x ≡Fi z, therefore z ∈ U (x, i) The following result is motivated by similar considerations for general topological lattices. Proposition 3.11. If L is a topological FLew -algebra and if U is an open subset of L then U ∧ L, U ∨ L, U L, and U → L are all open. Proof. It is well known for general lattices that U ∧ L, U ∨ L are open when U is [5, Fact 3]. To see that U L is open, let a ∈ U and x ∈ L, then there exists i ∈ I such that U (a, i) ⊆ U . Hence, U (a x, i) = U (a, i) U (x, i) ⊆ U (a, i) L ⊆ U L. Thus, U L is open. The proof that U → L is open is similar. We follow the idea of [10] to introduce the following concepts. Definition 3.12. A sequence {xn : n ∈ N} of elements of L is Cauchy if for each i ∈ I there exists a positive integer ni such that for all integers m, n > ni we have xn ≡Fi xm . Definition 3.13. x ∈ L is a limit of a sequence {xn : n ∈ N} of elements of L if for each i ∈ I there exists a positive integer ni such that for all integers n > ni we have xn ≡Fi x. If x is the unique limit of {xn : n ∈ N}, we shall write x = limn→∞ xn .
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The following can be proved using the standard argument. Remark 3.14. When L is Hausdorff, the limit of any sequence {xn : n ∈ N} is unique, if it exists. We provide a complete proof of the next result to illustrate the technique. Theorem 3.15. If L is complete then every Cauchy sequence in L has a unique limit. Proof. If L is complete, then it is Hausdorff and hence by Remark 3.14 limits are unique. Let {xn : n ∈ N} be Cauchy. For each i ∈ I there exists a positive integer ni such that for all integers m, n > ni we have xm ≡Fi xn . Let ni0 = min{n ∈ N : xm ≡Fi0 xn }. Now suppose that i0 < j ∈ I. For j we have a corresponding positive integer nj . Since Fj ⊆ Fi0 , we have xm ≡Fi xn for all positive integers m, n > nj . If i0 < j, we have ni0 < nj . Now, for each i ∈ I, let bi = xni + 1. Then we have a family (bi )i∈I of elements of L such that if i < j we have bi ≡Fi bj . Since L is complete, it follows that we can find x ∈ L with bi ≡Fi x for each ∈ I, that is xni0 +1 ≡Fi0 x. Thus for n > ni0 we have xn ≡Fi0 xni0 +1 ≡Fi0 x When X is a subset of a TFLew -algebra L, X will denote the closure of X in L. Proposition 3.16. Let F be a filter of L. 1. If Fi ⊆ F for some i ∈ I, then F is closed. 2. If F ⊆ {Fi : i ∈ I}, then F ⊆ {Fi : i ∈ I}. Hence {Fi : i ∈ I} is closed. Proof. (1) Suppose x ∈ F = i∈I a∈F U (a, i). Then x ∈ U (a, i) for some a ∈ F , that is, a → x ∈ Fi ⊆ F . Hence a, a → x ∈ F , therefore x ∈ F . (2) Let y ∈ F . Then for each i ∈ I, we have y ∈ U (a, i) for some a ∈ F, that is, a → y ∈ Fi . Since x ∈ F ⊆ Fi , we have that y ∈ Fi . Hence y ∈ i∈I Fi . Taking the case J = i∈I Fi we have that i∈I Fi is closed. The following result is easily checked. Proposition 3.17. Suppose that {Gj : j ∈ J} is another topology. Then the following are equivalent: (1) For each j ∈ J there exists i ∈ I with Fi ⊆ Gj , (2) The {Fi : i ∈ I} topology is finer than the {Gj : j ∈ J} topology, that is, each basic open set U (a, j) in the {Gj , j ∈ J} topology is open in the {Fi , i ∈ I} topology.
TOPOLOGICAL FLew -ALGEBRAS
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Recall that L is finitely cogenerated if for every family {Gj : j ∈ J} of filters of L such that j∈J Gj = {1}, there exists a finite subset K of J such that j∈K Gj = {1}. Recall [3], that a filter F of an FLew -algebra L is called prime if, x ∨ y ∈ F implies x ∈ F or y ∈ F ; for all x, y ∈ L. Proposition 3.18. If L is finitely cogenerated and Hausdorff, then every prime filter of L is closed. Proof. Suppose that F is a prime filter of L. Since j∈J Gj = {1}, there exists a finite subset K of J such that j∈K Gj = {1} ⊆ F . Since F is prime, by [3], lemma 3(ii) it follows that Gi ⊆ F for some i ∈ K. Hence F is closed by proposition 3.16(1). 4. Final remarks We have introduced and treated the basic properties of TFLew -algebras. A natural topic that is closely related to the completions as treated in section 3 is that of profiniteness. We hope to investigate profinite topological FLew algebras in the same spirit as profinite topological orthomodular lattices [5], and profinite MV-algebras [14]. Ackowledgments: The authors are grateful to the referees whose comments helped improve both the content and the presentation of the paper. References [1] L. Bˇehounek, P. Cintula, P. H´ ajek, Handbook of mathematical fuzzy logic 1, Studies in Logic 38(2011). [2] G. Bezhanishvili, P. J. Morandi, Profinite Heyting algebras and profinite completions of Heyting algebras, Georgian J. Math. bf 16 (2009), No. 1, 29-47. [3] C. Bu¸sneag, D. Piciu, The stable topology for residuated lattices, Soft Comput 16 (2012)1639-1655. [4] T. H. Choe, Zero-dimensional compact associative distributive universal algebras, Proc. Amer. Math. Soc. 42(1974) 607-613. [5] T. H. Choe, R. J. Greechie, Profinite orthomodular lattices, Proceedings of AMS 118(1993). [6] N. Galatos, Variety of residuated lattices, PhD thesis (2003). [7] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at substructural logics, bf 151 Studies in Logic and the foundations of Mathematics, Elsevier, Amsterdam, 2007. [8] B. Van Gasse, G. Deschrijver, C. Cornelis, and E. E. Kerre, Filters of residuated lattices and triangle algebras, Inf. Sciences 18016 (2010)3006-3020.
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[9] S. Ghorbani, A. Hasankhani, Some Properties of Quotient Topology on Residuated Lattices, PU. M. A. 21 (2010)15-26. [10] C. S. Hoo, Topological MV-algebras, Topology Appl. 81(1997)103-121. [11] P. Jipsen, C. Tsinakis, A Survey of Residuated Lattices, in: J. Martinez (ed) Ordered Algebraic Structures, Kluwer, (2002)19-56. [12] P. T. Johnstone, Stone spaces, Cambridge Studies in Adv. Math. 3, Cambridge University Press, Cambridge (1982). [13] L. D. Nel, Universal topological algebra needs closed topological categories, Topology Appl. 12(1981)321-330. [14] J. B. Nganou, Profinite MV-algebras and multisets, Order (2015) DOI 10.1007/s11083-014-9345-5. [15] H. Ono, Substructural Logics and Residuated Lattices- an Introduction, V. F. Hendricks and J. Malinowski (eds.), Trends in Logic 20 (2003)177–212. [16] D. P. Strauss, Topological lattices, Proc. London Math. Soc. 3 18 (1968)217-230. [17] H. Taqdir, Introduction to Topological Groups, Philadelphia: R.E. Krieger Pub. Co(1981). [18] E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag,1999. [19] O. Wyler, On the categories of general topology and topological algebras, Arch. Math. 22(1971)7-17. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: [email protected] Department of Mathematics, University of Yaounde I, Cameroon E-mail address: [email protected]