ChenLiDengAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 8, No. 2, 2007 pp. 309-317 Stone compactification of additive generalized-algebraic lattices Xueyou Chen, Qingguo Li and Zike Deng∗ Abstract. In this paper, the notions of regular, completely regular, compact additive generalized algebraic lattices ([7]) are introduced, and Stone compactification is constructed. The following theorem is also obtained. Theorem: An additive generalized algebraic lattice has a Stone com- pactification if and only if it is regular and completely regular. 2000 AMS Classification: 06B30, 06B35, 54D35, 54H10 Keywords: additivity, generalized way-below relation, lower homomorphism, upper adjoint. 1. Introduction The notions of directed sets, way-below relations, continuous lattices, al- gebraic lattices were introduced in [12], and applied in the study of domain theory, topological theory, lattice theory, etc.. As a generalization, D. Novak introduced the notions of generalized contin- uous lattices (M-continuous lattices) and generalized algebraic lattices in [15]. In the study of topological theory and lattice theory, many researchers are interested in the topological representation of a complete lattice. For example: suppose (X, T ) is a topological space, all open sets T of a topological space may be viewed as a frame, and a frame may also be viewed as an open sets lattice. For the theory of Frame (Locale), please refer to [13]. On the other hand, suppose (X, C) is a co-topological space. C is the set of all closed subsets. D. Drake, W. J. Thron and S. Papert considered C as a complete lattice (C, ∪, ∩, ∅, X)([11, 16]). But unfortunately the correspon- dence between complete lattices and T0-topological spaces is not one-to-one. ∗This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471035/A010104) and Natural Science Foundation of Shandong Province (Grant No. 2003ZX13). 310 X. Chen, Q. Li and Z. Deng To solve the problem, Deng also investigated generalized continuous lat- tices on the basis of [1, 11, 15, 16]. He introduced the notions of maximal systems of subsets, additivity property, homomorphisms, direct sums, lower sublattices in [5, 6, 9, 10]. Finally, the question was settled in [7, 8]. He ob- tained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of T0-topological spaces with continuous mappings. This paper is a sequel of [2, 7, 8]. In Section 2, we begin with an overview of generalized continuous lattices, Deng’s work, which surveys Preliminaries. In Section 3, we introduce the notions of regular, completely regular and com- pactness on an additive generalized algebraic lattice, and obtain a Stone com- pactification. 2. Preliminaries We introduce some notions for each area, i.e., generalized continuous lattices and additive generalized algebraic lattices. 2.1. Generalized Continuous Lattices. In [15], D. Novak introduced the notions of generalized way-below relations and systems of subsets. Let (P, ≤) be a complete lattice, ≺ is said to be a generalized way-below relation if (i) a ≺ b ⇒ a ≤ b, (ii) a ≤ b ≺ c ≤ d ⇒ a ≺ d. Obviously, it is a natural generalization of the notion of a way-below relation ([12]). M ⊆ 2P is said to be a system of subsets of P , if for a ∈ P , there exists S ∈ M , such that ↓ a =↓ S, where ↓ a = {b | b ≤ a}, ↓ S = ∪{↓ a | a ∈ S}. There are three kinds of common used system of subsets: (i) the system of all finite subsets, (ii) the system of all directed sets and (iii) the system of all subsets. By means of the notion of systems of subsets, he defined a generalized way- below relation. Suppose M is a system of subsets. For a, b ∈ P , a is said to be way-below b with respect to M , in symbols a ≺M b, if for every S ∈ M with b ≤ ∨S, then a ∈↓ S. Clearly ≺M is a generalized way-below relation induced by M ([15]). We will denote ≺M as ≺. (P, ≺) is called a generalized continuous lattice, if for every a ∈ P , we have a = ∨ ⇓ a, where ⇓ a = {b | b ≺ a}. a ∈ P is called a compact element, if a ≺ a. Let K(≺) = {a ∈ P | a ≺ a}. (P, ≺) is called a generalized algebraic lattice, if for every a ∈ P , we have a = ∨{↓ a ∩ K(≺)}. Further study, see [1, 17]. 2.2. Additive Generalized Algebraic Lattices. Suppose (P, ≺) is a generalized continuous lattice, Deng introduced the no- tion of a maximal system of subsets generated by ≺, that is, Stone compactification of additive generalized-algebraic lattices 311 M (≺) = {S ⊆ P | ∀a ∈ P with a ≺ ∨S, then a ∈↓ S}. Suppose (P, ≺) is an generalized algebraic lattice, Deng defined a new prop- erty: (P, ≺) is additivity, if for a, b, c ∈ P with a ≺ b ∨ c implies a ≺ b or a ≺ c ([7]). He investigated the connection between additive generalized algebraic lat- tices and T0-topological spaces as follows. From the one direction, suppose (P, ≺) is an generalized algebraic lattice, let X = K(≺), and T : P → 2X , T (a) =↓ a ∩ K(≺). If (P, ≺) is additivity, then T satisfies: (1) T (0) = ∅, (2) T (1) = X, (3) for S ∈ M (≺) = M (K(≺ )), T (∨S) = ∪T (S), (4) for S ⊆ P, T (∧S) = ∩T (S), (5) T (a ∨ b) = T (a) ∪ T (b). Let C = T (P ), then (X, C) is a T0 co-topological space, and (P, ≺) is iso- morphic to (X, C), see [7]. From the other direction, we assume (X, C) is a co-topological space, let Q = {{x}− | x ∈ X} be the collection of closure of all singletons. Clearly Q is a ∨−base for C, i.e., a ∈ C, a is a closed subset, we have a = ∨ ↓ a. M (Q) = {S | S ⊆ X, for a ∈ Q, a ≤ ∨S we have a ∈↓ S} is a system of subsets induced by Q, then (C, ≺M(Q)) is an additive generalized algebraic lattice with K(≺M(Q)) = Q. In this case, a ≺M(Q) b for a, b ∈ C if and only if a ⊆ {x}− for some x ∈ b. It is clearly that ≺M(Q) is the specialization order ([12]) which is essentially in topological and domain theory. Suppose (P1, ≺1), (P2, ≺2) are two generalized continuous lattices, and h : P1 → P2 is said to be a lower homomorphism if it preserves arbitrary joins and the generalized way-below relations. Thus a lower homomorphism h is residuated. Let g be its upper adjoint, we have (g, h) is a Galois connection ([7]). The lower homomorphism also corresponds to the closed mapping. So he obtained the equivalence between the category of additive generalized algebraic lattices with lower homomorphisms and the category of T0-topological spaces with continuous mappings in [7]. From the point of view of Deng’s work ([7, 8]), an additive generalized algebraic lattice is algebraic abstraction of a topological space. Thus topological theory may be directly constructed on it. The work will benefit the study of the theory of topological algebra and the possible application about additive generalized algebraic lattices. In [2], we constructed Tietze extension theorem. Furthermore, (C, ≺M(Q)) is an example of additive generalized algebraic lattice. For another example in commutative ring, see [7]. For other notions and results cited in this paper, please refer to [7, 15]. 3. Stone Compactification In the section, (P, ≺) denotes an additive generalized algebraic lattice. It is T0, but not T1 ([7]). K(≺) is the set of all compact elements of (P, ≺). Definition 3.1. For a ∈ P , a∗ = ∧{x|a ∨ x = 1} 312 X. Chen, Q. Li and Z. Deng Note 1. Since (P, ≺) is a complete lattice, we have a ∨ 1 = 1 for every a ∈ P , so the existence of a∗ is obvious. Proposition 3.2. (1) ∀a ∈ P , a ∨ a∗ = 1 (2) a ≺ b ⇒ b∗ ≤ a∗ (3) a ∧ b = 0 ⇒ a ≤ b∗ Proof. (1) ∀y ∈ K(≺), if y ≺ a, then y ≺ a ∨ a∗. If y 6≺ a, then ∀x ∈ {x | a ∨ x = 1}, y ≺ y ≤ 1 = a ∨ x. Since (P, ≺) is additive, we have y ≺ x, which implies y ≺ a∗. Hence y ≺ a ∨ a∗. Furthermore (P, ≺) is algebraic, 1 = ∨(↓ 1 ∩ k(≺)), we obtain a ∨ a∗=1. (2) It is clear. (3) ∀y ≺ a, if y = 0, certainly y ≺ b∗. If y 6= 0, a ∧ b = 0, so b ∧ y = 0, which implies y 6≺ b. By (1), we have y ≺ 1 = b ∨ b∗. Since (P, ≺) is additive, so y ≺ b∗. Thus a = ∨ ⇓ a = ∨{y | y ≺ a} ≤ b∗ � Note 2. On (P, ≺), ∀a ∈ P , a ∧ a∗ = 0 is false in general. We introduce the notion of regular on (P, ≺). Definition 3.3. (P, ≺) is said to be regular, if for x ∈ K(≺), b ∈ P , x 6≺ b, then x ∧ b = 0. Note 3. Let (X, T ) be a point-set topological space, if ∀z ∈ X, a closed set A ⊆ X, z 6∈ A if and only if {z}− 6⊆ A, which equivalent to {z}− 6≺ A according to the definition of ≺. If (X, T ) is regular, then there exist U, V two open sets, such that z ∈ U, A ⊆ V and U ∩ V = ∅. We obtain {z}− ∩ A = ∅. Otherwise if {z}− ∩ A 6= ∅, there exists y ∈ {z}− ∩ A, so y ∈ A ⊆ V . By y ∈ {z}−, we have {z} ∩ V 6= ∅, thus z ∈ V , a contradiction. Definition 3.3 coincides with the above definition when (X, C) is a co-topological space. The notion of compactness is defined as follows. Definition 3.4. (P, ≺) is said to be compact if for every D ⊆ P , ∧D = 0 implies that there exists a finite subset D0 ⊆ D satisfying ∧D0 = 0. That is to say, if D has the finite intersection property, then ∧D 6= 0. We introduce the notions of a scale, completely regular on (P, ≺). Definition 3.5. A family of elements 〈cα ∈ P | α ∈ [0, 1] & α is a rational number 〉 is called a scale of (P, ≺), if it satisfies: for α < β, we have cα ≺ cβ. For a, b ∈ P , if there exists a scale 〈cα〉, such that a ≤ c0, c1 ≤ b. We denote the relation by a � b. (P, ≺) is said to be completely regular, if ∀a ∈ P, a = ∧{b | a � b}. Stone compactification of additive generalized-algebraic lattices 313 Suppose (Pα, ≺α) is a family of additive generalized algebraic lattices, α ∈ Λ (a index set), then (ΠPα, ≺Π) is the direct product, and prα : ΠPα → Pα, ∀a = (aα) ∈ ΠPα, prα(a) = aα, prα is onto upper adjoint. qα : (Pα, ≺α) → (ΠPα, ≺Π) is the lower homomorphism of prα ([9]). By the definitions of prα and qα, we know that qα preserves the generalized way-below relation, and obtain the following proposition. Proposition 3.6. Suppose (Pα, ≺α) is regular, completely regular for every α ∈ Λ, then (ΠPα, ≺Π) is also regular, completely regular. Proof. It is trivial. � Since every inclusion mapping is a lower homomorphism, it is obvious that every lower sublattice of regular, completely regular (P, ≺) is also regular, com- pletely regular. Proposition 3.7 (Tychonoff product theorem). Suppose for every α ∈ Λ, (Pα, ≺α) is compact, then (ΠPα, ≺Π) is also com- pact. Proof. It is similar to Bourbaki’s proof ([14]). (1) Let B ⊆ ΠPα be the maximal with respect to the finite intersection property ([14]) (2) prα : ΠPα → Pα is the onto upper adjoint, then for some α ∈ Λ, {prα(b) | b ∈ B} also has the finite intersection property. Since (Pα, ≺α) is compact, by Definitions 3.4, ∧{prα(b) | b ∈ B} 6= 0, so there exists c ∈ K(≺α), c 6= 0, c ≺ ∧{prα(b) | b ∈ B}. (3) qα is the lower homomorphism of prα, so by c ≺α prα(b), we obtain qα(c) ≺ b for every b ∈ B, and qα(c) 6= 0, qα(c) ∈ ΠPα. Thus ∧B 6= 0, which shows that (ΠPα, ≺Π) is compact. � Suppose I = [0, 1], the topology on I induced by ρ(x, y) = |x − y|. CI denotes the family of all closed subsets, thus (I, CI ) is a co-topology on I. According to Proposition 4.2 ([7]), let Q = {{r}− | r ∈ [0, 1]}, M (Q) gen- erated by Q. CI ordered by inclusion relation, forms a complete lattice. The generalized way-below relation ≺I induced by M (Q), and M (≺I ) = M (Q). Then (CI , ≺I ) is an additive generalized algebraic lattice. By Definitions 3.3, 3.4 and 3.5, (CI , ≺I ) is regular, completely regular and compact. Furthermore, by Propositions 3.6 and 3.7, (ΠCI , ≺Π) is also regular, completely regular and compact. By [7] Lemma 4.5, the system of subsets M (≺I ) is the collection of classes of closed subsets such that the union of any class is still closed. i.e., ∨S = ∪S for every S ∈ M (≺I ), and ∪S ∈ CI . By the property of closed sets, for D ⊆ CI , we have ∧D = ∩D ∈ CI . Lemma 3.8. For a, b ∈ P , suppose a � b, then there exists a lower homomor- phism h : (P, ≺) → (CI , ≺I ), such that a ≤ g(0) and g(I) ≤ b. 314 X. Chen, Q. Li and Z. Deng Proof. The upper adjoint g : (CI , ≺I ) → (P, ≺) is first defined. Since a � b, then there exists a scale 〈cα〉, such that a ≤ c0, c1 ≤ b and cα ≺ cβ for α < β. This implies {cα} is an increasing function of α. For [α, β] ∈ CI , g([α, β]) = eα ∧ dβ , where eα = ∨r≥αcr, dα = ∨r≤αcr. By [5] Theorem 3, we obtain eα, dα ∈ M (≺). (1) For (CI , ≺I ), the closed interval is [α, β], and the elementary closed set Fλ = n⋃ i=1 [αi, βi], the closed set F = ∩Fλ. Since for every S ∈ M (≺I ), by [7] Lemma 4.5, ∨S = ∪S. So for every S ∈ M (≺I ), we have g(S) ∈ M (≺). (2) By ∨S = ∪S, we obtain g(∨S) = g(∪S) = ∨g(S) for every S ∈ M (≺I ), (3) Since for S ⊆ CI , ∧S = ∩S, we know that g also preserves arbitrary meets, i.e., g(∧S) = ∧g(S). By the above proof, g is an upper adjoint. Thus h : (P, ≺) → (CI , ≺I ) is a lower homomorphism. g(I) = g([0, 1]) = e0 ∧ d1 ≤ b g(0) = g({0}) = e0 ∧ d0 ≥ a. � Proposition 3.9 (Tychonoff embedding theorem). Suppose (P, ≺) is an additive generalized algebraic lattice, then (P, ≺) is regular, completely regular iff (P, ≺) is isomorphic to a lower sublattice of (ΠCI , ≺Π). Proof. By Proposition 3.6, (ΠCI , ≺Π) is regular, completely regular, and every lower sublattice of (ΠCI , ≺Π) is also regular, completely regular, so the proof is trivial On the other hand, suppose (P, ≺) is an additive generalized algebraic lat- tice, let S = {(gs, hs) | gs : (CI , ≺I ) → (P, ≺) is an upper adjoint, hs : (P, ≺) → (CI , ≺I ) is a lower homomorphism of gs}, S 6= ∅ Taking: H: (P, ≺) → (ΠCI , ≺Π) the direct product of (CI , ≺I ) by index set of S, ∀a ∈ P , H(a) = Πhs(a). By the property of {hs}, H is also a lower homomorphism, so G : (ΠCI , ≺Π) → (P, ≺) is the upper adjoint of H. We show (P, ≺) is isomorphic to a lower sublattice of (ΠCI , ≺Π), it suffices to prove H is one-to-one on K(≺). ∀x, y ∈ K(≺), x 6= y, then we may assume x 6≺ y. Since (P, ≺) is regular, so x ∧ y = 0, which follows that H(x) 6= H(y). Thus (P, ≺) is isomorphic a lower sublattice of (ΠCI , ≺Π), which generated by H(K(≺)), and H(K(≺)) ⊆ K(≺Π). � Proposition 3.10 (Stone compactification). Suppose (P, ≺) is regular, completely regular, then there exists a regular, completely regular compact additive generalized algebraic lattice (βP, ≺β ), such that (P, ≺) is isomorphic to a dense lower sublattice of (βP, ≺β ). Stone compactification of additive generalized-algebraic lattices 315 Proof. By Proposition 3.9, (P, ≺) is isomorphic to a lower sublattice of (ΠCI , ≺Π). Let (βP, ≺β ) be the closure of the lower sublattice, and the compactness of (βP, ≺β ) follows from Proposition 3.7. � In general, (βP, ≺β ) is said to be a Stone compactification of (P, ≺). Note 4. Clearly, if the generalized way-below relation ≺ satisfies the interpo- lation property, then (P, ≺) is completely regular by The Choice Axiom. As the end of this paper, we embark on an alternative description of (βP, ≺β ) by means of ideals of (P, ≺). Definition 3.11. I ⊆ P is said to be an ideal if (1) for any finite E ⊆ I, ∨E ∈ I, (2) z ∈ I, x ≤ z implies x ∈ I. Idl(P ) denotes all ideals of P , and certainly Idl(P ) is a complete lattice, the order is the inclusion order. ∀I ∈ Idl(P ), ↓ I = {J | J ≤ I}, where J ≤ I iff J ⊆ I Definition 3.12. For I, J ∈ Idl(P ), a binary relation on Idl(P ) is defined as: I ≺∗ J if and only if ∨I ≺ ∨J holds on (P, ≺). Lemma 3.13 ([15]). ≺∗ is a generalized way-below relation on Idl(P ). Proof. (1) I ≺∗ J if and only if ∨I ≺ ∨J holds on (P, ≺). Then ∀a ∈ I, a ≤ ∨I ≺ ∨J, so a ∈ J. that is, I ⊆ J, thus I ≤ J. (2) I1 ≤ I2 ≺ ∗ I3 ≤ I4, which implies that ∨I1 ≤ ∨I2 ≺ ∨I3 ≤ ∨I4 holds on (P, ≺). So we have ∨I1 ≺ ∨I4, thus I1 ≺ ∗ I4. � Lemma 3.14. Idl(P ) is algebraic. Proof. For I, J ∈ Idl(P ), I ≺∗ J implies ∨I ≺ ∨J on (P, ≺). Since (P, ≺) is algebraic, there exists c ∈ K(≺), such that ∨I ≤ c ≤ ∨J. Furthermore ↓ c ∈ Idl(P ). By c ∈ K(≺), so c ≺ c on (P, ≺), hence ↓ c ≺∗↓ c on Idl(P ). i.e, ↓ c ∈ K(≺∗) by Definition 3.11. Considering I ≤↓ c ≤ J and ↓ c ∈ K(≺∗), thus (Idl(P ), ≺∗) is algebraic. � Lemma 3.15. Idl(P ) is continuous. Proof. It is trivial ([4]). � Lemma 3.16. Idl(P ) is additive. Proof. For I ≺∗ J1 ∨ J2, where I, J1, J2 ∈ Idl(P ), then on (P, ≺), ∨I ≺ ∨(J1 ∨ J2) = (∨J1) ∨ (∨J2) holds. Since (P, ≺) is additive, it follows that ∨I ≺ ∨J1 or ∨I ≺ ∨J2, thus I ≺ ∗ J1 or I ≺ ∗ J2, which proves Lemma 3.16. � Proposition 3.17. (Idl(P ), ≺∗) is an additive generalized algebraic lattice. Proof. By Lemmas 3.14, 3.15, 3.16. � Lemma 3.18. For any regular (P, ≺), Idl(P ) is also regular. 316 X. Chen, Q. Li and Z. Deng Proof. It is obvious that on (Idl(P ), ≺∗), K(≺∗) = {↓ x | x ∈ K(≺)}. Then ∀ ↓ x ∈ K(≺∗), ∀J ∈ Idl(P ), if ↓ x 6≺∗ J, which implies x 6≺ ∨J by Definition 3.11. Since (P, ≺) is regular, x ∈ K(≺), ∨J ∈ P , x 6≺ ∨J, then x ∧ (∨J) = 0. So we obtain that ↓ x ∧ J = 0. It follows that Idl(P ) is regular. � For I ∈ Idl(P ), I is called completely regular, if ∀a ∈ I, there exists b ∈ I, such that a � b. Let R(P ) = {I is completely regular in Idl(P )}, then we have Lemma 3.19. Suppose (P, ≺) is completely regular, then (R(P ), ≺∗) is also completely regular. Proof. It is trivial. � Lemma 3.20. Suppose (P, ≺) is compact, then (R(P ), ≺∗) is also compact. Proof. For a family {Iα|α ∈ Λ} satisfying ∧Iα = 0. Since (P, ≺) is a complete lattice, Iα = ∨{↓ x | x ∈ Iα}, we may assume Iα =↓ aα. Then ∧Iα = ∧(↓ aα) =↓ (∧aα), thus ↓ (∧aα) = 0, it follows that ∧aα = 0. Furthermore (P, ≺) is compact, by Definition 3.4, there exist a1, a2, · · · , am satisfying m∧ i=1 ai = 0. By this, it is easy to prove m∧ i=1 (↓ ai) = 0. that is, m∧ i=1 Ii = 0. Thus (R(P ), ≺ ∗) is compact. � Proposition 3.21. Suppose (P, ≺) is compact, regular and completely regular, then (P, ≺) and R(P ) are isomorphic. Proof. By Lemmas 3.18, 3.19, 3.20, and h : P → R(P ), ∀a ∈ P , h(a) =⇓ a = {b | b ≺ a}, certainly h(a) ∈ R(P ). a ≺ b holds on (P, ≺) if and only if h(a) ≺∗ h(b) holds on (R(P ), ≺∗). Since (P, ≺) is continuous, ∀a ∈ P , a = ∨ ⇓ a, so (P, ≺) is embedded into R(P ), and h preserves the generalized way-below relation. It is trivial to prove h is one-to-one. � By Proposition 3.10, suppose (P, ≺) is compact, regular, completely regular, then (βP, ≺β ) and (R(P ), ≺ ∗) are also isomorphic By Propositions 3.10 and 3.21, the following theorem is also obtained. Theorem 3.22. An additive generalized algebraic lattice (P, ≺) has a Stone compactification iff it is regular, completely regular. Note 5. (1) According to [3], the class of generalized continuous lattices includes completely distributive lattices and traditional continuous lattices ([15]) as its special cases. (2) According to [4], the traditional algebraic lattice is generalized algebraic lattice ([4]), and completely distributive lattice is also generalized alge- braic lattice ([4]). Stone compactification of additive generalized-algebraic lattices 317 Acknowledgements We are grateful to the editor for his valuable comments and suggestions. References [1] H. J. Bandelt, M-distributive lattices, Arch Math 39 (1982), 436–444. [2] X. Chen, Q. Li, F. Long and Z. Deng, Tietze Extension Theorem on Additive Generalized Algebraic Lattice, Acta. Mathematica Scientia (A)(in Chinese), accepted. [3] Z. Deng, Generalized-continuous lattices I, J. Hunan Univ. 23, No. 3 (1996), 1–3. [4] Z. Deng, Generalized-continuous lattices II, J. Hunan Univ. 23, No. 5 (1996), 1–3. [5] Z. Deng, Homomorphisms of generalized-continuous lattices, J. Hunan Univ. 26, No. 3 (1999), 1–4. [6] Z. Deng, Direct sums and sublattices of generalized-continuous lattices, J. Hunan Univ. 28, No. 1 (2001), 1–4. [7] Z. Deng, Topological representation for generalized-algebraic lattices, (in W.Charles. 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Papert, Which distributive lattices are lattices of closed sets?, Proc. Cambridge. Phil. Soc. 55 (1959), 172–176. [17] Q. X. Xu, Construction of homomorphisms of M-continuous lattices, Trans. Amer. Math. Soc. 347 (1995), 3167–3175. Received December 2005 Accepted May 2006 Xueyou Chen (chenxueyou0@yahoo.com.cn) College of Mathematics and Information Science, Shandong University of Tech- nology, Zibo,shandong 255012, P. R. CHINA. Qingguo Li College of Mathematics and Economics, Hunan University, Changsha, Hunan 410012, P. R.CHINA. Zike Deng College of Mathematics and Economics, Hunan University, Changsha, Hunan 410012, P. R.CHINA.