- Email: [email protected]
Lattices of Irreducibly-derived Closed Sets
Available online at www.sciencedirect.com
Electronic Notes in Theoretical Computer Science 345 (2019) 233–247 www.elsevier.com/locate/entcs
Lattices of Irreducibly-derived Closed Sets Shuhua Su1,2 Qi Li3 Fuzhou Teachers’ College, East China University of Technology, Fuzhou, China
Abstract This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set ΓSI (X) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of CSI -continuous posets and CSI -prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y , X ≅ Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed. Keywords: Irreducible sets; Strong complete posets; CSI -continuity; CSI -prealgebra; Cartesian-closed
1
Introduction
Domain theory or the more general theory of posets embodies the combination of an order and a topology. One of the basic and important results is that a poset is continuous if and only if the lattice of the Scott-closed sets above it is a completely distributive lattice. However, for a non-continuous poset, little is known about the order-theoretic properties of the Scott-closed set lattice. To study the Scott-closed set lattice on a general non-continuous poset, Ho and Zhao introduced the concept of a C-continuous poset in [4], and proved that the Scott-closed set lattice on a poset is a C-prealgebraic lattice, especially a C-continuous lattice. With the help of the C-continuity, they obtained some good results, such as: (i) a complete lattice L is isomorphic to the Scott-closed set lattice for a complete semilattice P if and only if L is weak-stably C-algebraic; (ii) for any two complete semilattices X and 1
This work is supported by the National Natural Science Foundation of China (NO. 11861006 and NO. 11561002) 2 Email: [email protected] 3 Email: [email protected]
https://doi.org/10.1016/j.entcs.2019.07.026 1571-0661/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
234
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Y , X ≅ Y if and only if the Scott-closed set lattices above them are isomorphic. In an invited presentation at the 6th International Symposium in Domain Theory, J. D. Lawson emphasized the need to develop the core of domain theory in the wider context of T0 spaces instead of restricting to posets. Just as directed sets are to domains, irreducible sets play an important role in T0 spaces. A nonempty subset F of a T0 space X is irreducible if for any closed two sets A and B with F ⊆ A ⋃ B one has either F ⊆ A or F ⊆ B. Zhao and Ho [9] developed the core of domain theory in the context of T0 spaces by choosing the irreducible sets. They studied the irreducibly-derived topology constructed from any given topology on a set. Moreover, they obtained an interesting result that Scott topology of a poset is exactly the irreducibly-derived topology of its Alexandroff topology. Then Zhao and Xu established some classes of SCL-faithful dcpos and identified some classes of Cσ -unique dcpos based on Scott-closed set lattices.[10,11] Inspired by the works mentioned above, this paper is devoted to investigate the closed set lattices arising from the irreducibly-derived topology of Scott topology. This paper is arranged as follows. In Section 2, we recall some basic materials which will be used throughout this paper, and by the definition of a irreduciblyderived topology, we obtain the irreducibly-derived topology of Scott topology. In Section 3, we define the CSI -continuity on posets by SI-closed sets, and then study its properties. In Section 4, we define SI-dominated posets and study the lattice of SI-closed sets on the SI-dominated posets. We obtain a conclusion similar to the literature [4], that is, for any two SI-dominated posets X and Y , X ≅ Y if and only if the SI-closed set lattices above them are isomorphic. In Section 5, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.
2
Preliminaries
Given a topological space (X, τ ), a nonempty subset F of X is called an irreducible set if for closed sets A, B ⊆ X with F ⊆ A ∪ B one has either F ⊆ A or F ⊆ B. The set of all irreducible sets of X will be denoted by Irrτ (X). On irreducible sets, here are some elementary properties: Proposition 2.1 (Gierz, et al. [2]) For any topological space (X, τ ), it holds that: (1) Every singleton is irreducible; (2) A set is in Irrτ (X) if and only if its closure is in Irrτ (X); (3) The continuous image of an irreducible set is again irreducible; (4) Every directed set of X is irreducible. A subset U of a poset X is called Scott-open if (i) U = ↑U , and (ii) for any directed set D in X, ⋁ D ∈ U implies U ⋂ D ≠ ∅ whenever ⋁ D exists. The set of all Scott-open sets of X forms a topology (called Scott topology) on X, denoted by σ(X). A complement of a Scott-open set is called a Scott-closed set. We use Γ(X) to denote the set of all Scott-closed sets of X. Thus a subset C ⊆ X is Scott-closed if and only if (i) C = ↓C, and (ii) for any directed set D ⊆ C, if ⋁ D exists then
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
235
⋁ D ∈ C. σ(X) and Γ(X) are both complete and distributive lattices with respect to the inclusion relation. A subset A of a poset X is called convex if for any x < y < z, x, z ∈ A implies y ∈ A. Let X be a poset. The way-below relation ≪ on X is defined by x ≪ y for x, y ∈ X if for any directed set D with ⋁ D existing, b ≤ ⋁ D implies a ≤ d for some d ∈ D. A poset X is said to be continuous if for any x ∈ X, the set ↡x = {y ∈ X ∣ y ≪ x} is directed and x = ⋁ ↡x. According the definition of the irreducibly-derived topology in [9], we can obtain the irreducibly-derived topology from Scott topology on a poset, as described next. Definition 2.2 Let X be a poset. A subset C of X is called SI-closed if the following conditions are satisfied: (1) C = ↓C; (2) For any F ∈ Irrτ (X), F ⊆ C implies ⋁ F ∈ C whenever ⋁ F exists. The set of all SI-closed sets of X is denoted by ΓSI (X). A complement of a SI-closed set is called a SI-open set and the set of all SI-open sets of (X, τ ) is denoted by σSI (X). By Theorem 3.2 in [9], σSI (X) is a topology on X and is called a irreducibly-derived topology of σ, a SI-topology for short. Obviously, for any poset X, ΓSI (X) ⊆ Γ(X). Definition 2.3 (Ho, et al. [3]) A poset X is strongly complete if every irreducible set of X has a supremum. In this case we also say that X is a scpo. Obviously, every scpo is a dcpo. Definition 2.4 The SI-topology on a poset X is called lower hereditary if for every SI-closed set C, the SI-topology on C agrees with the relative SI-topology from X. Proposition 2.5 (Zhao and Ho [9]) A function f ∶ X → Y between posets X and Y is SI-continuous iff it is order-preserving and f (⋁ F ) = ⋁ f (F ) for any F ∈ Irrσ (X) with ⋁ F existing. Lemma 2.6 (Xu and Mao [6]) Let X be a poset and A ⊆ X. If s, t are two upper bounds of A with s ≤ t and A has a supremum in ↓t, denoted ⋁t A, then A has a supremum in ↓s, in this case ⋁s A = ⋁t A. Lemma 2.7 Let X be a poset. The following statements are equivalent: (1) The SI-topology on X is lower hereditary; (2) For any x ∈ X, the inclusion map from ↓x into X is SI-continuous; (3) Any minimal upper bound of any irreducible set of X is the supremum for that irreducible set; (4) For any x ∈ X and any irreducible set F , x = ⋁x F implies x = ⋁ F . Proof. (1)⇒(2): Follows from that any principal ideal is a SI-closed set.
236
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
(2)⇒(3): Let F be an irreducible set with a minimal upper bound b. Then b is the supremum of F in ↓b. Since the inclusion map from ↓b into X is continuous, it follows from Proposition 2.5 that b is the supremum of F in X . (3)⇒(1): Let E be a SI-closed set. A subset B that is SI-closed in E is easily verified to be SI-closed in X. Conversely suppose that A is SI-closed in X . Then A ∩ E ∈ Γ(E). To show that A ∩ E is SI-closed in E, let F be an irreducible set in A ∩ E that has the supremum b in E. Then ⋁E F = b ∈ E and b = ⋁ F by (3). It follows from the SI-closedness of A that b ∈ A. Hence b ∈ A ∩ E. This shows that A ∩ E is closed in the SI-topology of E. (3)⇒(4): Suppose that F is an irreducible set with x = ⋁x F . Then x is a minimal upper bound of F . By (3), we have ⋁ F = x. (4)⇒(3): Let y be a minimal upper bound of an irreducible set F . Then y = ⋁y F . By (4), we have y = ⋁ F . ◻ Applying Lemmas 2.7, we obtain the following corollary. Corollary 2.8 Every scpo has a lower hereditary SI-topology. Lemma 2.9 (Mao and Xu [5]) Let X be a poset, x ∈ X and A ⊆ ↓x = C. Then ⋁C A = ⋁ A whenever ⋁ A exists; If (X, ≤) is a semilattice, then ⋁ A = ⋁C A whenever ⋁C A exists, where ⋁C A denotes the supremum of A in the principal ideal ↓x = C. Applying Lemmas 2.7 and Lemma 2.9, we obtain the following corollary. Corollary 2.10 Every semilattice has a lower hereditary SI-topology. Proposition 2.11 Let X and Y be posets with lower hereditary SI-topologies. Then (1) Every convex subset A of X in the inherited order has also a lower hereditary SI-topology. In particular, every SI-closed set of X has a lower hereditary SItopology; (2) The product X × Y has also a lower hereditary SI-topology. Proof. (1) Let F ⊆ A be an irreducible set with a minimal upper bound z ∈ A. Then z is also a minimal upper bound of F in X because of the convexity of A and hence the supremum of F in X by Lemma 2.7. (2) Let F ⊆ X × Y be an irreducible set with a minimal upper bound z = (u, v). Then it is easy to see that u is a minimal upper bound of p1 (F ) and v is a minimal upper bound of p2 (F ), where p1 ∶ X × Y → X and p2 ∶ X × Y → Y are the projection maps. Since X and Y have the lower hereditary SI-topologies, we have ⋁ p1 (F ) = u and ⋁ p2 (F ) = v. Thus ⋁ F = (⋁ p1 (F ), ⋁ p2 (F )) = (u, v) = z. Therefore, X × Y has the lower hereditary SI-topology by Lemma 2.7. ◻
3
CSI -continuous posets
In this section, we define a new auxiliary relation on a poset X by SI-closed sets. Then we introduce the notions of CSI -continuous and CSI -algebraic posets, which
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
237
is important for us to study the properties of SI-closed set lattices. Definition 3.1 Let X be a poset and x, y ∈ X. We say that x is beneath y, denoted by x≺SI y, if for every nonempty SI-closed set C ⊆ X with ⋁ C existing, the relation ⋁ C ≥ y always implies that x ∈ C. We denote the set {x ∈ X ∶ x≺SI y} by ⇓SI y and the set {x ∈ X ∶ y≺SI x} by ⇑SI y. Remark 3.2 Obviously, for any poset (X, ≤), ≺SI and ≺ defined in [4] are quite different. As ΓSI (X) ⊆ Γ(X), x ≺ y implies that x ≺SI y. However, the converse is not true. If (X, ≤) is k-bounded sober defined in [8], then ≺SI =≺. Example 1. Let X = (N × (N ⋃{∞})) ⋃{⊺} and define x ≤ y if one of the following conditions holds: (1) y = ⊺; (2) x = (m1 , n1 ) and y = (m2 , n2 ), where m1 = m2 , n1 ≤ n2 (≤ ∞) or n2 = ∞, n1 ≤ m2 . Then (X, ≤) is a poset. By [7], we know that C = N × (N ⋃{∞}) is an irreducible Scott closed set and ⋁ C = ⊺. Hence it is easily checked that ⊺ ⊀ ⊺ and ⊺ ≺SI ⊺. Example 2. Let X = N ×(N ⋃{∞}) and define (m, n) ≤ (m0 , n0 ) if m = m0 , n ≤ n0 , or n0 = ∞ and n ≤ m0 . As was pointed out in [7], the Scott space (X, σ(X)) is k-bounded sober. Hence ≺SI =≺. Remark 3.3 Let X be a poset. Then (1) x≺SI y ⇒ x ≤ y; (2) x ≤ y≺SI z ≤ w implies x≺SI w; (3) For all x ∈ X, 0≺SI x whenever X is a pointed poset. Proposition 3.4 For a poset X, the following conditions are equivalent: (1) x ≺SI y; (2) x ∈ C for every SI-closed set C with y ≤ ⋁ C whenever ⋁ C exists; (3) x ∈ ⋂ J(y), where J(y) = {C ∈ ΓSI (X) ∶ y ≤ ⋁ C}. Corollary 3.5 Let X be a poset and x ∈ X, then ⇓SI x is a SI-closed set of X. Proof. By Proposition 3.4, we have ⇓SI x = ⋂ J(x), and thus ⇓SI x is a SI-closed set of X. ◻ Definition 3.6 A poset X is said to be CSI -continuous if it satisfies x = ⋁ ⇓SI x for all x ∈ X. A CSI -continuous poset which is also a complete lattice is called a CSI continuous lattice. Obviously, we have ⇓x ⊆ ⇓SI x ⊆ ↓x, ∀x ∈ X by Remark 3.2, and thus every C-continuous poset is also CSI -continuous. Proposition 3.7 For a complete lattice X, the following are equivalent: (1) (X, τ ) is CSI -continuous; (2) For each x ∈ X, the set ⇓SI x is the smallest SI-closed set C with x ≤ ⋁ C; (3) For each x ∈ X, there is a smallest nonempty SI-closed set C such that x ≤ ⋁ C; (4) The ⋁ map r = (C ↦ ⋁ C) ∶ ΓSI (X) → X has a lower adjoint; (5) The ⋁ map r ∶ ΓSI (X) → X preserves all existing infs; (6) For any collection {Ci ∶ i ∈ I} of SI-closed sets of X, the following equation holds: ⋀ ⋁ Ci = ⋁ ⋂ Ci . i∈I
i∈I
238
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. (1) ⇒ (2): Condition (1) holds if and only if for each x ∈ X, ⇓SI x ∈ J(x) by Definition 3.6. Thus (2) follows. Condition (2) trivially implies (3). (3) ⇒ (1): If J(x) has a smallest element M , then M ⊆ C for all C ∈ J(x) and thus M ⊆ ⋂ J(x) ⊆ M . Hence M = ⋂ J(x) = ⇓SI x. Thus (1),(2) and (3) are equivalent. (3)⇔(4): By the definition of Galois connection, the map r has a lower adjoint if and only if minr−1 (↑x) exists for all x. But minr−1 (↑x) is precisely the smallest element of J(x). (4) ⇒ (5): The ⋁ map preserves infs by Theorem O-3.3 in [2]. (5) ⇒ (4): Obviously, for any x ∈ X, r−1 (↑x) = J(x) ≠ ∅, which implies that the ⋁ map is cofinal. Thus the ⋁ map has a lower adjoint by O-3.4 in [2]. (5) ⇔ (6): Obviously. ◻ Proposition 3.8 A poset X is CSI -continuous if and only if X⊥ is CSI -continuous, where X⊥ is the poset obtained from X by adjoining a new bottom element. Proof. Suppose that X is CSI -continuous and x ∈ X⊥ . If x ∈ X, then there is a smallest nonempty SI-closed set Cx of X with ⋁ Cx ≥ x by Proposition 3.7. Thus Cx′ = Cx ⋃{⊥} is the smallest nonempty SI-closed set of X⊥ with ⋁ Cx′ ≥ x. If x =⊥, then by Remark 3.3, C⊥ = {⊥} is the smallest nonempty SI-closed set of X⊥ with ⋁ C⊥ ≥⊥. Thus X⊥ is CSI -continuous by Proposition 3.7. Conversely, suppose that X⊥ is CSI -continuous and x ∈ X. By Proposition 3.7, there is a smallest nonempty SI-closed subset Cx′ of X⊥ with ⋁ Cx′ ≥ x. Then Cx = Cx′ /{⊥} is the smallest nonempty SI-closed set Cx of X with ⋁ Cx ≥ x. Thus X is CSI -continuous by Proposition 3.7 again. ◻ Proposition 3.9 Let X and Y be posets and (X, σ(X)) ≅ (Y, σ(Y )). If X is CSI -continuous, then Y is also CSI -continuous. Proof. Obviously.
◻
Proposition 3.10 Let X be a CSI -continuous semilattice. Then every principal ideal of X is CSI -continuous. Proof. Suppose that X is a CSI -continuous semilattice. Then we claim that for C all x ∈ X and u ∈ ↓x = C, ⇓SI u ⊆ ⇓C SI u holds, where ⇓SI u denotes the subset ⇓SI u in C. In fact, for all v ∈ ⇓SI u and all A ∈ ΓSI (C) with ⋁C A ≥ u, we have A ∈ ΓSI (X) by Definition 2.2 and Corollary 2.10. It follows from Lemma 2.9 that x ≥ ⋁C A = ⋁ A ≥ u. Since v ∈ ⇓SI u, there is a v ∈ A. This shows that v ∈ ⇓C SI u. By the CSI -continuity of X and Lemma 2.9, u = ⋁ ⇓SI u = ⋁C ⇓SI u. Clearly, u is an upper bound of ⇓C SI u in the principal ideal C = ↓x. Suppose that t is any upper u in C. Then t is an upper bound of ⇓SI u in C. Thus u = ⋁ ⇓SI u ≤ t. bound of ⇓C SI This shows that ⋁C ⇓C SI u = u. Thus for all x ∈ X, the principal ideal C = ↓x is ◻ CSI -continuous. Proposition 3.11 Let X be a semilattice. If every principal ideal of X is CSI continuous, then X is CSI -continuous.
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
239
Proof. Suppose that for all x ∈ X, the principal ideal C = ↓x is CSI -continuous. C We claim that ⇓C SI x ⊆ ⇓SI x. In fact, for all y ∈ ⇓SI x and all A ∈ ΓSI (X) with ⋁ A = z ≥ x, the principal ideal B = ↓z is CSI -continuous by the hypothesis. By C Corollary 3.5 and 2.10, ⇓C SI x ∈ ΓSI (B) and thus ⇓SI x ∈ ΓSI (C). By Lemma 2.9, B B C we have ⋁B ⇓SI x = ⋁C ⇓SI x = x. Since y ∈ ⇓SI x, there is a u ∈ ⇓B SI x such that y ≤ u. Since A ∈ ΓSI (X) with existing ⋁ A = z ≥ x, we have A ∈ ΓSI (B) and B ⋁B A = ⋁ A = z ≥ x by Corollary 3.5 and 2.10 again. It follows from u ∈ ⇓SI x that C there is a a ∈ A such that a ≥ u ≥ y. This shows that y ∈ ⇓SI x and thus ⇓SI x ⊆ ⇓SI x. B By the CSI -continuity of C = ↓x and Lemma 2.9, we have x = ⋁B ⇓B SI x = ⋁ ⇓SI x. B Clearly x is an upper bound of ⇓SI x in X. Suppose that t is any upper bound of B ⇓SI x in X. Then t is an upper bound of ⇓B SI x in X. Thus x = ⋁ ⇓SI x ≤ t. This ◻ shows that ⋁ ⇓B SI x = x and hence X is CSI -continuous. By Propositions 3.10 and 3.11, we immediately have the following characterization of CSI -continuous posets by principal ideals. Theorem 3.12 Let X be a semilattice. Then X is CSI -continuous if and only if every principal ideal of X is CSI -continuous. Proposition 3.13 Let X be a CSI -continuous lattice. The for any collection {Fi ∶ i ∈ I} of finite subsets of X the following equation holds: ⋀ ⋁ Fi =
i∈I
⋁
⋂ f (i).
f ∈∏i∈I Fi i∈I
In particular, every CSI -continuous lattice space is distributive. Proof. Denote the left hand side (respectively, the right hand side) of the equation by a (respectively, b). It suffices to prove that a ≤ b. Let a = ⋀i∈I ⋁ Fi and x ≺SI a. For each i ∈ I, the set ↓Fi = ⋃y∈Fi ↓y is a SI-closed set by ↓y ∈ ΓSI (X), and x ≺SI a ≤ ⋁ Fi . Thus there is a di ∈ Fi with x ≤ di . Let f ∈ ∏i∈I Fi be defined by f (i) = di , i ∈ I. Then x ≤ ∏i∈I f (i) ≤ b. Therefore a = ⋁ ⇓SI a by the CSI -continuity of X and thus a ≤ b. ◻ Theorem 3.14 Let X be a complete lattice. Then the following are equivalent: (1) X is CSI -continuous and continuous; (2) X is completely distributive. Proof. It suffices to show that (1) ⇒ (2). Assume that X is both CSI -continuous and continuous. Since X is continuous, for each a ∈ X, a = ⋁ ↡a. Now for each x ≪ a, x = ⋁ ⇓SI x. It follows that a = ⋁{y ∈ X ∶ ∃x. y ≺SI x ≪ a}. Next suppose y ≺SI x ≪ a, we shall show that y ⊲ a. Let A ⊆ X with a ≤ ⋁ A. Construct the se E = {⋁ B ∶ B is a f inite subset of A}. Then E is directed and thus is irreducible and ⋁ E = ⋁ A ≥ a. Since x ≪ a, there is a finite subset B ⊆ A such that x ≤ ⋁ B = ⋁ ↓B. Note that the last set ↓B is SI-closed. So it follows from y ≺SI x that y ≤ d for some d ∈ B ⊆ A. This implies that y ⊲ a. Hence X is completely distributive. ◻ Lemma 3.15 For any poset X, if C ∈ Γ(ΓSI (X)), then ⋃ C ∈ Γ(X).
240
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. Let C ∗ = {C ∈ Γ(X) ∣ ∃V ∈ C s.t. C ⊆ V }, then C ⊆ C ∗ and ⋃ C = ⋃ C ∗ . Therefore, we only need to show C ∗ ∈ Γ(Γ(X)) by Proposition 4.1 in [4]. Obviously, C ∗ is a lower set of Γ(X). For any directed set D ⊆ Γ(X) with D ⊆ C ∗ , D∗ = {clσSI (D) ∣ D ∈ D} is a directed set of ΓSI (X). Moreover, for any D ∈ D, there is a V ∈ C such that D ⊆ V , thus clσSI (D) ⊆ V . Since C is a lower set, clσSI (D) ∈ C. Therefore, D ∗ ⊆ C, thus we have ⋁ΓSI (X) D∗ ∈ C. As ⋁Γ(X) D ⊆ ⋁Γ(X) D∗ ⊆ ◻ ⋁ΓSI (X) D∗ , ⋁Γ(X) D ∈ C ∗ . To sum up, C ∗ ∈ Γ(Γ(X)). Lemma 3.16 For any poset X, if F ∈ Irrσ (X), then F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Proof. Let A, B ∈ Γ(ΓSI (X)) such that F ⊆ A ⋃ B, then for any d ∈ F , ↓d ∈ A or ↓d ∈ B and thus F ⊆ (⋃ A) ⋃(⋃ B). By Lemma 3.15, ⋃ A, ⋃ B ∈ Γ(X), and so F ⊆ (⋃ A) or F ⊆ (⋃ B). Without loss of generality, we assume that F ⊆ ⋃ A and thus for any d ∈ F , there is a A ∈ A such that d ∈ A. Because A is a lower set, ↓d ⊆ A, ◻ and thus ↓d ∈ A. Therefore, F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Proposition 3.17 Let X be a poset and C ∈ ΓSI (ΓSI (X)). Then ⋁ΓSI (X) C = ⋃ C. Proof. Note that member of C is a SI-closed set of ΓSI (X). In order to prove the equation, it suffices to show that ⋃ C ∈ ΓSI (X). Obviously ⋃ C ∈ Γ(X) by Lemma 3.15. Now let F be any irreducible set of X contained in ⋃ C such that ⋁ F exists in X. We want to prove that ⋁ F ∈ C for some C ∈ C. By Lemma 3.16, we have that F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Moreover, F ⊆ C because C is a lower set in ΓSI (ΓSI (X)). Since C is a SI-closed set of ΓSI (X), ⋁ΓSI (X) F ∈ C. ◻ Note that ⋁ΓSI (X) F is precisely ↓ ⋁ F . Hence ⋁ F ∈ C for some C ∈ C. Definition 3.18 An element x of a poset X is called CSI -compact if x ≺SI x. We use κSI (X) to denote the set of all the CSI -compact elements of X. Obviously, every C-compact element defined in [4] is CSI -compact, thus κ(X) ⊆ κSI (X). Proposition 3.19 Let X be a lattice. (1) If r ∈ κSI (X), then r is co-prime; (2) If X is a completely distributive lattice, then every co-prime element is CSI compact; (3) If X is a complete lattice and κSI (X) ≠ ∅, then κSI (X) is a pointed scpo with respect to the order inherited from X. Proof. (1) Suppose r ∈ κSI (X) and r ≤ x ∨ y. Let C = ↓{x, y}. Then C is SI-closed and ⋁ C = x ∨ y. Hence r ∈ C which implies r ≤ x or r ≤ y, thus r is co-prime. (2) Let X be a completely distributive lattice and r ∈ X be co-prime. Note that the set β(r) = {x ∈ X ∶ x ⊲ r} is a directed set and thus β(r) is an irreducible set and ⋁ β(r) = r. Since x ⊲ r implies x ≺SI r, it follows from Corollary 3.5 that ⋁ β(r) ≺SI r, i.e., r ≺SI r. Hence r is CSI -compact. (3) Let F be an irreducible set in κSI (X). It suffices to show that ⋁ F ≺SI ⋁ F . So let C ∈ ΓSI (X) with ⋁ F ≤ ⋁ C. Thus d ≤ ⋁ C for all d ∈ F . Since F ⊆ κSI (X),
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
241
it follows that d ≺SI d for all d ∈ F and so F ⊆ C. Because C ∈ ΓSI (X), this implies that ⋁ F ∈ C and so ⋁ F ≺SI ⋁ F , i.e., ⋁ F ∈ κSI (X). Also 0 ≺SI 0 implies that 0 ∈ κSI (X). Hence κSI (X) is a pointed scpo with respect to the order inherited from X. ◻ Proposition 3.20 Let X be a poset and C be a nonempty SI-closed set of X. Then for each x ∈ C, ↓x ≺SI C holds in ΓSI (X). Proof. Let x ∈ C. Suppose C ∈ ΓSI (ΓSI (X)) with C ⊆ ⋁ΓSI (X) C. Then C ⊆ ⋃ C by Proposition 3.17. Hence there is a A ∈ C such that x ∈ A. This means ↓x ⊆ A, and thus ↓x ∈ C. Corollary 3.21 Let X be a poset. κ(ΓSI (X)).
Then for each x ∈ X, it holds that ↓x ∈
Definition 3.22 A poset X is said to be CSI -prealgebraic if for each a ∈ X, a = ⋁(↓a ⋂ κSI (X)). A CSI -prealgebraic poset X is CSI -algebraic if for any a ∈ X, ↓(↓a ⋂ κSI (X)) ∈ ΓSI (X). Remark 3.23 (1) Obviously every CSI -prealgebraic poset is CSI -continuous, since ↓a ⋂ κSI (X) ⊆ ⇓SI a ⊆ ↓a and a = ⋁(↓a ⋂ κSI (X)) = ⋁ ↓a. In this case, we call a CSI (pre)algebraic poset which is also a complete lattice a CSI -(pre)algebraic lattice. (2) Every CSI -prealgebraic poset is C-prealgebraic. Proposition 3.24 Let X be a poset and F ∈ Irrσ (X) with ⋁ F existing. If x≺SI y for all x ∈ F , then ⋁ F ≺SI y. Proof. Let C ∈ ΓSI (X) be nonempty such that ⋁ C exists with y ≤ ⋁ C. Since x≺SI y for all x ∈ F , it follows that F ⊆ C. Because C is SI-closed and F is irreducible, we have ⋁ F ∈ C. Thus ⋁ F ≺SI y. ◻ Proposition 3.25 In a poset X, the following statements are equivalent: (1) X is CSI -algebraic; (2) X is CSI -continuous, and x ≺SI y if and only if there is a k ∈ κSI (X) with x ≤ k ≤ y. Proof. (1) ⇒ (2): Assume (1) and x, y ∈ X with x ≺SI y. Since y = ⋁ C with the SI-closed set C = ↓(↓y ⋂ κSI (X)) by (1), there is a k ∈ ↓y ⋂ κSI (X) with x ≤ k, and thus x ≤ k ≤ y. Conversely, let x ≤ k ≺SI k ≤ y, then we have x ≺SI y. The CSI -continuity of X follows directly from Remark 3.23. (2) ⇒ (1): Assume (2) and let y ∈ X. Then y = ⇓SI y and ⇓SI y is SI-closed. For every x ≺SI y, there is a CSI -compact element k such that x ≤ k ≤ y by (2), so we conclude that y = ⋁(↓y ⋂ κSI (X)). Further, ↓(↓y ⋂ κSI (X)) ∈ ΓSI (X). In fact, we only show that for any irreducible set F with F ⊆ ↓y ⋂ κSI (X), ⋁ F ∈ κSI (X). Let C ∈ ΓSI (X) with ⋁ F ≤ ⋁ C. As, for any x ∈ F , x ≺SI x and x ≤ ⋁ F , x ≺SI ⋁ F and thus F ⊆ C. Then we have ⋁ F ∈ C by C ∈ ΓSI (X). ◻ Proposition 3.26 For any poset X, the lattice ΓSI (X) is CSI -prealgebraic.
242
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. This follows from Corollary 3.20 and the fact that C = ⋃x∈C ↓x = ◻ ⋁ΓSI (X) {↓x ∶ x ∈ C} holds for every C ∈ ΓSI (X). Corollary 3.27 For any poset X, the following statements are equivalent: (1) ΓSI (X) is a continuous lattice; (2) ΓSI (X) is a completely distributive lattice. Proof. (1) ⇔ (2): Clear from Theorem 3.14 and Proposition 3.26.
4
◻
SI-closed set lattices of SI-dominated poset
For two dcpos X and Y , Γ(X) ≅ Γ(Y ) ⇏ X ≅ Y , Ho et al. [3] exhibit a new class domDCPO of dominated dcpos for which the result holds. Similarly, we first define the SI-dominated posets and then study some properties of the lattice (ΓSI (X), ⊆) on it. Definition 4.1 Given C ′ , C ∈ ΓSI (X), we write C ′ ⊲ C if there is a x ∈ C such that C ′ ⊆ ↓x. We write ∇SI C for the set {C ′ ∈ ΓSI (X) ∣ C ′ ⊲ C} Clearly, an element C of ΓSI (X) is of the form ↓x if and only if C ⊲ C holds and C = ⋁ ∇SI C. Proposition 4.2 C ′ ⊲ C implies C ′ ≺SI C for all C ′ , C ∈ ΓSI (X). Proof. This holds by Proposition 3.20.
◻
Next, we give a condition for a poset X which guarantees the reverse implication. Definition 4.3 A poset X is called SI-dominated if for every SI-closed set C of X, the collection ∇SI C is SI-closed in ΓSI (X). Lemma 4.4 A poset X is SI-dominated if and only if C ′ ≺SI C implies C ′ ⊲ C for all C ′ , C ∈ ΓSI (X). Proof. If: Let C ∈ ΓSI (X). By assumption, we have ∇SI C = {C ′ ∈ ΓSI (X) ∣ C ′ ⊲ C} = {C ′ ∈ ΓSI (X) ∣ C ′ ≺SI C} and in Corollary 3.5 we showed that the latter is always SI-closed. Only if: Let C ′ ≺SI C. We know that ∇SI C is SI-closed and C = ⋁ ∇SI C. ◻ Therefore, it must be the case that C ′ ∈ ∇SI C. Proposition 4.5 Let X be a SI-dominated poset. Then C ∈ κSI (ΓSI (X)) if and only if C is a principal ideal, i.e., C = ↓x for some x ∈ X. Also, the principal ideal mapping ↓∶ X → κSI (ΓSI (X)), x ↦ ↓x is an order-isomorphism. Proof. By Corollary 3.21, it suffices to prove the “only if” part. Suppose C ∈ κ(ΓSI (X)), then ∇SI C is a SI-closed set of ΓSI (X) and ⋁ΓSI (X) ∇SI C = C. Since C ≺SI C, it follows that C ∈ ∇SI C. Thus C ⊆ ↓x for some x ∈ C. This means that C = ↓x. ◻ Theorem 4.6 For any two SI-dominated posets X and Y , the following statements are equivalent:
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
243
(1) X ≅ Y ; (2) σSI (X) ≅ σSI (Y ); (3) ΓSI (X) ≅ ΓSI (Y ). Similar to the proof of Lemma 5.1 in [4], we can obtain that for any complete semilattice and any C ∈ ΓSI (X), ∇SI C ∈ Γ(ΓSI (X)). Thus the following result can be checked easily. Lemma 4.7 Let X be a complete semilattice and C ∈ ΓSI (X). If E is an irreducible family in ∇SI C, then F = {⋁ E ∣ E ∈ E} is an irreducible set of X. Proof. Let B, D ∈ Γ(X) such that F ⊆ B ⋃ D, then for any E ∈ E, ⋁ E ∈ B or ⋁ E ∈ D and so E ⊆ ∇SI B ⋃ ∇SI D. As E is irreducible, E ⊆ ∇SI B or E ⊆ ∇SI D, which implies F ⊆ B or F ⊆ D. ◻ Proposition 4.8 All complete semilattices and all complete lattices are SIdominated. Proof. If C ′ ⊲ C for SI-closed sets of a complete semilattice X, then by definition C ′ ⊆ ↓x for some x ∈ C and hence ⋁ C ′ ∈ C. If E is an irreducible family in ∇SI C, then ⋁ΓSI (X) E ⊆ ↓ ⋁E∈E (⋁ E), and the element ⋁E∈E (⋁ E) ∈ C because C is a SI-closed set. ◻ Corollary 4.9 For any SI-dominated poset X, the lattice ΓSI (X) is CSI -algebraic. Corollary 4.10 Every completely distributive lattice is CSI -prealgebraic.
5
The category of strong completed posets
In this section, we show that the category of scpos is Cartesian-closed. Let SCP O be the category whose objects are scpos and morphisms the SIcontinuous maps (i.e., the monotone maps preserving sups of irreducible sets). Obviously, every single point set is a terminal object of SCP O. Let CP ALGSI be the category whose objects are the CSI -prealgebraic lattices and morphisms the lower adjoints which preserve the relation ≺SI . A category with terminal and finite products is called Cartesian-closed if for each pair (A, B) of objects there exists an object B A (sometimes be called the function space from A to B) and a morphism ev ∶ A × B A → B with the following universal property: for each morphism f ∶ A × C → B there exists a unique morphism f ∶ C → B A such that ev ○ (idA × f ) = f . Let {Xj ∣j ∈ J} be a nonempty family of scpos and X = ∏j∈J Xj the product of {Xj ∣j ∈ J}. Then for any j ∈ J, the jth-projection pj ∶ X → Xj is a full Scotcontinuous open mapping and pj (F ) ∈ Irrσ (Xj ) for any j ∈ J and F ∈ Irrσ (X). Moreover, it is easy to check that ⋁ F = (⋁ pj (F ))j∈J for any F ∈ Irrσ (X). Therefore, we have that the category SCP O is closed under finite products. Let X and Y be two scpos and [X → Y ]SI denote the set of all SI-continuous
244
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
maps from X to Y and define the order ≤ by ∀f, g ∈ [X → Y ]SI , f ≤ g ⇔ f (x) ≤ g(x), ∀x ∈ X. Then [X → Y ]SI is a poset and we can have the following results. Lemma 5.1 Let X and Y be two scpos and B ∈ Γ(Y ), then for any x ∈ X, B = {g ∈ [X → Y ]SI ∣ g(x) ∈ B} ∈ Γ([X → Y ]SI ). Proof. Obviously, B is a lower subset of [X → Y ]SI . Let D be a directed set of [X → Y ]SI and D ⊆ B. As [X → Y ]SI ⊆ [X → Y ] and [X → Y ] is a dcpo, D is also a directed set of [X → Y ]. Then ⋁ D exists and ⋁ D(x) = ⋁g∈D g(x). It is easy to check that {g(x) ∣ g ∈ D} is a directed set of Y and {g(x) ∣ g ∈ D} ⊆ B. Now, let f (x) = ⋁ D(x) for any x ∈ X. As B ∈ Γ(Y ), f (x) ∈ B. Moreover, for any F ∈ Irrσ (X), f (⋁ F ) = ⋁ g(⋁ F ) = ⋁ (⋁ g(F )) = ⋁ ( ⋁ g(x)) g∈D
g∈D
g∈D x∈F
= ⋁ ( ⋁ g(x)) = ⋁ f (x) = ⋁ f (F ), x∈F g∈D
x∈F
which implies that f ∈ B. To sum up, B ∈ Γ([X → Y ]SI ).
◻
Proposition 5.2 For any two scpos X and Y , [X → Y ]SI is a scpo. Proof. Let F ∈ Irrσ ([X → Y ]SI ) and A = {g(x) ∣ g ∈ F } for any x ∈ X. Suppose that A ⊆ B ⋃ C for any B, C ∈ Γ(Y ), then B = {g ∈ [X → Y ]SI ∣ g(x) ∈ B}, C = {g ∈ [X → Y ]SI ∣ g(x) ∈ C} ∈ Γ([X → Y ]SI ) and F ⊆ B ⋃ C. As F ∈ Irrσ ([X → Y ]SI ), F ⊆ B or F ⊆ C. Without losing generality, we might as well assume F ⊆ B and so A ⊆ B, which implies that A = {g(x) ∣ g ∈ F} ∈ Irrσ (Y ). Thus ⋁g∈F g(x) exists. Now let f (x) = ⋁g∈F g(x), then for any F ∈ Irrσ (X), ⋁ f (F ) = ⋁ f (x) = ⋁ ⋁ g(x) = ⋁ (⋁ g(F )) x∈F
x∈F g∈F
g∈F
= ⋁ g(⋁ F ) = f (⋁ F ). g∈F
To sum up, for any F ∈ Irrσ ([X → Y ]SI ), ⋁ F = f and ⋁ F ∈ [X → Y ]SI , therefore [X → Y ]SI is a scpo. ◻ Lemma 5.3 Suppose that F (resp., E) is an irreducible set of X (resp., Z). (1) For z0 ∈ Z, define F̃z0 ∈ X × Z by F̃z0 = {(x, z) ∣ x ∈ F, z = z0 }. Then F̃z0 is an irreducible set of X × Z and ⋁ F̃z0 = (⋁ F, z0 ). ̂x ∈ X × Z by E ̂x = {(x, z) ∣ z ∈ E, x = x0 }. Then E ̂x is (2) For x0 ∈ X, define E 0 0 0 ̂ an irreducible set of X × Z and ⋁ Ex0 = (x0 , ⋁ E). Proof. We only show (1) here, (2) is similar. Obviously, ⋁ F̃z0 = (⋁ p1 (F̃z0 ), ⋁ p2 (F̃z0 )) = (⋁ F, z0 ).
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
245
Let F̃z0 ⊆ A ⋃ B, where A, B ∈ Γ(X × Z), then F ⊆ p1 (A) ⋃ p1 (B) and z0 ∈ p2 (A) ⋃ p2 (B). As F ∈ Irrσ (X) and p1 (A), p1 (B) ∈ Γ(X), we have F ⊆ p1 (A) or F ⊆ p1 (B). Suppose that F ⊆ p1 (A), then ⎧ ⎪ ⎪A F̃z0 ⊆ ⎨ ⎪ ⎪ ⎩B
z0 ∈ p2 (A) z0 ∉ p2 (A).
This means that F̃z0 is an irreducible set of X × Z.
◻
Proposition 5.4 If f ∶ X × Z → Y is a SI-continuous mapping, then so is f ∶ Z → [X → Y ]SI , where f (z)(x) = f (x, z) for any z ∈ Z and any x ∈ X. Proof. Step 1: f is a well-defined mapping. We need to show that for all z0 ∈ Z, f (z0 ) = f (⋅, z0 ) ∶ X → Y is a SI-continuous mapping. Obviously, f (z0 ) is monotone. Suppose that F ∈ Irrσ (X). It is routine to show that f (z0 )(⋁ F ) = f (⋁ F, z0 ) = f (⋁ F̃z0 ) = ⋁ f (F̃z0 ) = ⋁ f (z0 )(F ). These show that f is a mapping. Step 2: f is a SI-continuous mapping. Let F ∈ Irrσ (Z), then f (⋁ F )(x) = f (x, ⋁ F ) = ⋁ f (x, F ) = ⋁{f (z)(x) ∣z ∈ F } = (⋁ f (F ))(x). ◻ Proposition 5.5 The evaluation map ev ∶ X × [X → Y ]SI → Y is SI-continuous. Proof. Let F ∈ Irrσ (X × [X → Y ]SI ) and ⋁ F = (x, f ), then x = ⋁ p1 (F ), f = ⋁ p2 (F ) = ⋁{g ∣ (a, g) ∈ F } and f (x) = f (⋁ p1 (F )) = ⋁ f (p1 (F )) = ⋁(f ○ p1 )(F ). Next, we only need to show f (x) = ev(⋁ F ) = ⋁ ev(F ). In fact, ⋁ ev(F ) = ⋁{g(a) ∣ (a, g) ∈ F } implies that for all (a, g) ∈ F , we have (a, g) ≤ ⋁ F = (x, f ), i.e., a ≤ x, g ≤ f and so g(a) ≤ f (x). This means that f (x) is an upper bound of ev(F ), i.e., ⋁ ev(F ) ≤ f (x). Let y ∈ Y be an upper bound of ev(F ), then for all (a, g) ∈ F , g(a) ≤ y and so f (a) ≤ y. Therefore f (x) = ⋁(f ○ p1 )(F ) = ⋁{f (a) ∣ (a, g) ∈ F } ≤ y. To sum up, we have f (x) = ev(⋁ F ) = ⋁ ev(F ).
◻
Theorem 5.6 SCPO is Cartesian-closed. Next, similar to Ho and Zhao’s approach in [4], we obtain an adjunction between SCPO and CP ALGSI . We remove the proofs of results since they are similar to the proofs in [4].
246
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Lemma 5.7 Let (g, d) be a Galois connection between two complete lattices M and N , where d ∶ M → N and g ∶ N → M . Then for any SI-closed set C of N , ↓g(C) is a SI-closed set of M . Lemma 5.8 Let (g, d) be a Galois connection between two complete lattices M and N , where d ∶ M → N and g ∶ N → M . If g preserves the sups of SI-closed sets, then d preserves the relation ≺SI . If M is CSI -continuous, then the converse conclusion is also true. By Proposition II-2.1 in [2], if f ∶ P → Q is a morphism in SCP O then f is continuous with respect to the irreducibly-derived topology. Thus the mapping f −1 ∶ ΓSI (Q) → ΓSI (P ) is well-defined and preserves arbitrary infs, so it is an upper adjoint. Similar to [4], we can obtain the following results Lemma 5.9 Let f ∶ P → Q be a morphism in SCPO and h ∶ ΓSI (P ) → ΓSI (Q) a lower adjoint of f −1 . Then h preserves the relation ≺SI . Proposition 5.10 (1) For all P, Q ∈ ob(SCP O) and all f ∈ HomSCP O (P, Q), let C(P ) = ΓSI (P ) and C(f ) = g, where g is a lower adjoint of f −1 , then C is a functor from the category SCP O to the category CP AlgSI , i.e., the following diagram. P C
ΓSI (P )
f
C(f )=g
/Q
C
/ ΓSI (Q)
(2) For all P, Q ∈ ob(CP AlgSI ) and all f ∈ HomCP AlgSI (P, Q), let K(P ) = κSI (P ) and K(f ) = f ∣κSI (P ) , then K is a functor from the category CP AlgSI to the category SCP O, i.e., the following diagram. P K
κSI (P )
f
/Q
C(f )=f ∣κSI (P )
K
/ κSI (Q)
Theorem 5.11 The functor K is a right adjoint to the functor C.
References [1] Davey, B.A., and H.A. Priestley, Introduction to Lattices and Order, second ed., Cambridge University Press, Cambridge, 2002. 8. [2] Gierz, G., K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott, Continuous Lattices and Domains. Cambridge University Press, 2003. [3] Ho Weng Kin, Jean Goubault-Larrecq, Achim Jung, and Xiaoyong Xi, The Ho-Zhao Problem, Logical Methods in Computer Science 14 1:7(2018) 1-19. [4] Ho Weng Kin, and Dongsheng Zhao, Lattices of Scott-closed sets, Comment. Math. Univ. Carolin. 2 (2009) 297-314. [5] Mao Xuxin, and Luoshan Xu, Order-theoretical Characterizations of Countably Approximating Posets, Int. J. Contemp. Math. Sciences 9 (2014) 447-454.
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
247
[6] Xu Luoshan, and Xuxin Mao, Strongly continuous posets and the local Scott topology, Journal of Mathematical Analysis and Applications 345 (2008) 816-824. [7] Zhao Bin, Jing Lu, and Kaiyun Wang, The Answer to a Problem Posed by Zhao and Ho, Acta Mathematica Sinica, English Series 34 (2018) 1-7. [8] Zhao Dongsheng, and Taihe Fan, Dcpo-completion of posets, Theoretical Computer Science 411 (2010) 2167-2173. [9] Zhao Dongsheng, and Weng Kin Ho, On topologies defined by irreducible sets, Journal of Logical and Algebraic Methods in Programming 84 (2015) 185-195. [10] Zhao Dongsheng, and Luoshan Xu, Uniqueness of directed complete posets based on Scott closed set lattices, Logical Methods in Computer Science 14 2:10(2018) 1-12, DOI: 10.23638/LMCS-14(2:10). [11] Zhao Dongsheng, and Luoshan Xu, Faithfulness of directed complete posets based on Scott closed set lattices, arXiv preprint arXiv:1607.03576, (2016) 1-8.
Electronic Notes in Theoretical Computer Science 345 (2019) 233–247 www.elsevier.com/locate/entcs
Lattices of Irreducibly-derived Closed Sets Shuhua Su1,2 Qi Li3 Fuzhou Teachers’ College, East China University of Technology, Fuzhou, China
Abstract This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set ΓSI (X) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of CSI -continuous posets and CSI -prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y , X ≅ Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed. Keywords: Irreducible sets; Strong complete posets; CSI -continuity; CSI -prealgebra; Cartesian-closed
1
Introduction
Domain theory or the more general theory of posets embodies the combination of an order and a topology. One of the basic and important results is that a poset is continuous if and only if the lattice of the Scott-closed sets above it is a completely distributive lattice. However, for a non-continuous poset, little is known about the order-theoretic properties of the Scott-closed set lattice. To study the Scott-closed set lattice on a general non-continuous poset, Ho and Zhao introduced the concept of a C-continuous poset in [4], and proved that the Scott-closed set lattice on a poset is a C-prealgebraic lattice, especially a C-continuous lattice. With the help of the C-continuity, they obtained some good results, such as: (i) a complete lattice L is isomorphic to the Scott-closed set lattice for a complete semilattice P if and only if L is weak-stably C-algebraic; (ii) for any two complete semilattices X and 1
This work is supported by the National Natural Science Foundation of China (NO. 11861006 and NO. 11561002) 2 Email: [email protected] 3 Email: [email protected]
https://doi.org/10.1016/j.entcs.2019.07.026 1571-0661/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
234
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Y , X ≅ Y if and only if the Scott-closed set lattices above them are isomorphic. In an invited presentation at the 6th International Symposium in Domain Theory, J. D. Lawson emphasized the need to develop the core of domain theory in the wider context of T0 spaces instead of restricting to posets. Just as directed sets are to domains, irreducible sets play an important role in T0 spaces. A nonempty subset F of a T0 space X is irreducible if for any closed two sets A and B with F ⊆ A ⋃ B one has either F ⊆ A or F ⊆ B. Zhao and Ho [9] developed the core of domain theory in the context of T0 spaces by choosing the irreducible sets. They studied the irreducibly-derived topology constructed from any given topology on a set. Moreover, they obtained an interesting result that Scott topology of a poset is exactly the irreducibly-derived topology of its Alexandroff topology. Then Zhao and Xu established some classes of SCL-faithful dcpos and identified some classes of Cσ -unique dcpos based on Scott-closed set lattices.[10,11] Inspired by the works mentioned above, this paper is devoted to investigate the closed set lattices arising from the irreducibly-derived topology of Scott topology. This paper is arranged as follows. In Section 2, we recall some basic materials which will be used throughout this paper, and by the definition of a irreduciblyderived topology, we obtain the irreducibly-derived topology of Scott topology. In Section 3, we define the CSI -continuity on posets by SI-closed sets, and then study its properties. In Section 4, we define SI-dominated posets and study the lattice of SI-closed sets on the SI-dominated posets. We obtain a conclusion similar to the literature [4], that is, for any two SI-dominated posets X and Y , X ≅ Y if and only if the SI-closed set lattices above them are isomorphic. In Section 5, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.
2
Preliminaries
Given a topological space (X, τ ), a nonempty subset F of X is called an irreducible set if for closed sets A, B ⊆ X with F ⊆ A ∪ B one has either F ⊆ A or F ⊆ B. The set of all irreducible sets of X will be denoted by Irrτ (X). On irreducible sets, here are some elementary properties: Proposition 2.1 (Gierz, et al. [2]) For any topological space (X, τ ), it holds that: (1) Every singleton is irreducible; (2) A set is in Irrτ (X) if and only if its closure is in Irrτ (X); (3) The continuous image of an irreducible set is again irreducible; (4) Every directed set of X is irreducible. A subset U of a poset X is called Scott-open if (i) U = ↑U , and (ii) for any directed set D in X, ⋁ D ∈ U implies U ⋂ D ≠ ∅ whenever ⋁ D exists. The set of all Scott-open sets of X forms a topology (called Scott topology) on X, denoted by σ(X). A complement of a Scott-open set is called a Scott-closed set. We use Γ(X) to denote the set of all Scott-closed sets of X. Thus a subset C ⊆ X is Scott-closed if and only if (i) C = ↓C, and (ii) for any directed set D ⊆ C, if ⋁ D exists then
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
235
⋁ D ∈ C. σ(X) and Γ(X) are both complete and distributive lattices with respect to the inclusion relation. A subset A of a poset X is called convex if for any x < y < z, x, z ∈ A implies y ∈ A. Let X be a poset. The way-below relation ≪ on X is defined by x ≪ y for x, y ∈ X if for any directed set D with ⋁ D existing, b ≤ ⋁ D implies a ≤ d for some d ∈ D. A poset X is said to be continuous if for any x ∈ X, the set ↡x = {y ∈ X ∣ y ≪ x} is directed and x = ⋁ ↡x. According the definition of the irreducibly-derived topology in [9], we can obtain the irreducibly-derived topology from Scott topology on a poset, as described next. Definition 2.2 Let X be a poset. A subset C of X is called SI-closed if the following conditions are satisfied: (1) C = ↓C; (2) For any F ∈ Irrτ (X), F ⊆ C implies ⋁ F ∈ C whenever ⋁ F exists. The set of all SI-closed sets of X is denoted by ΓSI (X). A complement of a SI-closed set is called a SI-open set and the set of all SI-open sets of (X, τ ) is denoted by σSI (X). By Theorem 3.2 in [9], σSI (X) is a topology on X and is called a irreducibly-derived topology of σ, a SI-topology for short. Obviously, for any poset X, ΓSI (X) ⊆ Γ(X). Definition 2.3 (Ho, et al. [3]) A poset X is strongly complete if every irreducible set of X has a supremum. In this case we also say that X is a scpo. Obviously, every scpo is a dcpo. Definition 2.4 The SI-topology on a poset X is called lower hereditary if for every SI-closed set C, the SI-topology on C agrees with the relative SI-topology from X. Proposition 2.5 (Zhao and Ho [9]) A function f ∶ X → Y between posets X and Y is SI-continuous iff it is order-preserving and f (⋁ F ) = ⋁ f (F ) for any F ∈ Irrσ (X) with ⋁ F existing. Lemma 2.6 (Xu and Mao [6]) Let X be a poset and A ⊆ X. If s, t are two upper bounds of A with s ≤ t and A has a supremum in ↓t, denoted ⋁t A, then A has a supremum in ↓s, in this case ⋁s A = ⋁t A. Lemma 2.7 Let X be a poset. The following statements are equivalent: (1) The SI-topology on X is lower hereditary; (2) For any x ∈ X, the inclusion map from ↓x into X is SI-continuous; (3) Any minimal upper bound of any irreducible set of X is the supremum for that irreducible set; (4) For any x ∈ X and any irreducible set F , x = ⋁x F implies x = ⋁ F . Proof. (1)⇒(2): Follows from that any principal ideal is a SI-closed set.
236
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
(2)⇒(3): Let F be an irreducible set with a minimal upper bound b. Then b is the supremum of F in ↓b. Since the inclusion map from ↓b into X is continuous, it follows from Proposition 2.5 that b is the supremum of F in X . (3)⇒(1): Let E be a SI-closed set. A subset B that is SI-closed in E is easily verified to be SI-closed in X. Conversely suppose that A is SI-closed in X . Then A ∩ E ∈ Γ(E). To show that A ∩ E is SI-closed in E, let F be an irreducible set in A ∩ E that has the supremum b in E. Then ⋁E F = b ∈ E and b = ⋁ F by (3). It follows from the SI-closedness of A that b ∈ A. Hence b ∈ A ∩ E. This shows that A ∩ E is closed in the SI-topology of E. (3)⇒(4): Suppose that F is an irreducible set with x = ⋁x F . Then x is a minimal upper bound of F . By (3), we have ⋁ F = x. (4)⇒(3): Let y be a minimal upper bound of an irreducible set F . Then y = ⋁y F . By (4), we have y = ⋁ F . ◻ Applying Lemmas 2.7, we obtain the following corollary. Corollary 2.8 Every scpo has a lower hereditary SI-topology. Lemma 2.9 (Mao and Xu [5]) Let X be a poset, x ∈ X and A ⊆ ↓x = C. Then ⋁C A = ⋁ A whenever ⋁ A exists; If (X, ≤) is a semilattice, then ⋁ A = ⋁C A whenever ⋁C A exists, where ⋁C A denotes the supremum of A in the principal ideal ↓x = C. Applying Lemmas 2.7 and Lemma 2.9, we obtain the following corollary. Corollary 2.10 Every semilattice has a lower hereditary SI-topology. Proposition 2.11 Let X and Y be posets with lower hereditary SI-topologies. Then (1) Every convex subset A of X in the inherited order has also a lower hereditary SI-topology. In particular, every SI-closed set of X has a lower hereditary SItopology; (2) The product X × Y has also a lower hereditary SI-topology. Proof. (1) Let F ⊆ A be an irreducible set with a minimal upper bound z ∈ A. Then z is also a minimal upper bound of F in X because of the convexity of A and hence the supremum of F in X by Lemma 2.7. (2) Let F ⊆ X × Y be an irreducible set with a minimal upper bound z = (u, v). Then it is easy to see that u is a minimal upper bound of p1 (F ) and v is a minimal upper bound of p2 (F ), where p1 ∶ X × Y → X and p2 ∶ X × Y → Y are the projection maps. Since X and Y have the lower hereditary SI-topologies, we have ⋁ p1 (F ) = u and ⋁ p2 (F ) = v. Thus ⋁ F = (⋁ p1 (F ), ⋁ p2 (F )) = (u, v) = z. Therefore, X × Y has the lower hereditary SI-topology by Lemma 2.7. ◻
3
CSI -continuous posets
In this section, we define a new auxiliary relation on a poset X by SI-closed sets. Then we introduce the notions of CSI -continuous and CSI -algebraic posets, which
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
237
is important for us to study the properties of SI-closed set lattices. Definition 3.1 Let X be a poset and x, y ∈ X. We say that x is beneath y, denoted by x≺SI y, if for every nonempty SI-closed set C ⊆ X with ⋁ C existing, the relation ⋁ C ≥ y always implies that x ∈ C. We denote the set {x ∈ X ∶ x≺SI y} by ⇓SI y and the set {x ∈ X ∶ y≺SI x} by ⇑SI y. Remark 3.2 Obviously, for any poset (X, ≤), ≺SI and ≺ defined in [4] are quite different. As ΓSI (X) ⊆ Γ(X), x ≺ y implies that x ≺SI y. However, the converse is not true. If (X, ≤) is k-bounded sober defined in [8], then ≺SI =≺. Example 1. Let X = (N × (N ⋃{∞})) ⋃{⊺} and define x ≤ y if one of the following conditions holds: (1) y = ⊺; (2) x = (m1 , n1 ) and y = (m2 , n2 ), where m1 = m2 , n1 ≤ n2 (≤ ∞) or n2 = ∞, n1 ≤ m2 . Then (X, ≤) is a poset. By [7], we know that C = N × (N ⋃{∞}) is an irreducible Scott closed set and ⋁ C = ⊺. Hence it is easily checked that ⊺ ⊀ ⊺ and ⊺ ≺SI ⊺. Example 2. Let X = N ×(N ⋃{∞}) and define (m, n) ≤ (m0 , n0 ) if m = m0 , n ≤ n0 , or n0 = ∞ and n ≤ m0 . As was pointed out in [7], the Scott space (X, σ(X)) is k-bounded sober. Hence ≺SI =≺. Remark 3.3 Let X be a poset. Then (1) x≺SI y ⇒ x ≤ y; (2) x ≤ y≺SI z ≤ w implies x≺SI w; (3) For all x ∈ X, 0≺SI x whenever X is a pointed poset. Proposition 3.4 For a poset X, the following conditions are equivalent: (1) x ≺SI y; (2) x ∈ C for every SI-closed set C with y ≤ ⋁ C whenever ⋁ C exists; (3) x ∈ ⋂ J(y), where J(y) = {C ∈ ΓSI (X) ∶ y ≤ ⋁ C}. Corollary 3.5 Let X be a poset and x ∈ X, then ⇓SI x is a SI-closed set of X. Proof. By Proposition 3.4, we have ⇓SI x = ⋂ J(x), and thus ⇓SI x is a SI-closed set of X. ◻ Definition 3.6 A poset X is said to be CSI -continuous if it satisfies x = ⋁ ⇓SI x for all x ∈ X. A CSI -continuous poset which is also a complete lattice is called a CSI continuous lattice. Obviously, we have ⇓x ⊆ ⇓SI x ⊆ ↓x, ∀x ∈ X by Remark 3.2, and thus every C-continuous poset is also CSI -continuous. Proposition 3.7 For a complete lattice X, the following are equivalent: (1) (X, τ ) is CSI -continuous; (2) For each x ∈ X, the set ⇓SI x is the smallest SI-closed set C with x ≤ ⋁ C; (3) For each x ∈ X, there is a smallest nonempty SI-closed set C such that x ≤ ⋁ C; (4) The ⋁ map r = (C ↦ ⋁ C) ∶ ΓSI (X) → X has a lower adjoint; (5) The ⋁ map r ∶ ΓSI (X) → X preserves all existing infs; (6) For any collection {Ci ∶ i ∈ I} of SI-closed sets of X, the following equation holds: ⋀ ⋁ Ci = ⋁ ⋂ Ci . i∈I
i∈I
238
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. (1) ⇒ (2): Condition (1) holds if and only if for each x ∈ X, ⇓SI x ∈ J(x) by Definition 3.6. Thus (2) follows. Condition (2) trivially implies (3). (3) ⇒ (1): If J(x) has a smallest element M , then M ⊆ C for all C ∈ J(x) and thus M ⊆ ⋂ J(x) ⊆ M . Hence M = ⋂ J(x) = ⇓SI x. Thus (1),(2) and (3) are equivalent. (3)⇔(4): By the definition of Galois connection, the map r has a lower adjoint if and only if minr−1 (↑x) exists for all x. But minr−1 (↑x) is precisely the smallest element of J(x). (4) ⇒ (5): The ⋁ map preserves infs by Theorem O-3.3 in [2]. (5) ⇒ (4): Obviously, for any x ∈ X, r−1 (↑x) = J(x) ≠ ∅, which implies that the ⋁ map is cofinal. Thus the ⋁ map has a lower adjoint by O-3.4 in [2]. (5) ⇔ (6): Obviously. ◻ Proposition 3.8 A poset X is CSI -continuous if and only if X⊥ is CSI -continuous, where X⊥ is the poset obtained from X by adjoining a new bottom element. Proof. Suppose that X is CSI -continuous and x ∈ X⊥ . If x ∈ X, then there is a smallest nonempty SI-closed set Cx of X with ⋁ Cx ≥ x by Proposition 3.7. Thus Cx′ = Cx ⋃{⊥} is the smallest nonempty SI-closed set of X⊥ with ⋁ Cx′ ≥ x. If x =⊥, then by Remark 3.3, C⊥ = {⊥} is the smallest nonempty SI-closed set of X⊥ with ⋁ C⊥ ≥⊥. Thus X⊥ is CSI -continuous by Proposition 3.7. Conversely, suppose that X⊥ is CSI -continuous and x ∈ X. By Proposition 3.7, there is a smallest nonempty SI-closed subset Cx′ of X⊥ with ⋁ Cx′ ≥ x. Then Cx = Cx′ /{⊥} is the smallest nonempty SI-closed set Cx of X with ⋁ Cx ≥ x. Thus X is CSI -continuous by Proposition 3.7 again. ◻ Proposition 3.9 Let X and Y be posets and (X, σ(X)) ≅ (Y, σ(Y )). If X is CSI -continuous, then Y is also CSI -continuous. Proof. Obviously.
◻
Proposition 3.10 Let X be a CSI -continuous semilattice. Then every principal ideal of X is CSI -continuous. Proof. Suppose that X is a CSI -continuous semilattice. Then we claim that for C all x ∈ X and u ∈ ↓x = C, ⇓SI u ⊆ ⇓C SI u holds, where ⇓SI u denotes the subset ⇓SI u in C. In fact, for all v ∈ ⇓SI u and all A ∈ ΓSI (C) with ⋁C A ≥ u, we have A ∈ ΓSI (X) by Definition 2.2 and Corollary 2.10. It follows from Lemma 2.9 that x ≥ ⋁C A = ⋁ A ≥ u. Since v ∈ ⇓SI u, there is a v ∈ A. This shows that v ∈ ⇓C SI u. By the CSI -continuity of X and Lemma 2.9, u = ⋁ ⇓SI u = ⋁C ⇓SI u. Clearly, u is an upper bound of ⇓C SI u in the principal ideal C = ↓x. Suppose that t is any upper u in C. Then t is an upper bound of ⇓SI u in C. Thus u = ⋁ ⇓SI u ≤ t. bound of ⇓C SI This shows that ⋁C ⇓C SI u = u. Thus for all x ∈ X, the principal ideal C = ↓x is ◻ CSI -continuous. Proposition 3.11 Let X be a semilattice. If every principal ideal of X is CSI continuous, then X is CSI -continuous.
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
239
Proof. Suppose that for all x ∈ X, the principal ideal C = ↓x is CSI -continuous. C We claim that ⇓C SI x ⊆ ⇓SI x. In fact, for all y ∈ ⇓SI x and all A ∈ ΓSI (X) with ⋁ A = z ≥ x, the principal ideal B = ↓z is CSI -continuous by the hypothesis. By C Corollary 3.5 and 2.10, ⇓C SI x ∈ ΓSI (B) and thus ⇓SI x ∈ ΓSI (C). By Lemma 2.9, B B C we have ⋁B ⇓SI x = ⋁C ⇓SI x = x. Since y ∈ ⇓SI x, there is a u ∈ ⇓B SI x such that y ≤ u. Since A ∈ ΓSI (X) with existing ⋁ A = z ≥ x, we have A ∈ ΓSI (B) and B ⋁B A = ⋁ A = z ≥ x by Corollary 3.5 and 2.10 again. It follows from u ∈ ⇓SI x that C there is a a ∈ A such that a ≥ u ≥ y. This shows that y ∈ ⇓SI x and thus ⇓SI x ⊆ ⇓SI x. B By the CSI -continuity of C = ↓x and Lemma 2.9, we have x = ⋁B ⇓B SI x = ⋁ ⇓SI x. B Clearly x is an upper bound of ⇓SI x in X. Suppose that t is any upper bound of B ⇓SI x in X. Then t is an upper bound of ⇓B SI x in X. Thus x = ⋁ ⇓SI x ≤ t. This ◻ shows that ⋁ ⇓B SI x = x and hence X is CSI -continuous. By Propositions 3.10 and 3.11, we immediately have the following characterization of CSI -continuous posets by principal ideals. Theorem 3.12 Let X be a semilattice. Then X is CSI -continuous if and only if every principal ideal of X is CSI -continuous. Proposition 3.13 Let X be a CSI -continuous lattice. The for any collection {Fi ∶ i ∈ I} of finite subsets of X the following equation holds: ⋀ ⋁ Fi =
i∈I
⋁
⋂ f (i).
f ∈∏i∈I Fi i∈I
In particular, every CSI -continuous lattice space is distributive. Proof. Denote the left hand side (respectively, the right hand side) of the equation by a (respectively, b). It suffices to prove that a ≤ b. Let a = ⋀i∈I ⋁ Fi and x ≺SI a. For each i ∈ I, the set ↓Fi = ⋃y∈Fi ↓y is a SI-closed set by ↓y ∈ ΓSI (X), and x ≺SI a ≤ ⋁ Fi . Thus there is a di ∈ Fi with x ≤ di . Let f ∈ ∏i∈I Fi be defined by f (i) = di , i ∈ I. Then x ≤ ∏i∈I f (i) ≤ b. Therefore a = ⋁ ⇓SI a by the CSI -continuity of X and thus a ≤ b. ◻ Theorem 3.14 Let X be a complete lattice. Then the following are equivalent: (1) X is CSI -continuous and continuous; (2) X is completely distributive. Proof. It suffices to show that (1) ⇒ (2). Assume that X is both CSI -continuous and continuous. Since X is continuous, for each a ∈ X, a = ⋁ ↡a. Now for each x ≪ a, x = ⋁ ⇓SI x. It follows that a = ⋁{y ∈ X ∶ ∃x. y ≺SI x ≪ a}. Next suppose y ≺SI x ≪ a, we shall show that y ⊲ a. Let A ⊆ X with a ≤ ⋁ A. Construct the se E = {⋁ B ∶ B is a f inite subset of A}. Then E is directed and thus is irreducible and ⋁ E = ⋁ A ≥ a. Since x ≪ a, there is a finite subset B ⊆ A such that x ≤ ⋁ B = ⋁ ↓B. Note that the last set ↓B is SI-closed. So it follows from y ≺SI x that y ≤ d for some d ∈ B ⊆ A. This implies that y ⊲ a. Hence X is completely distributive. ◻ Lemma 3.15 For any poset X, if C ∈ Γ(ΓSI (X)), then ⋃ C ∈ Γ(X).
240
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. Let C ∗ = {C ∈ Γ(X) ∣ ∃V ∈ C s.t. C ⊆ V }, then C ⊆ C ∗ and ⋃ C = ⋃ C ∗ . Therefore, we only need to show C ∗ ∈ Γ(Γ(X)) by Proposition 4.1 in [4]. Obviously, C ∗ is a lower set of Γ(X). For any directed set D ⊆ Γ(X) with D ⊆ C ∗ , D∗ = {clσSI (D) ∣ D ∈ D} is a directed set of ΓSI (X). Moreover, for any D ∈ D, there is a V ∈ C such that D ⊆ V , thus clσSI (D) ⊆ V . Since C is a lower set, clσSI (D) ∈ C. Therefore, D ∗ ⊆ C, thus we have ⋁ΓSI (X) D∗ ∈ C. As ⋁Γ(X) D ⊆ ⋁Γ(X) D∗ ⊆ ◻ ⋁ΓSI (X) D∗ , ⋁Γ(X) D ∈ C ∗ . To sum up, C ∗ ∈ Γ(Γ(X)). Lemma 3.16 For any poset X, if F ∈ Irrσ (X), then F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Proof. Let A, B ∈ Γ(ΓSI (X)) such that F ⊆ A ⋃ B, then for any d ∈ F , ↓d ∈ A or ↓d ∈ B and thus F ⊆ (⋃ A) ⋃(⋃ B). By Lemma 3.15, ⋃ A, ⋃ B ∈ Γ(X), and so F ⊆ (⋃ A) or F ⊆ (⋃ B). Without loss of generality, we assume that F ⊆ ⋃ A and thus for any d ∈ F , there is a A ∈ A such that d ∈ A. Because A is a lower set, ↓d ⊆ A, ◻ and thus ↓d ∈ A. Therefore, F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Proposition 3.17 Let X be a poset and C ∈ ΓSI (ΓSI (X)). Then ⋁ΓSI (X) C = ⋃ C. Proof. Note that member of C is a SI-closed set of ΓSI (X). In order to prove the equation, it suffices to show that ⋃ C ∈ ΓSI (X). Obviously ⋃ C ∈ Γ(X) by Lemma 3.15. Now let F be any irreducible set of X contained in ⋃ C such that ⋁ F exists in X. We want to prove that ⋁ F ∈ C for some C ∈ C. By Lemma 3.16, we have that F = {↓d ∶ d ∈ F } is an irreducible set of ΓSI (X). Moreover, F ⊆ C because C is a lower set in ΓSI (ΓSI (X)). Since C is a SI-closed set of ΓSI (X), ⋁ΓSI (X) F ∈ C. ◻ Note that ⋁ΓSI (X) F is precisely ↓ ⋁ F . Hence ⋁ F ∈ C for some C ∈ C. Definition 3.18 An element x of a poset X is called CSI -compact if x ≺SI x. We use κSI (X) to denote the set of all the CSI -compact elements of X. Obviously, every C-compact element defined in [4] is CSI -compact, thus κ(X) ⊆ κSI (X). Proposition 3.19 Let X be a lattice. (1) If r ∈ κSI (X), then r is co-prime; (2) If X is a completely distributive lattice, then every co-prime element is CSI compact; (3) If X is a complete lattice and κSI (X) ≠ ∅, then κSI (X) is a pointed scpo with respect to the order inherited from X. Proof. (1) Suppose r ∈ κSI (X) and r ≤ x ∨ y. Let C = ↓{x, y}. Then C is SI-closed and ⋁ C = x ∨ y. Hence r ∈ C which implies r ≤ x or r ≤ y, thus r is co-prime. (2) Let X be a completely distributive lattice and r ∈ X be co-prime. Note that the set β(r) = {x ∈ X ∶ x ⊲ r} is a directed set and thus β(r) is an irreducible set and ⋁ β(r) = r. Since x ⊲ r implies x ≺SI r, it follows from Corollary 3.5 that ⋁ β(r) ≺SI r, i.e., r ≺SI r. Hence r is CSI -compact. (3) Let F be an irreducible set in κSI (X). It suffices to show that ⋁ F ≺SI ⋁ F . So let C ∈ ΓSI (X) with ⋁ F ≤ ⋁ C. Thus d ≤ ⋁ C for all d ∈ F . Since F ⊆ κSI (X),
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
241
it follows that d ≺SI d for all d ∈ F and so F ⊆ C. Because C ∈ ΓSI (X), this implies that ⋁ F ∈ C and so ⋁ F ≺SI ⋁ F , i.e., ⋁ F ∈ κSI (X). Also 0 ≺SI 0 implies that 0 ∈ κSI (X). Hence κSI (X) is a pointed scpo with respect to the order inherited from X. ◻ Proposition 3.20 Let X be a poset and C be a nonempty SI-closed set of X. Then for each x ∈ C, ↓x ≺SI C holds in ΓSI (X). Proof. Let x ∈ C. Suppose C ∈ ΓSI (ΓSI (X)) with C ⊆ ⋁ΓSI (X) C. Then C ⊆ ⋃ C by Proposition 3.17. Hence there is a A ∈ C such that x ∈ A. This means ↓x ⊆ A, and thus ↓x ∈ C. Corollary 3.21 Let X be a poset. κ(ΓSI (X)).
Then for each x ∈ X, it holds that ↓x ∈
Definition 3.22 A poset X is said to be CSI -prealgebraic if for each a ∈ X, a = ⋁(↓a ⋂ κSI (X)). A CSI -prealgebraic poset X is CSI -algebraic if for any a ∈ X, ↓(↓a ⋂ κSI (X)) ∈ ΓSI (X). Remark 3.23 (1) Obviously every CSI -prealgebraic poset is CSI -continuous, since ↓a ⋂ κSI (X) ⊆ ⇓SI a ⊆ ↓a and a = ⋁(↓a ⋂ κSI (X)) = ⋁ ↓a. In this case, we call a CSI (pre)algebraic poset which is also a complete lattice a CSI -(pre)algebraic lattice. (2) Every CSI -prealgebraic poset is C-prealgebraic. Proposition 3.24 Let X be a poset and F ∈ Irrσ (X) with ⋁ F existing. If x≺SI y for all x ∈ F , then ⋁ F ≺SI y. Proof. Let C ∈ ΓSI (X) be nonempty such that ⋁ C exists with y ≤ ⋁ C. Since x≺SI y for all x ∈ F , it follows that F ⊆ C. Because C is SI-closed and F is irreducible, we have ⋁ F ∈ C. Thus ⋁ F ≺SI y. ◻ Proposition 3.25 In a poset X, the following statements are equivalent: (1) X is CSI -algebraic; (2) X is CSI -continuous, and x ≺SI y if and only if there is a k ∈ κSI (X) with x ≤ k ≤ y. Proof. (1) ⇒ (2): Assume (1) and x, y ∈ X with x ≺SI y. Since y = ⋁ C with the SI-closed set C = ↓(↓y ⋂ κSI (X)) by (1), there is a k ∈ ↓y ⋂ κSI (X) with x ≤ k, and thus x ≤ k ≤ y. Conversely, let x ≤ k ≺SI k ≤ y, then we have x ≺SI y. The CSI -continuity of X follows directly from Remark 3.23. (2) ⇒ (1): Assume (2) and let y ∈ X. Then y = ⇓SI y and ⇓SI y is SI-closed. For every x ≺SI y, there is a CSI -compact element k such that x ≤ k ≤ y by (2), so we conclude that y = ⋁(↓y ⋂ κSI (X)). Further, ↓(↓y ⋂ κSI (X)) ∈ ΓSI (X). In fact, we only show that for any irreducible set F with F ⊆ ↓y ⋂ κSI (X), ⋁ F ∈ κSI (X). Let C ∈ ΓSI (X) with ⋁ F ≤ ⋁ C. As, for any x ∈ F , x ≺SI x and x ≤ ⋁ F , x ≺SI ⋁ F and thus F ⊆ C. Then we have ⋁ F ∈ C by C ∈ ΓSI (X). ◻ Proposition 3.26 For any poset X, the lattice ΓSI (X) is CSI -prealgebraic.
242
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Proof. This follows from Corollary 3.20 and the fact that C = ⋃x∈C ↓x = ◻ ⋁ΓSI (X) {↓x ∶ x ∈ C} holds for every C ∈ ΓSI (X). Corollary 3.27 For any poset X, the following statements are equivalent: (1) ΓSI (X) is a continuous lattice; (2) ΓSI (X) is a completely distributive lattice. Proof. (1) ⇔ (2): Clear from Theorem 3.14 and Proposition 3.26.
4
◻
SI-closed set lattices of SI-dominated poset
For two dcpos X and Y , Γ(X) ≅ Γ(Y ) ⇏ X ≅ Y , Ho et al. [3] exhibit a new class domDCPO of dominated dcpos for which the result holds. Similarly, we first define the SI-dominated posets and then study some properties of the lattice (ΓSI (X), ⊆) on it. Definition 4.1 Given C ′ , C ∈ ΓSI (X), we write C ′ ⊲ C if there is a x ∈ C such that C ′ ⊆ ↓x. We write ∇SI C for the set {C ′ ∈ ΓSI (X) ∣ C ′ ⊲ C} Clearly, an element C of ΓSI (X) is of the form ↓x if and only if C ⊲ C holds and C = ⋁ ∇SI C. Proposition 4.2 C ′ ⊲ C implies C ′ ≺SI C for all C ′ , C ∈ ΓSI (X). Proof. This holds by Proposition 3.20.
◻
Next, we give a condition for a poset X which guarantees the reverse implication. Definition 4.3 A poset X is called SI-dominated if for every SI-closed set C of X, the collection ∇SI C is SI-closed in ΓSI (X). Lemma 4.4 A poset X is SI-dominated if and only if C ′ ≺SI C implies C ′ ⊲ C for all C ′ , C ∈ ΓSI (X). Proof. If: Let C ∈ ΓSI (X). By assumption, we have ∇SI C = {C ′ ∈ ΓSI (X) ∣ C ′ ⊲ C} = {C ′ ∈ ΓSI (X) ∣ C ′ ≺SI C} and in Corollary 3.5 we showed that the latter is always SI-closed. Only if: Let C ′ ≺SI C. We know that ∇SI C is SI-closed and C = ⋁ ∇SI C. ◻ Therefore, it must be the case that C ′ ∈ ∇SI C. Proposition 4.5 Let X be a SI-dominated poset. Then C ∈ κSI (ΓSI (X)) if and only if C is a principal ideal, i.e., C = ↓x for some x ∈ X. Also, the principal ideal mapping ↓∶ X → κSI (ΓSI (X)), x ↦ ↓x is an order-isomorphism. Proof. By Corollary 3.21, it suffices to prove the “only if” part. Suppose C ∈ κ(ΓSI (X)), then ∇SI C is a SI-closed set of ΓSI (X) and ⋁ΓSI (X) ∇SI C = C. Since C ≺SI C, it follows that C ∈ ∇SI C. Thus C ⊆ ↓x for some x ∈ C. This means that C = ↓x. ◻ Theorem 4.6 For any two SI-dominated posets X and Y , the following statements are equivalent:
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
243
(1) X ≅ Y ; (2) σSI (X) ≅ σSI (Y ); (3) ΓSI (X) ≅ ΓSI (Y ). Similar to the proof of Lemma 5.1 in [4], we can obtain that for any complete semilattice and any C ∈ ΓSI (X), ∇SI C ∈ Γ(ΓSI (X)). Thus the following result can be checked easily. Lemma 4.7 Let X be a complete semilattice and C ∈ ΓSI (X). If E is an irreducible family in ∇SI C, then F = {⋁ E ∣ E ∈ E} is an irreducible set of X. Proof. Let B, D ∈ Γ(X) such that F ⊆ B ⋃ D, then for any E ∈ E, ⋁ E ∈ B or ⋁ E ∈ D and so E ⊆ ∇SI B ⋃ ∇SI D. As E is irreducible, E ⊆ ∇SI B or E ⊆ ∇SI D, which implies F ⊆ B or F ⊆ D. ◻ Proposition 4.8 All complete semilattices and all complete lattices are SIdominated. Proof. If C ′ ⊲ C for SI-closed sets of a complete semilattice X, then by definition C ′ ⊆ ↓x for some x ∈ C and hence ⋁ C ′ ∈ C. If E is an irreducible family in ∇SI C, then ⋁ΓSI (X) E ⊆ ↓ ⋁E∈E (⋁ E), and the element ⋁E∈E (⋁ E) ∈ C because C is a SI-closed set. ◻ Corollary 4.9 For any SI-dominated poset X, the lattice ΓSI (X) is CSI -algebraic. Corollary 4.10 Every completely distributive lattice is CSI -prealgebraic.
5
The category of strong completed posets
In this section, we show that the category of scpos is Cartesian-closed. Let SCP O be the category whose objects are scpos and morphisms the SIcontinuous maps (i.e., the monotone maps preserving sups of irreducible sets). Obviously, every single point set is a terminal object of SCP O. Let CP ALGSI be the category whose objects are the CSI -prealgebraic lattices and morphisms the lower adjoints which preserve the relation ≺SI . A category with terminal and finite products is called Cartesian-closed if for each pair (A, B) of objects there exists an object B A (sometimes be called the function space from A to B) and a morphism ev ∶ A × B A → B with the following universal property: for each morphism f ∶ A × C → B there exists a unique morphism f ∶ C → B A such that ev ○ (idA × f ) = f . Let {Xj ∣j ∈ J} be a nonempty family of scpos and X = ∏j∈J Xj the product of {Xj ∣j ∈ J}. Then for any j ∈ J, the jth-projection pj ∶ X → Xj is a full Scotcontinuous open mapping and pj (F ) ∈ Irrσ (Xj ) for any j ∈ J and F ∈ Irrσ (X). Moreover, it is easy to check that ⋁ F = (⋁ pj (F ))j∈J for any F ∈ Irrσ (X). Therefore, we have that the category SCP O is closed under finite products. Let X and Y be two scpos and [X → Y ]SI denote the set of all SI-continuous
244
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
maps from X to Y and define the order ≤ by ∀f, g ∈ [X → Y ]SI , f ≤ g ⇔ f (x) ≤ g(x), ∀x ∈ X. Then [X → Y ]SI is a poset and we can have the following results. Lemma 5.1 Let X and Y be two scpos and B ∈ Γ(Y ), then for any x ∈ X, B = {g ∈ [X → Y ]SI ∣ g(x) ∈ B} ∈ Γ([X → Y ]SI ). Proof. Obviously, B is a lower subset of [X → Y ]SI . Let D be a directed set of [X → Y ]SI and D ⊆ B. As [X → Y ]SI ⊆ [X → Y ] and [X → Y ] is a dcpo, D is also a directed set of [X → Y ]. Then ⋁ D exists and ⋁ D(x) = ⋁g∈D g(x). It is easy to check that {g(x) ∣ g ∈ D} is a directed set of Y and {g(x) ∣ g ∈ D} ⊆ B. Now, let f (x) = ⋁ D(x) for any x ∈ X. As B ∈ Γ(Y ), f (x) ∈ B. Moreover, for any F ∈ Irrσ (X), f (⋁ F ) = ⋁ g(⋁ F ) = ⋁ (⋁ g(F )) = ⋁ ( ⋁ g(x)) g∈D
g∈D
g∈D x∈F
= ⋁ ( ⋁ g(x)) = ⋁ f (x) = ⋁ f (F ), x∈F g∈D
x∈F
which implies that f ∈ B. To sum up, B ∈ Γ([X → Y ]SI ).
◻
Proposition 5.2 For any two scpos X and Y , [X → Y ]SI is a scpo. Proof. Let F ∈ Irrσ ([X → Y ]SI ) and A = {g(x) ∣ g ∈ F } for any x ∈ X. Suppose that A ⊆ B ⋃ C for any B, C ∈ Γ(Y ), then B = {g ∈ [X → Y ]SI ∣ g(x) ∈ B}, C = {g ∈ [X → Y ]SI ∣ g(x) ∈ C} ∈ Γ([X → Y ]SI ) and F ⊆ B ⋃ C. As F ∈ Irrσ ([X → Y ]SI ), F ⊆ B or F ⊆ C. Without losing generality, we might as well assume F ⊆ B and so A ⊆ B, which implies that A = {g(x) ∣ g ∈ F} ∈ Irrσ (Y ). Thus ⋁g∈F g(x) exists. Now let f (x) = ⋁g∈F g(x), then for any F ∈ Irrσ (X), ⋁ f (F ) = ⋁ f (x) = ⋁ ⋁ g(x) = ⋁ (⋁ g(F )) x∈F
x∈F g∈F
g∈F
= ⋁ g(⋁ F ) = f (⋁ F ). g∈F
To sum up, for any F ∈ Irrσ ([X → Y ]SI ), ⋁ F = f and ⋁ F ∈ [X → Y ]SI , therefore [X → Y ]SI is a scpo. ◻ Lemma 5.3 Suppose that F (resp., E) is an irreducible set of X (resp., Z). (1) For z0 ∈ Z, define F̃z0 ∈ X × Z by F̃z0 = {(x, z) ∣ x ∈ F, z = z0 }. Then F̃z0 is an irreducible set of X × Z and ⋁ F̃z0 = (⋁ F, z0 ). ̂x ∈ X × Z by E ̂x = {(x, z) ∣ z ∈ E, x = x0 }. Then E ̂x is (2) For x0 ∈ X, define E 0 0 0 ̂ an irreducible set of X × Z and ⋁ Ex0 = (x0 , ⋁ E). Proof. We only show (1) here, (2) is similar. Obviously, ⋁ F̃z0 = (⋁ p1 (F̃z0 ), ⋁ p2 (F̃z0 )) = (⋁ F, z0 ).
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
245
Let F̃z0 ⊆ A ⋃ B, where A, B ∈ Γ(X × Z), then F ⊆ p1 (A) ⋃ p1 (B) and z0 ∈ p2 (A) ⋃ p2 (B). As F ∈ Irrσ (X) and p1 (A), p1 (B) ∈ Γ(X), we have F ⊆ p1 (A) or F ⊆ p1 (B). Suppose that F ⊆ p1 (A), then ⎧ ⎪ ⎪A F̃z0 ⊆ ⎨ ⎪ ⎪ ⎩B
z0 ∈ p2 (A) z0 ∉ p2 (A).
This means that F̃z0 is an irreducible set of X × Z.
◻
Proposition 5.4 If f ∶ X × Z → Y is a SI-continuous mapping, then so is f ∶ Z → [X → Y ]SI , where f (z)(x) = f (x, z) for any z ∈ Z and any x ∈ X. Proof. Step 1: f is a well-defined mapping. We need to show that for all z0 ∈ Z, f (z0 ) = f (⋅, z0 ) ∶ X → Y is a SI-continuous mapping. Obviously, f (z0 ) is monotone. Suppose that F ∈ Irrσ (X). It is routine to show that f (z0 )(⋁ F ) = f (⋁ F, z0 ) = f (⋁ F̃z0 ) = ⋁ f (F̃z0 ) = ⋁ f (z0 )(F ). These show that f is a mapping. Step 2: f is a SI-continuous mapping. Let F ∈ Irrσ (Z), then f (⋁ F )(x) = f (x, ⋁ F ) = ⋁ f (x, F ) = ⋁{f (z)(x) ∣z ∈ F } = (⋁ f (F ))(x). ◻ Proposition 5.5 The evaluation map ev ∶ X × [X → Y ]SI → Y is SI-continuous. Proof. Let F ∈ Irrσ (X × [X → Y ]SI ) and ⋁ F = (x, f ), then x = ⋁ p1 (F ), f = ⋁ p2 (F ) = ⋁{g ∣ (a, g) ∈ F } and f (x) = f (⋁ p1 (F )) = ⋁ f (p1 (F )) = ⋁(f ○ p1 )(F ). Next, we only need to show f (x) = ev(⋁ F ) = ⋁ ev(F ). In fact, ⋁ ev(F ) = ⋁{g(a) ∣ (a, g) ∈ F } implies that for all (a, g) ∈ F , we have (a, g) ≤ ⋁ F = (x, f ), i.e., a ≤ x, g ≤ f and so g(a) ≤ f (x). This means that f (x) is an upper bound of ev(F ), i.e., ⋁ ev(F ) ≤ f (x). Let y ∈ Y be an upper bound of ev(F ), then for all (a, g) ∈ F , g(a) ≤ y and so f (a) ≤ y. Therefore f (x) = ⋁(f ○ p1 )(F ) = ⋁{f (a) ∣ (a, g) ∈ F } ≤ y. To sum up, we have f (x) = ev(⋁ F ) = ⋁ ev(F ).
◻
Theorem 5.6 SCPO is Cartesian-closed. Next, similar to Ho and Zhao’s approach in [4], we obtain an adjunction between SCPO and CP ALGSI . We remove the proofs of results since they are similar to the proofs in [4].
246
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
Lemma 5.7 Let (g, d) be a Galois connection between two complete lattices M and N , where d ∶ M → N and g ∶ N → M . Then for any SI-closed set C of N , ↓g(C) is a SI-closed set of M . Lemma 5.8 Let (g, d) be a Galois connection between two complete lattices M and N , where d ∶ M → N and g ∶ N → M . If g preserves the sups of SI-closed sets, then d preserves the relation ≺SI . If M is CSI -continuous, then the converse conclusion is also true. By Proposition II-2.1 in [2], if f ∶ P → Q is a morphism in SCP O then f is continuous with respect to the irreducibly-derived topology. Thus the mapping f −1 ∶ ΓSI (Q) → ΓSI (P ) is well-defined and preserves arbitrary infs, so it is an upper adjoint. Similar to [4], we can obtain the following results Lemma 5.9 Let f ∶ P → Q be a morphism in SCPO and h ∶ ΓSI (P ) → ΓSI (Q) a lower adjoint of f −1 . Then h preserves the relation ≺SI . Proposition 5.10 (1) For all P, Q ∈ ob(SCP O) and all f ∈ HomSCP O (P, Q), let C(P ) = ΓSI (P ) and C(f ) = g, where g is a lower adjoint of f −1 , then C is a functor from the category SCP O to the category CP AlgSI , i.e., the following diagram. P C
ΓSI (P )
f
C(f )=g
/Q
C
/ ΓSI (Q)
(2) For all P, Q ∈ ob(CP AlgSI ) and all f ∈ HomCP AlgSI (P, Q), let K(P ) = κSI (P ) and K(f ) = f ∣κSI (P ) , then K is a functor from the category CP AlgSI to the category SCP O, i.e., the following diagram. P K
κSI (P )
f
/Q
C(f )=f ∣κSI (P )
K
/ κSI (Q)
Theorem 5.11 The functor K is a right adjoint to the functor C.
References [1] Davey, B.A., and H.A. Priestley, Introduction to Lattices and Order, second ed., Cambridge University Press, Cambridge, 2002. 8. [2] Gierz, G., K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott, Continuous Lattices and Domains. Cambridge University Press, 2003. [3] Ho Weng Kin, Jean Goubault-Larrecq, Achim Jung, and Xiaoyong Xi, The Ho-Zhao Problem, Logical Methods in Computer Science 14 1:7(2018) 1-19. [4] Ho Weng Kin, and Dongsheng Zhao, Lattices of Scott-closed sets, Comment. Math. Univ. Carolin. 2 (2009) 297-314. [5] Mao Xuxin, and Luoshan Xu, Order-theoretical Characterizations of Countably Approximating Posets, Int. J. Contemp. Math. Sciences 9 (2014) 447-454.
S. Su, Q. Li / Electronic Notes in Theoretical Computer Science 345 (2019) 233–247
247
[6] Xu Luoshan, and Xuxin Mao, Strongly continuous posets and the local Scott topology, Journal of Mathematical Analysis and Applications 345 (2008) 816-824. [7] Zhao Bin, Jing Lu, and Kaiyun Wang, The Answer to a Problem Posed by Zhao and Ho, Acta Mathematica Sinica, English Series 34 (2018) 1-7. [8] Zhao Dongsheng, and Taihe Fan, Dcpo-completion of posets, Theoretical Computer Science 411 (2010) 2167-2173. [9] Zhao Dongsheng, and Weng Kin Ho, On topologies defined by irreducible sets, Journal of Logical and Algebraic Methods in Programming 84 (2015) 185-195. [10] Zhao Dongsheng, and Luoshan Xu, Uniqueness of directed complete posets based on Scott closed set lattices, Logical Methods in Computer Science 14 2:10(2018) 1-12, DOI: 10.23638/LMCS-14(2:10). [11] Zhao Dongsheng, and Luoshan Xu, Faithfulness of directed complete posets based on Scott closed set lattices, arXiv preprint arXiv:1607.03576, (2016) 1-8.