A Stone-Type Duality for St0 Stratified Alexandrov L-Topological Spaces

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Fuzzy Sets and Systems 282 (2016) 1–20 www.elsevier.com/locate/fss

A Stone-type duality for sT0 stratiﬁed Alexandrov L-topological spaces

∗ Wei Yao a, , Sang-Eon Han b

a Department of Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018, PR China b Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-City Jeonbuk 561-756, Republic of Korea Received 7 November 2013; received in revised form 25 December 2014; accepted 26 December 2014 Available online 30 December 2014

Abstract We show that for every complete lattice A, both the set of completely prime elements and the set of completely coprime elements are one-to-one corresponding to CH(A), the set of complete lattice homomorphisms from A to the two-element lattice 2. A complete lattice is called completely generated, a cg-lattice for short, if it is generated by the set of completely prime elements, or equivalently, by the set of completely coprime elements. Then we restudy the duality between the category of T0 Alexandrov topological spaces and the category of cg-lattices by means of CH(A). With these preparations, for a frame L as the truth value table, we introduce sT0 separation axiom for stratified Alexandrov L-topological spaces, and finally establish a duality between the category of sT0 stratified Alexandrov L-topological spaces and the category of completely generated complete L-ordered sets. We also investigate some properties of the sT0 axiom, for example, it is hereditary by closed subspaces and productive. © 2014 Elsevier B.V. All rights reserved.

Keywords: Topology; Category theory; Duality; Complete (co)prime element; Complete lattice homomorphism; Completely generated lattice; (Complete) L-ordered set; Alexandrov L-topology; sT0 space

1. Introduction

In mathematics, a topological duality refers to a kind of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the name Stone duality, since they form a natural generalization of the famous Stone Representation Theorem for Boolean algebras [34]. Stone-type dualities also provide the foundation for pointfree topology and are exploited in theoretical computer science for the study of formal semantics [1,17]. The famous Papert–Papert–Isbell adjunction [14,26] between topological spaces and locales provides an appro- priate environment in which to develop both point-set topology and the theory of locales (pointfree topology). This

* Corresponding author. Tel.: +86 311 81668514. E-mail addresses: [email protected] (W. Yao), [email protected] (S.-E. Han). http://dx.doi.org/10.1016/j.fss.2014.12.012 0165-0114/© 2014 Elsevier B.V. All rights reserved. 2 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 adjunction also gives rise to the concept of sobriety, which plays an important role in several Stone Representation Theorems [17]. But there are a lot of interesting spaces which are not sober. For example posets taken with the Alexandrov topology are not always sober [17]. Domains, that is continuous dcpos,1 equipped with the Scott topology are sober spaces [9], but Johnstone [16] showed that not every dcpo with the Scott topology was sober. Therefore, it is interesting in both mathematics and theoretical computer science to study Stone duality for spaces which need not be sober. For example in [6], Bonsangue, Jacobs and Kok defined and studied observation frames and used them to establish dualities for T0 spaces, T1 spaces, open compact spaces, core compact spaces and T0 Alexandrov topological spaces, etc. Precisely in [6], it is shown that the duality for T0 Alexandrov topological spaces are those complete lattices order-generated by completely prime elements. Since for a frame A, the set of frame homomorphisms from A to the two-element lattice 2 ={0, 1} is bijective to the set of prime elements or completely prime filters of A [17], we could guess that for a complete lattice A, the completely coprime elements are closely related to some special maps from A to 2. Explicitly, for a complete lattice A, we would like to replace completely prime elements of A by certain special homomorphisms from A to 2 and then restudy the duality between T0 Alexandrov topological spaces and those complete lattices mentioned above. Such a restudy can help us to generalize the classical results into lattice-valued setting. For example, in locale theory in lattice-valued setting, by using L-frame homomorphisms instead of certain fuzzy version of prime elements, Yao [38] established a duality between spatial L-frames (defined by L-orders) and modified L-sober spaces, for L a fixed frame as the truth value table. ,In this paper we try to generalize the duality between T0 Alexandrov topological spaces and those complete lattices order-generated by completely prime elements [6] into lattice-valued setting. Firstly, we show that for every complete lattice A, both the set of completely prime elements and the set of completely coprime elements are one-to-one corresponding to CH(A), the set of complete lattice homomorphisms from A to 2. We show that for every complete lattice, it is order-generated by the set of completely prime elements if and only if it is order-generated by the set of completely coprime elements. Hence, we have reasons to call such a complete lattice a completely generated lattice, a cg-lattice for short. Then we restudy the duality between the category of T0 Alexandrov topological spaces and the category of cg-lattices by replacing the set of completely prime elements with CH(A). For lattice-valued setting with a fixed frame L as the truth value table, we construct an adjunction between the category of stratified Alexandrov L-topological spaces and the category of complete L-ordered sets. This adjunction induces a duality between the category of sT0 stratified Alexandrov L-topological spaces and the category of completely generated complete L-ordered sets, as desired.

2. Preliminaries

In this section, we will recall some basic concepts and results on categories, lattices, (fuzzy) topology and L-ordered sets, which will be used through out this paper.

2.1. Category theory

For category theory, we refer to [3]. Foro tw objects A and B in a category C, we would like to use [A, B]C to denote the set of C-morphism from A to B, by |C| the class of C-objects and by Mor(C) the class of C-morphisms. Let F : A −→ B be a functor and B ∈|B|. A pair (u, A) with A ∈|A| and u : B −→ F(A) ∈ Mor(B) is called universal for B with respect to F , provided that for every A ∈|A| and every B-morphism f : B −→ F(A), there exists a unique A-morphism f : A −→ A such that F(f) ◦ u = f . Dually, a pair (A, u) with A ∈|A| and u : F(A) −→ B ∈ Mor(B) is called co-universal for B with respect to F , provided that for every A ∈|A| and every B-morphism f : F(A) −→ B there exists a unique A-morphism f : A −→ A such that u ◦ F(f) = f . Let F : A −→ B and G : B −→ A be two functors. F is called a left adjoint of G (or G a right adjoint of F ) or (F, G) is an adjunction between A and B, in symbols F G : A B, if for every A ∈|A|, there exists a universal pair (uA, F(A))with respect to G, or equivalently, for every B ∈|B|, there exists a co-universal pair (G(B), uB ) with respect to F .

1 A poset is called a directed-complete poset [9] (a dcpo, for short) if every directed subset has a join. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 3

An equivalence between two categories A and B is a pair of functors F : A −→ B and G : B −→ A such that there ∼ ∼ are natural isomorphisms F ◦ G = idB and G ◦ F = idA.

2.2. Lattices, and fuzzy topology

For lattices and fuzzy topology, we recommend [9,12]. For a poset (P, ≤), its opposite (P op, ≤op) is defined as: for sets P = P op, and for ordered relation, a ≤ b iff b ≤op a. For a bounded lattice L, we always use ⊥L and L to denote the least (bottom) element and the largest (top) element, called the zero and the unit respectively. In a lattice L, an element a ∈ L is called coprime if for all b, c ∈ L, the inequality a ≤ b ∨c always implies a ≤ b or a ≤ c. The set of non-zero coprime elements of L is denoted by J(L). A complete lattice L is called completely distributive [9,29,30] if it satisfies the complete distributivity law, that is, (CD1) aij = aif(i) ∈ ∈ ∈ ∈ i I j Ji f i∈I Ji i I or (CD2) aij = aif(i) ∈ ∈ ∈ ∈ i I j Ji f i∈I Ji i I holds for all Si ={aij | j ∈ Ji} ⊆ L(i∈ I). It is well-known that completely distributivity is a self-dual concept, that is, a complete lattice (A, ≤) is completely distributive iff (Aop, ≤op) is completely distributive. 2 Foro tw elements a, b ∈ A, a is called wedge-below b [29,30], in symbols a ¡ b, if for every subset S of L, b ≤ S always implies a ≤ s for some s ∈ S. A complete lattice L is completely distributive iff b = β(b) holds ∗ for every b ∈ L, where β(b) ={a ∈ L | a ¡ b} [29,30]. Put β (b) = β(b) ∩ J(L), we have that if L is completely ∗ distributive then b = β (b) for every b ∈ L [36]. : −→ → : X −→ Y → = Let L be a complete lattice. For a map f X Y , define fL L L by fL (A)(y) f(x)=y A(y) ∀ ∈ X ← : Y −→ X ( A L ), called the L-valued Zadeh function or forward L-powerset operator; and define fL L L by ← = ◦ ∀ ∈ Y fL (B) B f(B L ), called the backward L-powerset operator [32]. A topological space is called Alexandrov if open sets are closed under arbitrary intersections, or equivalently, every point x has a minimal open set V(x) [17]. A topological space is called T0 if different points have different (open) neighborhood systems. For a poset P , the family of upper sets (resp., lower sets) is a topology on P , which we would like to call it the upper topology (resp., lower topology)3 for convenience. It is well-known that the upper (resp., lower) topology is generated by the basis {↑x | x ∈ P } (resp., {↓x | x ∈ P }), and both the upper topology and the lower topology are T0 Alexandrov topologies [17]. For a set X and an element a ∈ L, we use aX to denote the constant L-subset with the value a. Suppose that ⊗ is a binary operation on a lattice L. For a ∈ L and S ∈ LX, the notation a ⊗ S means the L-subset of X sending every x ∈ X to a ⊗ S(x). A stratified L-topology on a set X is a subfamily δ ⊆ LX that satisfies

(OS) aX ∈ δ for every a ∈ L; (O1) for all A, B ∈ δ, A ∧ B ∈ δ; { | ∈ } ⊆ ∈ (O2) for every Ai i I δ, i Ai δ.

A stratiﬁed L-topology is called Alexandrov if

2 The wedge below relation is also called a well-below relation [41], a totally-below relation [4], a long-below relation [42], a long-way-below relation [11], a completely-below relation [35], a superway-below relation [7] by some authors and is written as ≪, or ρ in literatures. 3 The upper topology and the lower topology in this paper are not the same as that in traditional sense. In [9], for a poset P , the upper topology (resp., lower topology) on P is the topology generated by the subbasis {P −↓x | x ∈ P } (resp., {P −↑x | x ∈ P }), which is coarser than the upper topology (resp., lower topology) in this paper in general. While in [2], the topology generated by the subbasis {P −↓x | x ∈ P } is called a weak lower topology. 4 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 { | ∈ } ⊆ ∈ (O3) for every Ai i I δ, i Ai δ. : −→ ← ∈ A map f (X, δ) (Y, ξ) between two stratiﬁed L-topological spaces is called continuous if fL (B) δ for every B ∈ ξ. We denote AL-Top as the category of stratiﬁed Alexandrov L-topological spaces with continuous maps as morphisms. For L = 2, A2-Top is written as ATop for simpleness.

2.3. (Complete) L-ordered sets

A complete lattice L is called a complete Heyting algebra or a frame, if the operation of binary meets is distributive ∈ ⊆ ∧ = ∧ over arbitrary joins, that is, for every x L and every S L, we have x S s∈S x s, or equivalently, there is a derived operation → on L such that for all a, b, c ∈ L, a ∧b ≤ c iff a ≤ b → c. It is well-known that, every completely distributive lattice is a frame. A frame homomorphism between two complete lattices is a map preserves ﬁnite meets and arbitrary joins. Note that for a frame homomorphism f : A −→ B, we have f( A) = B , f(⊥A) =⊥B (since the top element is the meet of the empty set and the bottom element is the join of the empty set). Complete Heyting algebras are always taken as the truth value table in lattice-valued mathematics since the operations ∧ and → have some remarkable logic meanings. Properties of (complete) Heyting algebras can be found in many literatures, e.g. [9,12,17].

Deﬁnition 2.1. (See [5,8,38].) An L-relation e on A is an L-subset of A × A. An L-relation e on A is called an L-order or a fuzzy order if for all a, b, c ∈ A,

(E1) e(a, a) = 1; (E2) e(a, b) ∧ e(b, c) ≤ e(a, c); (E3) e(a, b) = e(b, a) = 1 implies a = b.

The, pair (A e) is called an L-ordered set or a fuzzy poset [5,8,38].

,Let (A eA) and (B, eB ) be two L-ordered sets. We call a map f : A −→ B monotone if eA(a, b) ≤ eB (f (a), f(b)) for all a, b ∈ A. Let L-Ord be the category of L-ordered sets with monotone maps as morphisms.

Example 2.2. (1) Deﬁne eL : L × L −→ L by eL(a, b) = a → b for all a, b ∈ L. Then eL is an L-order on L. X (2) Let X be a set. For all A, B ∈ L , the subsethood degree [10] (cf. page 82 in [5]) of A in B is deﬁned by = → : X × X −→ X subX(A, B) x∈X A(x) B(x). Then subX L L L is an L-order on L .

A Deﬁnition 2.3. (See [5,8,38].) (1) Let (A, e) be an L-ordered set, a0 ∈ A and let S ∈ L . = ∈ = → (1a) The element a0 is called a join of S, in symbols a0 S, if for every b A, e(a0, b) a∈A S(a) e(a, b). = ∈ = → (1b) The element a0 is called a meet of S, in symbols a0 S, if for every b X, e(b, a0) a∈A S(a) e(b, a).

(2) An L-ordered set (A, e) is called complete or a complete L-ordered set if for every S ∈ LA, S (or equivalently, S) exists [38]. = ∧ = → Example 2.4. (1) The L-ordered set (L, eL) is complete, where S a∈L S(a) a and S a∈L S(a) a for every S ∈ LL; (2) Suppose δ is a stratiﬁed L-topology on X, then (δ, subX) is a complete L-ordered set, where S = S ∧ S = S → U∈δ (U) U (Example 3.4 in [38]), and if δ is Alexandrov then it is easy to show that U∈δ (U) U for every S ∈ Lδ.

Suppose that (A, e) is an L-ordered set. Deﬁne a binary relation ≤e on A by x ≤e y if e(x, y) = 1. Then (A, ≤e) is a classical poset, denoted by |A| for short. It is known that if (A, e) is a complete L-ordered set, then |A| is a complete lattice [38]. The following lemma is routine, which will be used in Section 5. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 5

Lemma 2.5. Let X be a set and let δ be a stratiﬁed Alexandrov L-topology on a set Y . If f : X −→ δ is a map, then ∈ X → = ∧ → = → for every S L , fL (S) x∈X f(x) S(x) and fL (S) x∈X f(x) S(x).

Proof. By Example 2.4(2), → = → ∧ = ∧ = ∧ fL (S) fL (S)(F ) F S(x) F S(x) f(x) F ∈δ F ∈δ f(x)=F x∈X and → = → → = → = → 2 fL (S) fL (S)(F ) F S(x) F S(x) f(x). F ∈δ F ∈δ f(x)=F x∈X

Deﬁnition 2.6. Let (A, eA) and (B, eB ) be two L-ordered sets and let f : A −→ B, g : B −→ A be two monotone maps. The pair (f, g) is called a fuzzy adjunction between A and B if eB (f (a), b) = eA(a, g(b)) for all a ∈ A, b ∈ B, where f is called the fuzzy left adjoint of g and dually g the fuzzy right adjoint of f .

: −→ → ← Example 2.7. (See [40].) Let f X Y be an ordinal map. Then (fL , fL ) is a fuzzy adjunction between X Y (L , subX) and (L , subY ).

Theorem 2.8. (See Theorem 3.5 in [40].) Let f : (A, eA) −→ (B, eB ) and g : (B, eB ) −→ (A, eA) be two maps between L-ordered sets. Then

= → (1) If A is complete, then f is monotone and has a fuzzy right adjoint if and only if f( S) fL (S) for every S ∈ LA. = → (2) If B is complete, then g is monotone and has a fuzzy left adjoint if and only if g(T) gL (T ) for every T ∈ LB .

3. Completely (co)prime element and complete lattice homomorphisms

In this section, we will study completely prime elements, completely coprime elements and complete lattice homomorphisms as well as their relations in complete lattices. Suppose that A is a complete lattice. An element a ∈ A is called completely coprime if a ¡ a [24]. That is, a ∈ A is completely coprime if for every subset S ⊆ A, a ≤ S always implies that a ≤ s for some s ∈ S. Dually, an element a ∈ A is called completely prime if for every subset S ⊆ A, S ≤ a always implies that s ≤ a for some s ∈ S. Let CJ(A) (resp., CM(A)) be the set of completely coprime (resp., completely prime) elements of A. We call a complete lattice A a cj-lattice (resp., cm-lattice) if a = {c ∈ CJ(A) | c ≤ a} (resp., a = {c ∈ CM(A) | a ≤ c}).

Proposition 3.1. (1) Every cj-lattice (resp., cm-lattice) is completely distributive. (2) Every ﬁnite distributive lattice is a cj-lattice (resp., cm-lattice).

Proof. (1) If x ∈ CJ(A) and x ≤ a, then it is easily seen that x ¡ a. That is to say, a = x ∈ CJ(A) x ≤ a ≤ {x ∈ A | x ¡ a}≤ {x ∈ A | x ≤ a}=a. Hence, A is completely distributive. (2) In finite cases, complete distributivity is the same as (finite) distributivity, and an element is coprime iff it is completely coprime. Since in every completely distributive lattice, the set of coprime elements is join-dense, we get that every finite distributive lattice is a cj-lattice. Since both complete distributivity and finite distributivity are self-dual concepts, (1) and (2) also hold for every cm-lattice. 2

In lattice theory, special elements are closely related to special maps. For example, given a frame A, every prime element can be uniquely characterized by a frame homomorphism from A to 2 [9]. 6 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20

A map between two complete lattice is called a complete lattice homomorphism if it preserves joins and meets of every subsets. We denote CH(A) as the set of complete lattice homomorphisms from a complete lattice A to 2. We will show that both CJ(A) and CM(A) can be uniquely characterized by CH(A).

Proposition 3.2. (CH(A), ≤) =∼ (CJ(A), ≤)op. −1 Proof. Step 1. Define f : CH(A) −→ CJ(A) by f(p) = p ({1}). For every p ∈CH(A), it is easily seen that −1 { } ≤ = ≤ = f(p)is the least element of p ( 1 ). If f(p) S, then 1 p(f (p)) p( S) s∈S p(s), which implies that p(s) = 1, or equivalently, f(p) ≤ s for some s ∈ S. Hence, f(p) ∈ CJ(A). That is to say, f is a well-defined map. Step 2. Define g : CJ(A) −→ CH(A) by g(a)(x) = 1iff a ≤ x, that is g(a) is the characteristic function of ↑a. It is a routine to show that g(a) ∈ CH(A) for every a ∈ CJ(A). Hence, g is a well-defined map. Step 3. (f ◦ g)(a) = x ∈ A g(a)(x) = 1 = {x ∈ A | a ≤ x}=a and − (g ◦ f )(p) =↑f(p)= x ∈ A p 1 {1} ≤ x = x ∈ A p(x) = 1 = p.

Step 4. For all a, b ∈ CJ(A), a ≤op b iff b ≤ a iff ↑a ⊆↑b iff g(a) ≤ g(b). 2

In [24], it is shown that CJ(A) and CM(A) are one-to-one corresponding. This result also can be implied by Proposition 3.2.

Corollary 3.3. (See also in [24].) (CJ(A), ≤) =∼ (CM(A), ≤).

Proof. Replacing (A, ≤) with (Aop, ≤op) in Proposition 3.2 (in fact for sets, A and Aop are the same), we have (CH(Aop), ≤op) =∼ (CJ(Aop), ≤op)op. Since for sets, CH(Aop) = CH(A) and CJ(Aop) = CM(A). Then (CH(A), ≤op) =∼ (CM(A), ≤op)op and CH(A), ≤ =∼ CH(A), ≤op op =∼ CM(A), ≤op =∼ CM(A), ≤ op. Hence, (CJ(A), ≤) =∼ (CM(A), ≤). 2

Remark 3.4. (1) As has already been claimed in [24], for q ∈ CJ(A), the corresponding completely prime element of = \↑ ∈ A is computed as Rv(q) (A q) and dually, for p CM(A), the corresponding completely coprime element of A is computed as Ru(p) = (A\↓p), where the notations Rv and Ru are taken from [15]. (2) The transformations between (CH(A), ≤) and (CM(A), ≤) are f −1({0}) ∈ CM(A) for every f ∈ CH(A), and A\↓a ∈ CH(A) for every a ∈ CM(A). (3) By Corollary 3.3, for every ﬁnite lattice, the set of prime elements and the set of coprime element are one-to-one corresponding. This fact has already been claimed in [15].

Similar to complete distributivity and ﬁnite distributivity, we can show that the concept of cj-lattices (resp., cm- lattices) is self-dual.

Proposition 3.5. A complete lattice is a cj-lattice iff it is a cm-lattice. That is, the concept of a cj-lattice (resp., cm-lattice) is self-dual. Proof. Suppose that A is a cj-lattice. For a ∈ A, we shall show that a = {x ∈ CM(A) | a ≤ x}. Put b = {x ∈ CM(A) | a ≤ x}. We only need to show that a ≥ b. Since A is a cj-lattice, it sufﬁcient to show that for every y ∈ CJ(A), if y ≤ b then y ≤ a. In fact, for y ∈ CJ(A) ∩↓b, we have (A\↑y) ∈ CM(A). If y a, then a ∈ A\↑y and a ≤ (A\↑y). By the deﬁnition of b and by Remark 3.4(1), we have b ≤ (A\↑y) and then y ≤ (A\↑y). Since y is completely coprime, we have y ≤ z for some z ∈ A\↑y. But it is impossible. Hence, A is a cm-lattice. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 7

Conversely, if A is a cm-lattice, then Aop is a cj-lattice and then Aop is a cm-lattice, and consequently, A is a cj-lattice. 2

wFrom no on, a cm-lattice or a cj-lattice will be called a completely generated lattice, a cg-lattice for short. Let CLat be the category of complete lattices with complete lattice homomorphisms and CGLat be the full subcategory of CLat consisted of cg-lattices. It is easy to show that every complete lattice homomorphism preserves both the top and the bottom elements.

4. A duality between T0 Alexandrov topological spaces and cg-lattices: a restudy

A duality between the category of T0 Alexandrov topological spaces and the category of cg-lattices deﬁned by using completely prime elements has already been established in [6]. In this section, we will restudy this duality by replacing CM(A) with CH(A). This approach can help us in generalizing this duality into lattice-valued setting.

Proposition 4.1. Suppose that f : (X, T ) −→ (Y, S ) be a continuous map between two Alexandrov topological spaces. Then f −1 : S −→ T is a complete lattice homomorphism.

Proof. This is trivial since both (X, T ) and (Y, S ) are closed under arbitrary unions and intersections, which are preserved by f −1 : 2Y −→ 2X. 2

Wew no get a functor V : ATop −→ CLatop given by − V(X, T ) = (T , ⊆) and V(f ) = f 1 op. Conversely,

Proposition 4.2. Suppose that A is a complete lattice and equip CH(A) with the family Φ(A) := {Φ(a) | a ∈ A}, where Φ(a) ={p ∈ CH(A) | p(a) = 1}. Then (CH(A), Φ(A)) (sometimes we just write it as CH(A) for simpleness) is an Alexandrov topological space. rProof. Fo every subset {ai | i ∈ I} ⊆ A,

Φ(ai) = p ∈ CH(A) ∃i ∈ I, p(ai) = 1 = p ∈ CH(A) p ai = 1 = Φ ai i i i and

Φ(ai) = p ∈ CH(A) ∀i ∈ I, p(ai) = 1 = p ∈ CH(A) p ai = 1 = Φ ai . i i i

Clearly, Φ(⊥A) =∅and Φ( A) = CH(A). 2

Theorem 4.3. Suppose that g : A −→ B is a complete lattice homomorphism. Deﬁne G(g) : CH(B) −→ CH(A) by G(g)(p) = p ◦ g(∀p ∈ CH(B)). Then G(g) : CH(B) −→ CH(A) is a continuous map.

Proof. It is easily seen that G(g) is a well-deﬁned map. For every a ∈ L, − G(g) 1 Φ(a) = p ∈ CH(B) (p ◦ g)(a) = 1 = Φ g(a) . Hence, G(g) : CH(B) −→ CH(A) is a continuous map. 2

Wew no get a functor G : CLatop −→ ATop given by G(A) = CH(A), Φ(A) and G(g)(p) = p ◦ gop.

Theorem 4.4. The pair V G : ATop CLatop is an adjunction. 8 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20

Proof. The proof is similar to that of the Papert–Papert–Isbell adjunction [14,26], and also is a special case of Theo- rem 5.5 in this paper. Here we leave it to the readers. 2

Like the Papert–Papert–Isbell adjunction, the adjunction V G also can induce an equivalence between a subcategory of ATop and a subcategory of CLatop.

Proposition 4.5. The lattice of open sets of an Alexandrov topological space is a cg-lattice.

Proof. It is sufﬁcient to show that, in an Alexandrov topological space (X, T ), K ⊆ X is a completely coprime T ⊆ = ∈ = element of ( , ) iff K V(x) for some x X. In fact, every open set U can be written as U x∈U V(x). If U is a completely coprime element in (T , ⊆), then U = V(x) for some x ∈ U. Conversely, for every x ∈ X, if ⊆ { | ∈ } ⊆ T ∈ ∈ ⊆ V(x) i Ui for Ui i I , which implies that x Ui0 for some i0 I and then V(x) Ui0 by the minimality of V(x). That is to say, every V(x)is completely coprime. 2

Proposition 4.6. For every complete lattice A, Φ(A) exactly is the upper topology on CH(A), thus it is homeomorphic to the lower topology on (CJ(A), ≤). Therefore, CH(A) is a T0 space.

Proof.r Fo every a ∈ L, it is easily seen that Φ(a) is an upper set of CH(A). Conversely, since the family {↑p | p ∈ CH(A)} is a basis of the upper topology, we only need to show that, there exists a ∈ L such that ↑p = Φ(a) for every p ∈ CH(A). In fact, let a = p−1({1}). Then a is the least element of p−1({1}), and we have q ∈↑p iff p ≤ q iff q(a) = 1iff q ∈ Φ(a). Hence, ↑p = Φ(a). 2

Letp ATo 0 be subcategory of ATop consisted of T0 spaces and let CGLat be the subcategory of CLat consisted of cg-lattices. By Propositions 4.5 and 4.6, the functors V : ATop −→ CLatop and G : CLatop −→ ATop can be : −→ op : op −→ restricted to V ATop0 CGLat and G CGLat ATop0, respectively. As expected, we will show that the op restricted pair of functors (V, G) is an equivalence between CGLat and ATop0, that is a duality between ATop0 and CGLat.

Proposition 4.7. Suppose that (X, T ) is a T0 Alexandrov topological space. Then the space CH(T ) is homeomorphic to X.

Proof. By Proposition 4.6, it is equivalent to show that CJ(T ) equipped with the lower topology is homeomorphic to (X, T ). Deﬁne V : X −→ CJ(T ) by V(x) = V(x). Step 1. By the proof of Proposition 4.5, V is a surjective map. For all x, y ∈ X, if V(x) = V(y), by the minimality of open neighborhoods, we have that x and y possess a same open neighborhood system. It follows that x = y since X is a T0 space. VStep 2. is continuous. Suppose that U ⊆ CJ(T ) is a lower set. Then V−1(U) ={z ∈ X | V(z) ∈ U}. We need to show that for every x ∈ V−1(U), it holds that V(x) ⊆ V−1(U). In fact, if y ∈ V(x), then y ∈ V(y) ⊆ V(x) ∈ U, which implies that V(y) ∈ U, and y ∈ V−1(U) since U is a lower set. Hence, V(x) ⊆ V−1(U). ∈ T = ={ | ∈ } VStep 3. is open. Suppose that U . Then U x∈U V(x). We need to show that V(U) V(x) x U is a lower set of (CJ(T ), ⊆). In fact, suppose that V(y) ⊆ V(x) ∈ V(U) for some x ∈ U. We have y ∈ V(y) ⊆ V(x) ⊆ U. Hence, V(y) ∈ V(U). 2

Lemma 4.8. A complete lattice A is a cg-lattice if and only if for all a, b ∈ A, Φ(a) = Φ(b) implies a = b.

Proof. Necessity: We only need to show that if Φ(a) ⊆ Φ(b) then a ≤ b. Suppose that x ∈ CJ(A), if x ≤ a, then ↑x ∈ CH(A) (here ↑x is considered as its characteristic function) and a ∈↑x, which implies that ↑x ∈ Φ(a) ⊆ Φ(b) and then x ≤ b. Since A is a cg-lattice, we have a ≤ b. Sufﬁciency: We need to show that, for every a ∈ A, a = {x ∈ CJ(A) | x ≤ a}. Put b = {x ∈ CJ(A) | x ≤ a}. For p ∈ CH(A), put c = p−1({1}), we have c ∈ CJ(A) and W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 9

p ∈ Φ(b) iff p(b) = 1iffc ≤ b iff ∃x ∈ CJ(A) ∩↓a s.t. c ≤ x iff p(a) = 1iffp ∈ Φ(a). Hence, a = b = {x ∈ CJ(A) | x ≤ a}. 2

Proposition 4.9. Suppose that A is a cg-lattice, then A =∼ Φ(A).

Proof. Clearly, Φ : A −→ Φ(A) is a well-deﬁned surjective complete lattice homomorphism. By Lemma 4.8, Φ also is injective. 2

By Theorem 4.4, Propositions 4.6, 4.7 and 4.9, we have

op Theorem 4.10. ATop0 is equivalent to CGLat .

5. A lattice-valued generalization of the duality between ATop0 and CGLat

In this section, the truth value table L is assumed to be a complete Heyting algebra. We will generalize the results in Section 4 into lattice-valued setting. Suppose f : (A, e1) −→ (B, e2) is a map between two complete L-ordered sets. If f preserves joins and meets S = → S S = → S S ∈ A of every L-subsets, that is, f( ) fL ( ) and f( ) fL ( ) hold for every L , then f is called a complete L-ordered set homomorphism. It is easy to show that, if f : (A, e1) −→ (B, e2) is a complete L-ordered set homomorphism, then f :|A| −→ | B| is a complete lattice homomorphism (cf. Section 3 in [39]). Let L-CLat be the category of complete L-ordered sets with complete L-ordered set homomorphisms. It is a routine to show that L-CLat is a subcategory of L-Ord.

: −→ ← : Y −→ X Proposition 5.1. For every map f X Y , the map fL (L , subY ) (L , subX) is a morphism in L-CLat.

← : Y −→ X → : X −→ Y Proof. Since fL (L , subY ) (L , subX) is the fuzzy right adjoint of fL (L , subX) (L , subY ), we ← : Y −→ X S ∈ (LX) have that fL (L , subY ) (L , subX) preserves meets of L-subsets. For L , by Lemma 2.5, ← → S = S ∧ ← fL L ( ) (V ) fL (V ) ∈ Y VL = S(V ) ∧ (V ◦ f) V ∈LY = S(V ) ∧ V ◦ f V ∈LY = ← S ∧ fL (V ) V ∈ Y V L = ← S 2 fL .

For (X, δ) a stratiﬁed Alexandrov L-topological space, since (δ, subX) is a complete L-ordered set, we get a map VL :|AL-Top| −→ | L-CLat|, (X, δ) → (δ, subX). By Proposition 5.1, VL is a functor indeed, that is

Proposition 5.2. The assignment

V : AL-Top −→ L-CLatop, L : −→ −→ ← op : −→ f (X, δ) (Y, η) fL (δ, subX) (η, subY ) is a functor. 10 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20

Conversely, for (A, e) a complete L-ordered set, put

CHL(A) ={p : A −→ L | p is a morphism in L-CLat}.

For every a ∈ A, deﬁne ΦL(a) : CHL(A) −→ L by p → p(a). Then

Proposition 5.3. ΦL(A) ={ΦL(a) | a ∈ A} is a stratiﬁed Alexandrov L-topology on CHL(A).

∈ S = ∈ A = S ∈ = Proof. (OS) For every λ L, put λA L and a0 A. Then ΦL(a0) λCHL(A). In fact, for every p ∈ CHL(A), = = S = → S = ∧ S = ∧ = ΦL(a0)(p) p(a0) p pL ( ) p(a) (a) λ p(a) λ. a∈A a∈A :| | −→ = = | | Notice that p A L is morphism in CLat and a∈A p(a) p( A) L, where A is the top element of A . Let {ai | i ∈ I} ⊆ L.

(O2)r Fo every p ∈ CHL(A), ΦL ai (p) = p ai = p(ai) = ΦL(ai)(p). i i i i = ∈ Hence, i ΦL(ai) ΦL( i ai) ΦL(A).

(O3)r Fo every p ∈ CHL(A), ΦL ai (p) = p ai = p(ai) = ΦL(ai)(p). i i i i = ∈ 2 Hence, i ΦL(ai) ΦL( i ai) ΦL(A).

op op Proposition 5.4. Let f : (A, eA) −→ (B, eB ) be a morphism in L-CLat , that is, f : (B, eB ) −→ (A, eA) is a op morphism in L-CLat. Then GL(f ) : CHL(A) −→ CHL(B), p → p ◦ f is a continuous map. rProof. Fo every b ∈ B and every p ∈ CHL(A), ← = = ◦ op = ◦ op = op GL(f ) L ΦL(b) (p) ΦL(b) GL(f )(p) ΦL(b) p f p f (b) ΦL f (b) (p) and then ← = op ∈ 2 GL(f ) L ΦL(b) ΦL f (b) ΦL(A).

op By Propositions 5.3 and 5.4, we get a functor GL : L-CLat −→ AL-Top sending f : (A, eA) −→ (B, eB ) to op GL(f ) : CHL(A) −→ CHL(B) (p → p ◦ f ). As expected, we have

op Theorem 5.5. VL GL : AL-Top L-CLat .

op Proof.r Fo every (B, eB ) ∈|L-CLat |, we know that GL(B) = CHL(B), ΦL(B) ∈|AL-Top|. op We will show that there exists an L-CLat -morphism uB : (VL ◦ GL)(B) −→ B which is co-universal with respect to GL. = op : ◦ −→ ∈ op : −→ ∈ Step 1. Put uB ΦL (VL GL)(B) B, we have uB Mor(L-CLat ), that is, ΦL B (ΦL(B), subB ) Mor(L-CLat). For every S ∈ LB , W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 11

(1.1) For every p ∈ CHL(B), S = S = → S = S ∧ = S ∧ ΦL (p) p pL ( ) (b) p(b) (b) ΦL(b)(p). b∈B b∈B Then by Lemma 2.5, we have, S = S ∧ = → S ΦL (b) ΦL(b) (ΦL)L ( ). b∈B

(1.2) For every p ∈ CHL(B), S = S = → S = S → = S → ΦL (p) p pL ( ) (b) p(b) (b) ΦL(b)(p). b∈B b∈B Then by Lemma 2.5, we have S = S → = → S ΦL (b) ΦL(b) (ΦL)L ( ). b∈B op op Step 2. For every (X, δ) ∈|AL-Top| and every g : VL(X, δ) −→ (B, eB ) ∈ Mor(L-CLat ), or equivalently, g : (B, eB ) −→ (δ, subX) ∈ Mor(L-CLat), there is a unique f : (X, δ) −→ CHL(B) such that uB ◦ VL(f ) = g. op (2.1) Deﬁne f : X −→ CHL(B) by f (x)(b) = g (b)(x) (∀x ∈ X, b ∈ B). Then f is a continuous map. Indeed, (a) f is a map, that is, for every x ∈ X, f(x) : B −→ L ∈ Mor(L-CLat). For every S ∈ LB , by Lemma 2.5, we have f(x) S = gop S (x) = op → S g L ( ) (x) = S(b) ∧ gop(b) (x) ∈ b B = S(b) ∧ gop(b)(x) ∈ bB = S(b) ∧ f (x)(b) ∈ b B = → S f(x) L ( ) and f(x) S = gop S (x) = op → S g L ( ) (x) = S(b) → gop(b) (x) ∈ b B = S(b) → gop(b)(x) ∈ bB = S(b) → f (x)(b) ∈ b B = → S f(x) L ( ).

Hence, f(x) ∈ CHL(B). 12 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20

(b) f ∈ Mor(AL-Top). Indeed, for every ΦL(b) ∈ ΦL(B) and every x ∈ X, ← = = = op fL ΦL(b) (x) ΦL(b) f(x) f (x)(b) g (b)(x). ← = op ∈ Thus, fL (ΦL(b)) g (b) δ. = ← op : −→ = op ◦ = ← ◦ (2.2) By VL(f ) (fL ) δ ΦL(B) and uB (ΦL) , the equality uB VL(f ) g is equivalent to that fL op ΦL = g , which is valid by (2.1)(b). Finally, the uniqueness of f is clear. 2

Now, we will use the standard technique as that in Papert–Papert–Isbell adjunction to construct the duality induced by the adjunction VL GL. X On one hand, for every x ∈ X, deﬁne ΨL(x) : L −→ L by ΨL(x)(U) = U(x).

Proposition 5.6. Let (X, δ) be a stratiﬁed Alexandrov L-topological space. Then the following statements are equivalent,

(1) ΨL : X −→ CHL(δ) is a bijection. (2) ΨL : (X, δ) −→ (CHL(δ), ΦL(δ)) is a homeomorphism.

Proof. We only need to show (1) ⇒ (2). Since ΨL is a bijection, it sufﬁces to show that the map ΨL : (X, δ) −→ (CHL(δ), ΦL(δ)) is open and continuous. Firstly, for every U ∈ δ and every p = ΨL(x) ∈ CHL(δ) (x ∈ X), we have → = → = = = (ΨL)L (U)(p) (ΨL)L (U) ΨL(x) U(x) p(U) ΦL(U)(p). → = ∈ ∈ ∈ Then (ΨL)L (U) ΦL(U) ΦL(δ). Therefore, ΨL is open. Secondly, for every ΦL(U) ΦL(δ) (U δ) and every x ∈ X, we have ← = = = (ΨL)L ΦL(U) (x) ΦL(U) ΨL(x) ΨL(x)(U) U(x). ← = ∈ 2 Then (ΨL)L (ΦL(U)) U δ. Therefore, ΨL is continuous.

We call a stratiﬁed Alexandrov L-topological space an sT0 space if ΨL : X −→ CHL(δ) is a bijection.

Proposition 5.7. If (A, e) is a complete L-ordered set, then (CHL(A), ΦL(A)) is an sT0 space.

Proof. We need to show that ΨL : CHL(A) −→ CHL(ΦL(A)) is a bijection. Firstly, for every p, q ∈ CHL(A), if p = q, then there exists a ∈ A such that p(a) = q(a). Then for ΦL(a) ∈ ΦL(A), we have ΨL(p) ΦL(a) = ΦL(a)(p) = p(a) = q(a) = ΦL(a)(q) = ΨL(q) ΦL(a) .

Thus, ΨL is injective. Secondly, for every q ∈ CHL(ΦL(A)), put p = q ◦ ΦL, by the proof of Step 1 in Theorem 5.5, ΦL : (A, e) −→ (ΦL(A), subA) is an L-CLat-morphism and then p ∈ CHL(A). In the following, we will show that ΨL(p) = q. In fact, for every U = ΦL(a) with a ∈ A, we have ΨL(p)(U) = ΦL(a)(p) = p(a) = q ΦL(a) = q(U).

Hence, ΨL is surjective. 2

On the other hand,

Proposition 5.8. Let (A, e) be a complete L-ordered set. Then the followings are equivalent,

(1) ΦL : A −→ ΦL(A) is injective. (2) ΦL : (A, e) −→ (ΦL(A), subA) is an isomorphism in L-CLat. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 13

Proof. Since ΦL : A −→ ΦL(A) is surjective, we only need to show that, if ΦL : A −→ ΦL(A) is bijective, then A ΦL : (A, e) −→ (ΦL(A), subA) is an isomorphism in L-CLat. In fact, let U ∈ L . For every c ∈ L, it is easy to show → U = U ∈ that (ΦL)L ( )(ΦL(c)) (c) since ΦL is a bijection. For every p CHL(A), again since ΦL is a bijection and by Lemma 2.5, we have → U = U ∧ = ∧ U = → U = U = U (ΦL)L ( ) (p) (c) ΦL(c)(p) p(c) (c) pL ( ) p ΦL (p). c∈A c∈A → U = U Hence, (ΦL)L ( ) ΦL( ). Again by Lemma 2.5, → U = U → (ΦL)L ( ) (p) (c) ΦL(c)(p) c∈A = U → = → U = U = U (c) p(c) pL ( ) p ΦL (p). c∈A → U = U 2 Hence, (ΦL)L ( ) ΦL( ).

A complete L-ordered set is called completely generated if it satisﬁes the conditions in Proposition 5.8.

Proposition 5.9. If (X, δ) is a stratiﬁed Alexandrov L-topological space, then (δ, subX) is a completely generated complete L-ordered set. rProof. Fo U = V in δ, there exists x ∈ X such that U(x) = V(x). Put p = ΨL(x) ∈ CHL(δ), we have

ΦL(U)(p) = p(U) = ΨL(x)(U) = U(x)= V(x)= ΨL(x)(V ) = p(V ) = ΦL(V ). 2

By AL-Tops0, we denote the full subcategory of AL-Top consisted of sT0 stratiﬁed Alexandrov L-topological spaces, and by L-CGLat, the full subcategory of L-CLat consisted of completely generated complete L-ordered sets. By Theorem 5.5 and Propositions 5.6–5.9, we have

op Theorem 5.10. AL-Tops0 is equivalent to L-CGLat .

6. Discussions and remarks

In this section, we will discuss the relation between posets and cg-lattices, the relations among sT0 axiom, L-T0 axiom and fuzzy sobriety, and some further properties of sT0 stratiﬁed Alexandrov L-topological spaces.

6.1. Categorical relations between posets and cg-lattices

In 1966, McCord [25] and Steiner [33] independently observed the isomorphism between the category Pos of posets : −→ and the category ATop0 of T0 Alexandrov topological spaces. The related functor Γ Pos ATop0 is deﬁned as Γ(P) = (P, Γ(P)) to be the upper topological space on (P, ≤) and the functor Ω : ATop0 −→ Pos is deﬁned as − Ω(X, T ) = (X, ≤T ), where ≤T is the specialization order [9] on X given by x ≤T y iff x ∈{y} . Consequently, ATop0 and Pos are isomorphic through the pair of functors (Ω, Γ). Now, combining the isomorphism functors (Ω, Γ)and the equivalence functors (V, G) together, we get a duality between Pos and CGLat given by: the functor − E : Pos −→ CGLatop by E(P ) = Γ(P),⊆ and E(f ) = f 1 op, and the functor K : CGLatop −→ Pos by K(A) = CH(A), ≤ and K(g) = ()◦ gop. 14 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20

This duality is, in some sense, a little bit like the duality between the category POID of posets with maps under which inverse images of ideals are ideals and the category AlgDomG of algebraic domains with Scott-continuous maps having a lower adjoint [9]. The duality between Pos and CGLat can also be generalized into lattice-valued setting. Some preliminary attempts have been made in [18,37].

6.2. Relations between sT0 spaces, L-T0 spaces and modiﬁed L-sober spaces

The classical T0 axiom has been generalized to L-topological spaces by many scholars (see [13,19–23,27,31], etc.). Among them, an L-topology (X, δ) is called L-T0 space [31] if for any two points x, y ∈ X, if U(x) = U(y) holds for every U ∈ δ, then x = y. Rodabaugh [31] demonstrated that the L-T0 axiom categorically behaved exactly like the classical T0 axiom in his generalizations of the Isbell and Stone Representation Theorems and of the Stone-Cechˇ compactification reflector. Lowen and Srivastava [22,23] proved that the L-T0 axiom categorically behaved exactly like the classical T0 axiom with regard to the epireflective hull of the Sierpinski space in saturated I -topological spaces. Also, Yao [39] showed that an L-topological space is L-T0 iff the specialization order is an L-order, as a generalization of the corresponding classical result. In classical situation, the duality of completely generated complete lattices are T0 Alexandrov topological spaces, while the duality of completely generated complete L-ordered sets are sT0 stratified Alexandrov L-topological spaces. A natural question is that, what is the relation between L-T0 axiom and sT0 axiom? By the definition of sT0 spaces, it is easy to show that every sT0 stratified Alexandrov L-topological space is an L-T0 space. But the inverse is not true as the following Example 6.2 shows. Firstly, we need a lemma.

Lemma 6.1. Suppose that (L, ) is a Boolean algebra with be the negation. Then for two stratiﬁed Alexandrov L-topological spaces (X, δ) and (Y, η), a map F : δ −→ η is a morphism in L-CLat iff F : δ −→ η is a morphism in CLat with F(aX) = aY for every a ∈ L. Proof. The necessity: For S ⊆ δ, we have S = χS and S = χS . Then by Lemma 2.5, we have, S = = → = ∧ = F F χS FL (χS ) F(A) χS (A) F(A) A∈δ A∈δ and S = = → = → = F F χS FL (χS ) χS (A) F(A) F(A). A∈δ A∈δ = ∈ And similar to Proposition 5.3, we can show that F(aX) aY for every a L. S ∈ δ S = S ∧ S = S → = S ∨ The sufﬁciency: Let L , we have A∈δ (A) A and A∈δ (A) A A∈δ( (A)) A. Then by Lemma 2.5, we have S = S ∧ = S ∧ = S ∧ = → S F F (A) A F (A) F(A) (A) F(A) FL ( ) A∈δ A∈δ A∈δ and S = S ∨ = S ∨ = S → = → S F F (A) A F (A) F(A) (A) F(A) FL ( ). A∈δ A∈δ A∈δ Hence, F is a morphism in L-CLat. 2

The following example is a modiﬁcation of Example 4.6 in [39]. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 15

Example 6.2. Let X ={x, y} and L ={0, a, b, 1} be the simplest non-trivial Boolean algebra. Deﬁne a stratiﬁed L-topology δ (of course Alexandrov) on X ordered as in Fig. 1. The map p in Fig. 2 is a complete lattice homomorphism from δ to L. The values at every point in Fig. 2 are those under p of the corresponding open set in (X, δ) at the same place in Fig. 1. By Lemma 6.1, we know that p is a complete L-ordered set homomorphism from (δ, subX) to (L, eL) (considering L as a one-point L-topological space), which obviously does not equal to ΨL(z) for any z ∈ X. Therefore (X, δ) is not an sT0 space, but an L-T0 space.

Fig. 1. An L-topology δ on X.

Fig. 2. p ∈ CHL(δ).

In classical situation, sober spaces are a kind of spaces between T0 spaces and T2 spaces, that is, sobriety can be considered as a separation axiom which is stronger than T0 axiom and weaker than T2 axiom. It is also a link among topology theory, order theory and domain theory, and is very important in several Stone Representation Theorems [9,17]. Up to now, there are three basic fuzzy versions of sobriety: L-sobriety, modified L-sobriety and ιL-sobriety [28]. In [39], Yao has done a detail investigation of the relations among these three fuzzy sobriety. Therein, the modified L-sobriety is the best one since it possesses the most properties of the classical sobriety than the others, for example the category of modified L-sober spaces is dually equivalent to the category of spatial L-frames, and every fuzzy domain equipped with the fuzzy Scott topology is modified L-sober. For a stratified L-topological space (X, δ), let ptL(δ) be the family of L-frame homomorphisms, that are maps p : (δ, subX) −→ (L, eL) that preserving finite meets of δ and joins of L-subsets of δ, or equivalently, p : (δ, ≤) −→ (L, ≤) is a frame homomorphism preserving the constant L-subsets [39]. An L-topological space (X, δ) is called modified L-sober if ΨL : X −→ ptL(δ) is a bijection. Clearly for a stratified Alexandrov L-topological space (X, δ), we have {ΨL(x) | x ∈ X} ⊆ CHL(δ) ⊆ ptL(δ). Hence, we have that, for every stratified Alexandrov L-topological space, modified L-sobriety always implies sT0 axiom, but of course not vice versa even for the crisp setting. For a summary, we draw a diagram to show the relations among sT0 spaces, L-T0 spaces and modified L-sober spaces as follows: - - Modified L-sober ××sT0 L-T0

6.3. Properties of sT0 stratiﬁed Alexandrov L-topological spaces

Let (Y, δ) be an L-topological space and let X ⊆ Y . Then the family δ|X ={A|X | A ∈ δ} is an L-topology on X, X where A|X ∈ L is the restriction of A ∈ δ on X, that is, the L-subset sending x ∈ X to A(x). In this case, (X, δ|X) 16 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 is called a subspace of (Y, δ). It is easy to show that, if (Y, δ) is stratified and Alexandrov, then so is the subspace (X, δ|X). | Lemma 6.3. Suppose that (Y, δ) is a stratified Alexandrov L-topological space and (X, δ X) is a subspace of (Y, δ). | For A ∈ Lδ, define A∈ L(δ X) by A(V ) = A(U). Then A= ( A)| and A= ( A)| . U|X=V X X Proof. A= A(V ) ∧ V = A(U) ∧ V ∈ | ∈ | | = V δ X V δ X U X V

= A(U) ∧ (U|X) = A(U) ∧ U = A X U∈δ U∈δ X and A= A(V ) → V = A(U) → V ∈ | ∈ | | = V δ X V δ X U X V

= A(U) → (U|X) = A(U) → U = A . 2 X U∈δ U∈δ X

Theorem 6.4. Suppose that (Y, δ) is a stratiﬁed Alexandrov L-topological space and X is a closed subspace of Y (i.e., X ∈ δ). If (Y, δ) is sT0, then so is (X, δ|X).

δ Proof. Suppose that g : δ|X −→ L ∈ CHL(δ|X). Deﬁne g : δ −→ L by g(U) = g(U|X). Then for every A ∈ L , by Lemma 6.3,

A = A = A = → A g g g gL ( ) X = g(V ) ∧ A(V )

V ∈δ|X = g(V ) ∧ A(U) ∈ | | = Vδ X U X V = g(U|X) ∧ A(U) U∈δ = g(U) ∧ A(U) U∈δ = → A gL ( ) and

A = A = A = → A g g g gL ( ) X = A(V ) → g(V )

V ∈δ|X = A(U) → g(V ) ∈ | | = Vδ X U X V = A(U) → g(U|X) U∈δ = A(U) → g(U) U∈δ = → A gL ( ). W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 17

That is to say, g ∈ CHL(δ), and then g = ΨL(x) for a unique x ∈ Y . Since X ∈ δ, we have

X (x) = ΨL(x)( X ) = g( X ) = g( X |X) = g(0X) = 0.

It follows that x ∈ X. Hence, it is easily seen that g = ΨL(x). 2

Example 6.5. Let Y ={x, y, u, v} and L ={0, a, b, 1} be the same as that in Example 6.2. Deﬁne a stratiﬁed Alexan- drov L-topology σ on Y ordered as in Fig. 3. The map p in Fig. 4 is a complete lattice homomorphism from δ to L. The values at every point in Fig. 4 are those under p of the corresponding open sets in (Y, σ) at the same place in Fig. 3. By Lemma 6.1, we know that p is a complete L-ordered set homomorphism from (σ, subY ) to (L, eL). In4, Fig. the value λ has two possibilities 1or a, and the value η also has two possibilities 1or b. The other values are completely determined by λ and η. Hence, there are four elements in CHL(σ ), that is, |CHL(σ )| = 4. Since {ΨL(y) | y ∈ Y } ⊆ CHL(σ ) and Y exactly has four elements, we have {ΨL(y) | y ∈ Y } = CHL(σ ). Therefore 4 ΨL : Y −→ CHL(σ ) is a bijection and (Y, σ)is an sT0 space.

Fig. 3. The L-topology δ on X.

Fig. 4. p ∈ CHL(δ).

It is easy to check that (X, δ) in Example 6.2 is a subspace of (Y, η). Combining this result and Example 6.5 together, we see that a subspace of an sT0 stratified Alexandrov L-topological space need not be sT0. Hence, sT0 axiom is not hereditary, although it is hereditary by closed subspaces (Theorem 6.4). Readers may have doubt in their mind, why classical T0 axiom is hereditary but sT0 axiom is not? The reason is that, the definition of sT0 axiom is somewhat like the definition of modified L-sobriety. In fact, it should be considered as a counterpart of modified L-sobriety in Alexandrov case, rather than a generalization of the classical T0 axiom. We know that, both the classical sobriety and the modified L-sobriety are only hereditary by closed subspaces [9,39]. In the following, we will show that, if L is completely distributive, then sT0 axiom is productive.

Proposition 6.6. Let L be a completely distributive lattice. Suppose that {(Xi, δi) | i ∈ I} is a family of Alexandrov = S ={ |∀ ∈ ∈ } L-topological spaces. Let X i Xi and let ΠiAi i I, Ai δi . Then the L-topology δ on X generated by the family S as a basis is Alexandrov, which is precisely the product object in AL-Top.

Proof. Step 1. S is closed under meets. Suppose that {ΠiAij | j ∈ J } ⊆ S. For every x = (xi)i∈I ∈ X, we have

4 In fact, ΨL(x) is corresponding to λ = η = 1; ΨL(y) is corresponding to λ = a, η = b; ΨL(u) is corresponding to λ = 1, η = b; and ΨL(v) is corresponding to λ = a, η = 1. 18 W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 ΠiAij (x) = (ΠiAij )(x) = Aij (xi) = Aij (xi) = Πi Aij (x). j∈J j∈J j∈J i∈I i∈I j∈J j = ∈ ∈ That is to say, j∈J ΠiAij Πi( j Aij ). Since every δi is Alexandrov, we have j Aij δi for every i I . ∈ S Hence, j∈J ΠiAij , as desired. { | ∈ } ⊆ ∈ Step 2.δ is closed under meets. Suppose that Uk k K δ. For every k K, there exists a set Jk such that U = (Π A ) . Then k j∈Jk i i j Uk = (ΠiAi)kj = (ΠiAi)kf (k) . ∈ ∈ ∈ ∈ ∈ k K k K j Jk f l∈K Jl k K ∈ 2 By Step 1, k∈K Uk δ. Hence, δ is Alexandrov.

By Proposition 6.6, we also get that, if every (Xi, δi) is stratiﬁed, then so is their product.

Theorem 6.7. Let L be a completely distributive lattice. A product of sT0 stratiﬁed Alexandrov L-topological spaces is an sT0 space.

Proof. Suppose that {(Xi, δi) | i ∈ I} is a family of sT0 stratiﬁed Alexandrov L-topological space. Put X = ΠiXi and S ={ΠiAi |∀i ∈ I, Ai ∈ δ}. Then S is a basis of the product topology of {(Xi, δi) | i ∈ I} on X. Suppose that g ∈ CHL(δ). For every i ∈ I , since the ith projection pi : (X, δ) −→ (Xi, δi) is a continuous map, it follows that ← : −→ ◦ ← : −→ ◦ ← ∈ (pi)L δi δ is a complete L-ordered set homomorphism and then so is g (pi)L δi L. That is, g (pi)L ∈ ∈ ◦ ← = CHL(δi) for every i I . Since every (Xi, δi) is sT0, there exists a unique xi Xi such that g (pi)L ΨL(xi). For x = (xi)i∈I , ΨL(x)(ΠiAi) = Πi(Ai) (x) = Ai(xi) = ΨL(xi)(Ai) i i = ◦ ← = ◦ = ◦ = g (pi)L (Ai) g(Ai pi) g Ai pi g(ΠiAi). i i i

That is, g = ΨL(x). 2

In a category C, let A and B be two C-objects. B is called a retract [3] of A, if there are two morphisms f : B → A and g : A −→ B such that g ◦ f = idB .

Proposition 6.8. A retract of an sT0 stratiﬁed Alexandrov L-topological space in AL-Top0 is also sT0.

Proof. Let (Y, σ) be a retract of an sT0 stratiﬁed Alexandrov L-topological space (X, δ) in AL-Top0. Then there exist two continuous maps f : X −→ Y and g : Y −→ X such that f ◦ g = idY .

(1) ΨY is injective: Let x, y ∈ Y satisfy B(x) = B(y) for every B ∈ σ . For every A ∈ δ, we have A ◦ g ∈ σ and, therefore, A(g(x)) = A(g(y)). By the arbitrariness of A, we have g(x) = g(y) and, therefore, x = f(g(x)) = f(g(y)) = y. ∈ ◦ ← ∈ =: −1 ◦ ← ∈ = (2) ΨY is surjective: For every p CHL(σ ), we have, p gL CHL(δ) and x (ΨX) (p gL ) X with y ∈ = = ∈ ∈ ← ∈ f(x) Y . We show that ΨY (y) p or B(y) p(B) for every B σ . In fact, for every B σ , fL (B) δ and = = ← = ◦ ← ← = ◦ ◦ ← = 2 B(y) B f(x) ΨX(x) fL (B) p gL fL (B) p (f g)L (B) p(B).

Suppose that f, g : A −→ B are two morphisms in a category C. An equalizer [3] of the pair of f and g is a C-object E such that there exists a C-morphism e : E → A with the conditions that: (Eq1) f ◦ e = g ◦ e; (Eq2) for every C-morphism e : E → A with f ◦ e = g ◦ e, there exists a unique C-morphism h : E → E such that e = e ◦ h. W. Yao, S.-E. Han / Fuzzy Sets and Systems 282 (2016) 1–20 19

Proposition 6.9. The equalizer of two continuous maps from an sT0 stratiﬁed Alexandrov L-topological space to an L-T0-space is sT0.

Proof. Suppose (X, δ) is an sT0 stratiﬁed Alexandrov L-topological space and (Y, σ) is an L-T0 space. Let f, g : X −→ Y be two continuous maps and let (K, δK ) be an equalizer of f and g, that is, K ={x ∈ X | f(x) = g(x)} and δK = δ|K. Then (K, δK ) is a stratiﬁed Alexandrov L-T0 space. If p : δK −→ L is a complete L-ordered set ◦ ← : −→ : −→ ∈ homomorphism, then so is p iL δX L, where i K X is the inclusion. Then there exists x X such that = ◦ ← ΨL(x) p iL . It follows that = ◦ ← = ◦ ← ◦ ← = ◦ ◦ ← = ◦ ◦ ← = ΨL f(x) ΨL(x) fL p iL fL p (f i)L p (g i)L ΨL g(x) and therefore, f(x) = g(x) since Y is L-T0. Altogether, we obtain that x ∈ K. That is to say, ΨL : K −→ CHL(δK ) is surjective. Since (X, δ) is L-T0, ΨL : K −→ CHL(δK ) is also injective. Hence, K is sT0. 2

7. Conclusions

For every complete lattice A, after showing one-to-one correspondences among the set of completely prime elements, the set of completely coprime elements and the set of complete lattice homomorphisms from A to 2, we restudy the duality between the category of T0 Alexandrov spaces and completely generated lattices. Using a technique similar to Papert–Papert–Isbell adjunction and its induced equivalence, we establish a duality between the category of sT0 stratiﬁed Alexandrov L-topological spaces and the category of completely generated complete L-ordered sets. For the new sT0 axiom, we show that it has some similar properties as modiﬁed L-sobriety, for example, it is hereditary by closed subspaces and productive.

Acknowledgements

The ﬁrst author is supported by the NNSF of China (11201112) and the Foundation of Hebei Province (Y2012020, A2013208175, A2014403008, BRII210). The second author is supported by Basic Science Research Program through the NRF of Korea funded by the Ministry of Education, Science and Technology (2013R1A1A4A01007577).

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