Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 582105 |

Xiu-Yun Wu, Li-Li Xie, "On 𝜢-Layer Ideal Topologies", Advances in Fuzzy Systems, vol. 2012, Article ID 582105, 8 pages, 2012.

On 𝜢-Layer Ideal Topologies

Academic Editor: Katsuhiro Honda
Received08 Apr 2012
Accepted22 Jun 2012
Published29 Aug 2012


The purpose of this paper is to study theory of two different kinds of 𝛼-layer order-preserving operator space, namely, πœ”π›Ό-opos and πœ”βˆ—π›Ό(β„‘)-opos. The former kind of space is formed by 𝛼-layer function in L-fuzzy order-preserving operator space. The later kind of space is derived by local 𝛼-remote neighborhood function, which is related with πœ”π›Ό-opos and 𝛼-ideal. We study characteristic properties of the two kinds of spaces, respectively, and give some applications to show the intimate relations under two different πœ”βˆ—π›Ό(β„‘)-oposs.

1. Introduction

In general topology, Vaidyanathaswamy firstly defined concepts of local function and its derived ideal topology from initial topology and ideal [1]. Some interesting extensive works were done by JankoviΔ‡ and Hamlett [2]. After M. E. Abd El-Monsef introduced the concept of 𝐼-open set, many researchers were devoted to research on local semitopology. There are many local semiopen sets, such as 𝛽-𝐼-open set, strong 𝛽-𝐼-open set, and 𝛿-𝐼-open set [3–5]. All of these local semiopen sets are given by comparing interior and closure operators in the initial topology and its ideal derived topology.

By utilizing q-neighborhood which is mentioned in [6], Sarkar generalized the concepts of local function and derived topology into fuzzy topology in 1997 [7].

As there is a layer structure in 𝐿-fuzzy topology, fuzzy local functions and their derived fuzzy ideal topology must be more complex. Hence, in this paper, we will analyze the ideal topological properties in terms of layer structure of 𝐿-fuzzy topology and reveal the inner relations between 𝛼-layer ideal space and fuzzy ideal topological space.

In the first part of the paper, we establish the theory of πœ”π›Ό-opos. We introduce the concept of 𝛼-layer function in 𝐿-fuzzy order-preserving operator space. Then, based on its basic properties, we form πœ”π›Ό-opos. We also prove it preserves many good properties. In the second part of the paper, we establish the theory of πœ”βˆ—π›Ό(β„‘)-opos. It is a 𝛼-layer space with local topological properties. We introduce the concept of local 𝛼-remote neighborhood function via an πœ”π›Ό-opos and an 𝛼-ideal. On the basis of its basic properties, we form πœ”βˆ—π›Ό(β„‘)-opos. It is finer than the old one. We observe the structures of local 𝛼-remote neighborhood functions under different 𝛼-ideals and πœ”π›Ό-opos as well as the relations of the correspondent πœ”βˆ—π›Ό-opos. We also obtain some equivalent conditions of πœ”βˆ—π›Ό-opos under compatibility of 𝛼-ideal and πœ”π›Ό-opos. Finally, as an application, we define four kinds of connectivity and reveal their inner relations.

2. Preliminaries

In a general topological space (𝑋,𝜏), β„βŠ‚π‘‹ is an ideal, π‘ˆβŠ‚π‘‹. The concept of local function of π‘ˆ with respect to 𝜏 and ℐ is given by π‘ˆβˆ—={π‘₯βˆˆπ‘‹βˆΆπ‘ˆβˆ©π‘‰βˆ‰β„,π‘‰βˆˆπ’©(π‘₯)}, in which 𝒩(π‘₯) is the neighborhood system of π‘₯ [1].

In this paper, an lattice 𝐿 is called a completely distributive lattice with an order reserving involutionβ€².𝑋,π‘Œ will always denote nonempty crisp sets, A mapping π΄βˆΆπ‘‹β†’πΏ is called an 𝐿-fuzzy set.𝐿𝑋 is the set of all 𝐿-fuzzy sets on 𝑋. An element π‘’βˆˆπΏ is called an irreducible element in 𝐿, if π‘βˆ¨π‘ž=𝑒 implies 𝑝=𝑒 or π‘ž=𝑒, where 𝑝,π‘žβˆˆπΏ. The set of all nonzero irreducible elements in 𝐿 will be denoted by 𝑀(𝐿) (see [1]). If π‘₯βˆˆπ‘‹, π›Όβˆˆπ‘€(𝐿), then π‘₯𝛼 is called a molecule in 𝐿𝑋. The set of all molecules in 𝐿𝑋 is denoted by π‘€βˆ—(𝐿𝑋). If π΄βˆˆπΏπ‘‹, π›Όβˆˆπ‘€(𝐿), take 𝐴[𝛼]={π‘₯βˆˆπ‘‹βˆΆπ΄(π‘₯)β‰₯𝛼}. If πΈβŠ‚π‘‹, the complement of 𝐸, denoted by 𝐸′, and 𝐸′=π‘‹βˆ’πΈ={π‘¦βˆˆπ‘‹βˆΆπ‘¦βˆ‰πΈ} [8].

An 𝐿-fuzzy order-preserving operator space and some related conceptions are given in the following.

Let 𝑋 be an nonempty set. An operator πœ”βˆΆπΏπ‘‹β†’πΏπ‘‹ is called a 𝐿-fuzzy order preserving operator in 𝐿𝑋, if it satisfies (1) πœ”(1𝑋)=1𝑋, (2) for all 𝐴,π΅βˆˆπΏπ‘‹ and 𝐴≀𝐡 implies πœ”(𝐴)β‰€πœ”(𝐡). A set π΄βˆˆπΏπ‘‹ is called an πœ”-set, if πœ”(𝐴)=𝐴. The set of all πœ”-sets in 𝐿𝑋 is denoted by Ξ©. And (𝐿𝑋,Ξ©) is called an order-preserving operator space (briefly, 𝐿-𝑓opos). A molecule π‘₯π›Όβˆˆπ‘€βˆ—(𝐿𝑋),π‘ƒβˆˆΞ©, 𝑃 is called an πœ”-remote neighborhood of π‘₯𝛼, if π‘₯π›Όβˆ‰π‘ƒ. The set of all πœ”-remote neighborhood of π‘₯𝛼 is denoted by πœ”πœ‚(π‘₯𝛼). Let π‘₯π›Όβˆˆπ‘€βˆ—(𝐿𝑋),π΄βˆˆπΏπ‘‹, π‘₯𝛼 is called an πœ”-adherent point of 𝐴, if for all π‘ƒβˆˆπœ”πœ‚(π‘₯𝛼), π΄βˆ‰π‘ƒ. The union of all πœ”-adherent points of 𝐴 is called the πœ”-closure of 𝐴, denoted by π΄βˆ’πœ”. A set π΄βˆˆπΏπ‘‹ is called πœ”-closed, if π΄βˆ’πœ”=𝐴. The set of all πœ”-closed sets in 𝐿𝑋 is denoted by Ξ©βˆ’πœ”. Ξ©βˆ’πœ” is finite union and infinite intersection preserving [9].

Similar concepts in a general topology are defined as follows.

Let 𝑋 be a nonempty set, and let 𝒫(𝑋) be the family of all subsets of 𝑋. An operator πœŽβˆΆπ‘‹β†’π‘‹ is called an order-preserving operator in 𝑋, if it satisfies (1) 𝜎(𝑋)=𝑋, (2) for all π΄βŠ‚π΅βŠ‚π‘‹ implies 𝜎(𝐴)βŠ‚πœŽ(𝐡). A set π΄βŠ‚π‘‹ is called a 𝜎-set, if 𝜎(𝐴)=𝐴. The set of all 𝜎-sets in 𝑋 is denoted by Ξ”. (𝑋,Ξ”) is called an order-preserving operator space on 𝑋 (briefly, opos). Let π‘₯βˆˆπ‘‹,π‘ƒβˆˆΞ”, 𝑃 is called a 𝜎-remote neighborhood of π‘₯, if there is π‘„βŠ‚π‘‹, such that π‘₯βˆ‰π‘„,π‘ƒβŠ‚π‘„. The set of all 𝜎-remote neighborhood of π‘₯ is denoted by πœŽπœ‚(π‘₯). Let π‘₯βˆˆπ‘‹,π΄βŠ‚π‘‹, π‘₯ is called a 𝜎-adherent point of 𝐴, if for all π‘ƒβˆˆπœŽπœ‚(π‘₯), π΄βŠ„π‘ƒ. The union of all 𝜎-adherent points of 𝐴 is called the 𝜎-closure of 𝐴, denoted by π΄βˆ’πœŽ. A set π΄βŠ‚π‘‹ is called 𝜎-closed, if π΄βˆ’πœŽ=𝐴. The set of all 𝜎-closed sets in 𝑋 is denoted by Ξ”βˆ’πœŽ. Ξ”βˆ’πœŽ is finite union and infinite intersection preserving.

Let (𝑋,Ξ”) be an opos, and let 𝐿 be an fuzzy lattice, for all π›ΌβˆˆπΏ. A fuzzy set π΄βˆΆπΏβ†’π‘‹ is called a lower continuous function, if {π‘₯βˆˆπ‘‹βˆΆπ΄(π‘₯)≀𝛼}βˆˆΞ”βˆ’πœŽ. Then, the set of all the lower continuous functions, denoted by πœ”πΏ(Ξ”), consists an 𝐿-fuzzy cotopology in 𝐿𝑋. The space (𝐿𝑋,πœ”πΏ(Ξ”)) is called the induced 𝐿-𝑓opos by (𝑋,Ξ”).

A nonempty subfamily β„‘ of 𝐿𝑋 is called an 𝛼-ideal, if β„‘ satisfies the following conditions:(1)π΄βˆˆβ„‘, and 𝐡[𝛼]βŠ‚π΄[𝛼] implies π΅βˆˆβ„‘,(2)𝐴,π΅βˆˆβ„‘ implies π΄βˆ¨π΅βˆˆβ„‘.

It is easy to check if π΅βˆˆπΏπ‘‹, π΄βˆˆβ„‘, such that 𝐡[𝛼]=𝐴[𝛼], then π΅βˆˆβ„‘. Moreover, β„‘ of 𝐿𝑋 is an 𝛼-idea if and only if β„‘[𝛼]={𝐴[𝛼]βˆΆπ΄βˆˆβ„‘} is an ideal on 𝑋. If π΅βˆˆπΏπ‘‹, π΄βˆˆβ„‘, such that 𝐡[𝛼]=𝐴[𝛼], then π΅βˆˆβ„‘. If β„‘,𝔍 are two 𝛼-ideals, then β„‘βˆ¨π”={πΌβˆ¨π½βˆΆπΌβˆˆβ„‘,π½βˆˆπ”} and β„‘βˆ©π” are 𝛼-ideals too.

In this paper, if π”‰βŠ‚πΏπ‘‹, we denote 𝔉[𝛼]={π΄βˆΆπ΄βˆˆπ”‰} and π”ξ…ž={π΄ξ…žβˆΆπ΄βˆˆπ”‰}.

3. πœ”π›Ό-Closed Set and πœ”π›Ό-opos

In this section, we list out the main results in [10], which we will use in the following sections. Proofs of the theorems in the following can be found in [10] as well.

Definition 1. Let (𝐿𝑋,Ξ©) be an 𝐿-𝑓opos, π›Όβˆˆπ‘€(𝐿). An operator πœ”π›ΌβˆΆπΏπ‘‹β†’πΏπ‘‹ is defined by, for all π΄βˆˆπΏπ‘‹,πœ”π›Όξ—ξ€½(𝐴)=πΊβˆˆΞ©βˆ’πœ”βˆΆπΊ[𝛼]βŠƒπ΄[𝛼]ξ€Ύ.(1)

Theorem 2. Let (𝐿𝑋,Ξ©) be an 𝐿-𝑓opos, π›Όβˆˆπ‘€(𝐿). 𝐴,π΅βˆˆπΏπ‘‹, then the following statements hold:(1)𝐴[𝛼]βŠ‚π΅[𝛼] implies πœ”π›Ό(𝐴)β‰€πœ”π›Ό(𝐡);(2)𝐴[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼];(3)πœ”π›Ό(𝐴∨𝐡)=πœ”π›Ό(𝐴)βˆ¨πœ”π›Ό(𝐡);(4)πœ”π›Ό(πœ”π›Ό(𝐴))=πœ”π›Ό(𝐴).

Definition 3. Let (𝐿𝑋,Ξ©) be an 𝐿-𝑓opos, π›Όβˆˆπ‘€(𝐿). A set π΄βˆˆπΏπ‘‹ is called πœ”π›Ό-closed, if πœ”π›Ό(𝐴)[𝛼]=𝐴[𝛼]. The set of all πœ”π›Ό-closed sets in 𝐿𝑋 is denoted by πœ”π›Ό(Ξ©). (𝐿𝑋,πœ”π›Ό(Ξ©)) is called an πœ”π›Ό-order-preserving operator space (briefly, πœ”π›Ό-opos).

Theorem 4. Let (𝐿𝑋,Ξ©) be an 𝐿-𝑓opos, π›Όβˆˆπ‘€(𝐿). Then,(1)1π‘‹βˆˆπœ”π›Ό(Ξ©);(2)if {π΄π‘–βˆΆπ‘–=1,2,…,𝑛}βŠ‚πœ”π›Ό(Ξ©), then ⋁𝑛𝑖=1π΄π‘–βˆˆπœ”π›Ό(Ξ©);(3)if {π΄π‘–βˆΆπ‘–βˆˆπΌ}βŠ‚πœ”π›Ό(Ξ©), then βˆ§π‘–βˆˆπΌπ΄π‘–βˆˆπœ”π›Ό(Ξ©).

Remark 5. The theorem shows that πœ”π›Ό is finite union and infinite intersection preserving.

Corollary 6. Let (𝐿𝑋,Ξ©) be an 𝐿-𝑓opos, π›Όβˆˆπ‘€(𝐿). Then, πœ”π›Ό(Ξ©) consists an 𝐿-fuzzy cotopology on 𝐿𝑋.

Lemma 7. Let (𝑋,Ξ”) be an opos, (𝐿𝑋,πœ”πΏ(Ξ”)) be an 𝐿-𝑓opos induced by (𝑋,Ξ”) and let πΈβŠ‚π‘‹. Then, πΈβˆˆΞ”βˆ’πœŽ if and only if πœ’πΈβˆˆ(πœ”πΏ(Ξ”))βˆ’πœ”.

Theorem 8. Let (𝑋,Ξ”) be an opos, (𝐿𝑋,πœ”πΏ(Ξ”)) an 𝐿-𝑓opos induced by (𝑋,Ξ”). π΄βˆˆπΏπ‘‹, π›Όβˆˆπ‘€(𝐿). Then, π΄βˆˆπœ”π›Ό(πœ”πΏ(Ξ”)) if and only if 𝐴[𝛼]βˆˆΞ”βˆ’πœŽ.

Theorem 9. Let (𝑋,Ξ”) be an opos, (𝐿𝑋,πœ”πΏ(Ξ”)) is an 𝐿-𝑓opos induced by (𝑋,Ξ”). Then, π΄βˆˆπœ”πΏ(Ξ”) if and only if for all π›Όβˆˆπ‘€(𝐿), π΄βˆˆπœ”π›Ό(πœ”πΏ(Ξ”)).

Definition 10. Let (𝐿𝑋,πœ”1𝛼(Ξ©1)), (πΏπ‘Œ,πœ”2𝛼(Ξ©2)) be πœ”1𝛼-opos, πœ”2𝛼-opos, respectively. An 𝐿-fuzzy homomorphism π‘“β†’βˆΆπΏπ‘‹β†’πΏπ‘Œ is called (πœ”1𝛼(Ξ©1),πœ”2𝛼(Ξ©2))-continuous, if for all π΅βˆˆπœ”2𝛼(Ξ©2), then 𝑓←(𝐡)βˆˆπœ”1𝛼(Ξ©1).

Theorem 11. Let (𝐿𝑋,πœ”1𝛼(Ξ©1)), (πΏπ‘Œ,πœ”2𝛼(Ξ©2)) be πœ”1𝛼-opos, πœ”2𝛼-opos, respectively. π‘“β†’βˆΆπΏπ‘‹β†’πΏπ‘Œ is an 𝐿-fuzzy mapping. Then, the following statements are equivalent:(1)𝑓→ is (πœ”1𝛼(Ξ©1),πœ”2𝛼(Ξ©2))-continuous;(2)for all π΅βˆˆπΏπ‘Œ, πœ”1𝛼(𝑓←(𝐡))[𝛼]βŠ‚π‘“β†(πœ”2𝛼(𝐡))[𝛼];(3)for all π΄βˆˆπΏπ‘‹, 𝑓→(πœ”1𝛼(𝐴))[𝛼]βŠ‚πœ”2𝛼(𝑓→(𝐴))[𝛼].

Definition 12. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, π‘₯π›Όβˆˆπ‘€βˆ—(𝐿𝑋),π΄βˆˆπœ”π›Ό(Ξ©). 𝐴 is called an πœ”π›Ό-closed remote neighborhood of π‘Ž, if π‘₯βˆ‰π΄[𝛼]. The set of all πœ”π›Ό-closed remote neighborhood of π‘₯𝛼 will be denoted by πœ‚πœ”π›Ό(π‘₯𝛼). An fuzzy point π‘₯𝛼 is called πœ”π›Ό-adherent point of π΅βˆˆπΏπ‘‹, if for every π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼), 𝐡([𝛼])βŠ„π‘ƒ([𝛼]).

Remark 13. π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼)⇔(𝑃[𝛼])β€²βˆˆπ’©(π‘₯), where 𝒩(π‘₯) is the neighborhood system of π‘₯ in (𝑋,πœ”π›Ό(Ξ©)[𝛼]). Hence, (πœ‚πœ”π›Ό(π‘₯𝛼)[𝛼])β€²=𝒩(π‘₯).

Theorem 14. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, π‘₯π›Όβˆˆπ‘€βˆ—(𝐿𝑋),π΄βˆˆπΏπ‘‹, then(1)π‘₯𝛼 is an πœ”π›Ό-adherent point of π΄βˆˆπΏπ‘‹ if and only if π‘₯βˆˆπœ”π›Ό(𝐴)[𝛼],(2)πœ”π›Ό(𝐴) is the union of all πœ”π›Ό-adherent points of 𝐴.

4. πœ”π›Ό-β„‘-Closed Sets

Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. β„‘βŠ‚πΏπ‘‹ be an 𝛼-ideal. π‘₯π›Όβˆˆπ‘€βˆ—(𝐿𝑋), πœ‚πœ”π›Ό(π‘₯𝛼) be the πœ”π›Ό-remote family of π‘₯𝛼. For any π΄βˆˆπΏπ‘‹, take π΄βˆ—ξ€·β„‘,πœ”π›Όξ€Έ=ξ˜ξ€½π‘₯π›ΌβˆΆπ΄[𝛼]βŠ„π‘ƒ[𝛼]βˆͺ𝐼[𝛼],βˆ€π‘ƒβˆˆπœ‚πœ”π›Όξ€·π‘₯𝛼.,πΌβˆˆβ„‘(2) Then, π΄βˆ—(β„‘,πœ”π›Ό) is called the local πœ”π›Ό-remote function of 𝐴 with respect to β„‘ and πœ”π›Ό(Ξ©), simply denoted by π΄βˆ—(β„‘) or π΄βˆ—.

Remark 15. If 𝐿=[0,1], πœ” is the closure operator, then (π΄βˆ—[𝛼])β€² is the local function of 𝐴[𝛼] in (𝑋,πœ”π›Ό(Ξ©)[𝛼]), and π’œβˆ—=β‹π›ΌβˆˆπΏ(π΄βˆ—[𝛼])β€² is the fuzzy local function of 𝐴 in [7]. Furthermore, we have (π΄βˆ—[𝛼])β€²=(𝐴[𝛼])β€²βˆ—.

Theorem 16. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. β„‘,π”βŠ‚πΏπ‘‹ be two 𝛼-ideals. 𝐴,π΅βˆˆπΏπ‘‹. Then, the following statements hold.(1)β„‘[𝛼]={βˆ…}β‡’π΄βˆ—(β„‘)[𝛼]=πœ”π›Ό(𝐴)[𝛼], β„‘[𝛼]=𝒫(𝑋)β‡’π΄βˆ—(β„‘)[𝛼]=βˆ….(2)𝐴[𝛼]βŠ‚π΅[𝛼]β‡’π΄βˆ—(β„‘)[𝛼]βŠ‚π΅βˆ—(β„‘)[𝛼].(3)β„‘[𝛼]βŠ‚π”[𝛼]β‡’π΄βˆ—(𝔍)[𝛼]βŠ‚π΄βˆ—(β„‘)[𝛼].(4)π΄βˆ—(β„‘)[𝛼]=πœ”π›Ό(π΄βˆ—(β„‘))[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼].(5)(π΄βˆ—(β„‘))βˆ—(β„‘)[𝛼]βŠ‚π΄βˆ—(β„‘)[𝛼].(6)π΄βˆˆβ„‘β‡’π΄βˆ—(β„‘)[𝛼]=βˆ….(7)(𝐴∨𝐡)βˆ—(β„‘)[𝛼]=π΄βˆ—(β„‘)[𝛼]βˆͺπ΅βˆ—(β„‘)[𝛼].(8)(π΄βˆ—βˆ’π΅βˆ—)[𝛼]=((π΄βˆ’π΅)βˆ—βˆ’π΅βˆ—)[𝛼]βŠ‚(π΄βˆ’π΅)βˆ—[𝛼].(9)π΅βˆˆβ„‘β‡’(𝐴∨𝐡)βˆ—[𝛼]=π΄βˆ—[𝛼]=(π΄βˆ’π΅)βˆ—[𝛼].(10)π‘ƒβˆˆπœ”π›Ό(Ξ©)β‡’π‘ƒβˆ—[𝛼]βŠ‚π‘ƒ[𝛼].

Proof. (1) Suppose β„‘[𝛼]={βˆ…}, by Theorem 14, we have π‘₯π›Όβ‰€π΄βˆ—ξ€·β„‘,πœ”π›Όξ€ΈβŸΊπ‘₯π›Όβ‰€ξ˜ξ€½π‘₯π›ΌβˆΆπ΄[𝛼]βŠ„π‘ƒ[𝛼],βˆ€π‘ƒβˆˆπœ‚πœ”π›Όξ€·π‘₯π›Όξ€Έξ€ΎβŸΊπ‘₯π›Όβ‰€πœ”π›Ό(𝐴).(3)
If π‘‹βˆˆβ„‘[𝛼], then there must be π΅βˆˆβ„‘, such that 𝐴[𝛼]=𝐡[𝛼]={βˆ…}. Therefore, π΄βˆ—(β„‘)[𝛼]=βˆ….
(2), (3), and (6) Easy.
(4) π‘₯βˆ‰πœ”π›Ό(𝐴)[𝛼] implies πœ”π›Ό(𝐴)βˆˆπœ‚π›Ό(π‘₯𝛼). By 𝐴[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼], we have π‘₯βˆ‰π΄βˆ—[𝛼]. So π΄βˆ—[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼]. Thus, πœ”π›Ό(𝐴)βˆ—[𝛼]βŠ‚πœ”π›Ό(πœ”π›Ό(𝐴))[𝛼]=πœ”π›Ό(𝐴)[𝛼]. Besides, π‘₯βˆˆπœ”π›Ό(π΄βˆ—)[𝛼] implies for all π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼), π΄βˆ—[𝛼]βŠ„π‘ƒ[𝛼]. Take π‘¦βˆˆπ΄βˆ—[𝛼]βˆ’π‘ƒ[𝛼], so π‘ƒβˆˆπœ‚πœ”π›Ό(𝑦𝛼) and for all π‘„βˆˆπœ‚πœ”π›Ό(𝑦𝛼),πΌβˆˆβ„‘, 𝐴[𝛼]βŠ„π‘„[𝛼]βˆͺ𝐼[𝛼]. Particularly, let 𝑃=𝑄. So 𝐴[𝛼]βŠ„π‘ƒ[𝛼]βˆͺ𝐼[𝛼]. This means π‘₯βˆˆπ΄βˆ—[𝛼]. Hence, πœ”π›Ό(π΄βˆ—)[𝛼]βŠ‚π΄βˆ—[𝛼]. Therefore, πœ”π›Ό(π΄βˆ—)[𝛼]=π΄βˆ—[𝛼].
(5) By (4), (π΄βˆ—(β„‘))βˆ—(β„‘)[𝛼]=(πœ”π›Ό(π΄βˆ—(β„‘)))βˆ—(β„‘)[𝛼]=πœ”π›Ό(πœ”π›Ό(π΄βˆ—(β„‘)))[𝛼]=(πœ”π›Ό(π΄βˆ—(β„‘)))[𝛼]=π΄βˆ—(β„‘)[𝛼].
(7) By (2), (𝐴∨𝐡)βˆ—(β„‘)[𝛼]βŠƒπ΄βˆ—(β„‘)[𝛼]βˆͺπ΅βˆ—(β„‘)[𝛼]. Conversely, if and only if π‘₯βˆˆπ΄βˆ—(β„‘)[𝛼]βˆͺπ΅βˆ—(β„‘)[𝛼], then for all π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘, 𝐴[𝛼]βŠ„π‘ƒ[𝛼]βˆͺ𝐼[𝛼], and 𝐡[𝛼]βŠ„π‘ƒ[𝛼]βˆͺ𝐼[𝛼]. Hence, (𝐴∨𝐡)[𝛼]βŠ„π‘ƒ[𝛼]βˆͺ𝐼[𝛼]. This implies π‘₯∈(𝐴∨𝐡)βˆ—(β„‘)[𝛼]. So (𝐴∨𝐡)βˆ—(β„‘)[𝛼]βŠ‚π΄βˆ—(β„‘)[𝛼]βˆͺπ΅βˆ—(β„‘)[𝛼].
(8) For each π‘₯∈(π΄βˆ—βˆ’π΅βˆ—)[𝛼], we may remark π‘₯∈(π΄βˆ’π΅)βˆ—[𝛼]. If not, there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘, such that (π΄βˆ’π΅)[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]. Thus, 𝐴[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]βˆͺ𝐡[𝛼]. Since π‘₯βˆ‰π΅βˆ—[𝛼], there are π‘„βˆˆπœ‚πœ”π›Ό(π‘₯𝛼),π½βˆˆβ„‘, 𝐡[𝛼]βŠ‚π‘„[𝛼]βˆͺ𝐽[𝛼]. This shows 𝐴[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]βˆͺ𝐡[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]βˆͺ𝑄[𝛼]βˆͺ𝐽[𝛼]=(π‘ƒβˆ¨π‘„)[𝛼]βˆͺ(𝐼∨𝐽)[𝛼].(4) Consequently, π‘₯βˆ‰π΄βˆ—[𝛼]. But this contradicts with π‘₯∈(π΄βˆ—βˆ’π΅βˆ—)[𝛼]. Therefore, (π΄βˆ—βˆ’π΅βˆ—)[𝛼]βŠ‚((π΄βˆ’π΅)βˆ—βˆ’π΅βˆ—)[𝛼]. The reverse inclusion is obvious. So (π΄βˆ—βˆ’π΅βˆ—)[𝛼]=((π΄βˆ’π΅)βˆ—βˆ’π΅βˆ—)[𝛼].
(9) Obviously, (𝐴∨𝐡)βˆ—[𝛼]=π΄βˆ—[𝛼]βˆͺπ΅βˆ—[𝛼]=π΄βˆ—[𝛼]. And (π΄βˆ’π΅)βˆ—[𝛼]βŠ‚π΄βˆ—[𝛼]. Conversely, π‘₯βˆ‰(π΄βˆ’π΅)βˆ—[𝛼], there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘, (π΄βˆ’π΅)[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]. So 𝐴[𝛼]=(π΄βˆ’π΅)[𝛼]βˆͺ𝐡[𝛼]βŠ‚π‘ƒ[𝛼]βˆͺ𝐼[𝛼]βˆͺ𝐡[𝛼]=𝑃[𝛼]βˆͺ(𝐼∨𝐡)[𝛼]. Thus, π‘₯βˆ‰π΄βˆ—[𝛼]. Then, (9) holds.
(10) Suppose there is π‘₯βˆˆπ‘ƒβˆ—[𝛼]βˆ’π‘ƒ[𝛼]. So π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼). By 𝑃[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼] for each πΌβˆˆβ„‘, we have π‘₯βˆ‰π‘ƒ[𝛼], a contradiction.

Theorem 17. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘βŠ‚πΏπ‘‹ be an ideal. π΄βˆˆπΏπ‘‹. Take πœ”βˆ—π›Ό(𝐴)=π΄βˆ¨π΄βˆ—, then(1)πœ”βˆ—π›Ό(𝐴)[𝛼]=βˆ…, if and only if 𝐴[𝛼]=βˆ…,(2)πœ”βˆ—π›Ό(𝐴∨𝐡)[𝛼]=πœ”βˆ—π›Ό(𝐴)[𝛼]βˆͺπœ”βˆ—π›Ό(𝐡)[𝛼], (3)πœ”βˆ—π›Ό(πœ”βˆ—π›Ό(𝐴))[𝛼]=πœ”βˆ—π›Ό(𝐴)[𝛼],(4)(πœ”βˆ—π›Ό(π΄βˆ—))βˆ—[𝛼]=πœ”βˆ—π›Ό(π΄βˆ—)[𝛼]=(πœ”βˆ—π›Ό(𝐴))βˆ—[𝛼]=(πœ”π›Ό(π΄βˆ—))βˆ—[𝛼]=π΄βˆ—[𝛼].

Proof. According to (5), (7) in Theorem 16, the proof is trivial.
By Theorem 17, we know if and only if we take πœ”βˆ—π›Ό(β„‘,Ξ©)={π΄βˆˆπΏπ‘‹βˆΆπœ”βˆ—π›Ό(𝐴)[𝛼]=𝐴[𝛼]}, then πœ”βˆ—π›Ό(β„‘,Ξ©) (simply denoted by πœ”βˆ—π›Ό(β„‘)) consists an 𝐿-fuzzy cotopology on 𝐿𝑋. So it is called πœ”π›Ό-cotopology formed by the ideal β„‘ and πœ”π›Ό(Ξ©). The pair (𝐿𝑋,πœ”βˆ—π›Ό(β„‘,Ξ©)), simply denoted by (𝐿𝑋,πœ”βˆ—π›Ό(β„‘)), is called πœ”βˆ—π›Ό(β„‘)-π‘œπ‘π‘œπ‘ . By Theorem 16 (1), we have β„‘[𝛼]={βˆ…}β‡’πœ”βˆ—π›Ό(𝐴)[𝛼]=πœ”π›Ό(𝐴)[𝛼], thus πœ”βˆ—π›Ό(β„‘,Ξ©)=πœ”π›Ό(Ξ©), and β„‘[𝛼]=𝒫(𝑋)β‡’πœ”βˆ—π›Ό(𝐴)[𝛼]=𝐴[𝛼], thus πœ”βˆ—π›Ό(β„‘,Ξ©)=𝐿𝑋. But in general, {βˆ…}βŠ‚β„‘βŠ‚π’«(𝑋), 𝐴[𝛼]βŠ‚πœ”βˆ—π›Ό(𝐴)[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼], consequently, πœ”π›Ό(Ξ©)βŠ‚πœ”βˆ—π›Ό(β„‘,Ξ©)βŠ‚πΏπ‘‹. Moreover, if and only if πœ”π›Ό(Ξ©)[𝛼]={βˆ…,𝑋}, then πœ”βˆ—π›Ό(Ξ©)={𝐴∢𝐴[𝛼]βˆˆβ„‘[𝛼]}.

Example 18. Let 𝑋={π‘₯,𝑦},𝐿={0,1/2,1}. An 𝐿-fuzzy set πΊβˆˆπΏπ‘‹ satisfying 𝐺(π‘₯)=π‘Ž,𝐺(𝑦)=𝑏 will be denoted by (π‘Ž,𝑏). Let Ξ©={(0,0),(0,1/2),(1,1/2),(1,1)} be a cotopology on 𝐿𝑋. 𝛼=1/2. Put πœ”π›Όβ‹€(𝐴)={𝐸∈Ω∢𝐸[𝛼]βŠƒπ΄[𝛼]}. Then, πœ”π›Ό(Ξ©)={π΄βˆΆπœ”π›Ό(𝐴)[𝛼]=𝐴[𝛼]} consists an πœ”π›Ό-cotopology, and πœ‚πœ”π›Ό(π‘₯𝛼)={(0,0),(0,1/2),(0,1)},πœ‚πœ”π›Ό(𝑦𝛼)={(0,0)}. Totally, there are four kinds of 𝛼-ideals in 𝐿𝑋, namely, 𝔏={(0,0)}, 𝔐=𝐿𝑋, β„‘={(0,0),(0,1/2),(0,1)} and 𝔍={(0,0),(1/2,0),(1,0)}. 𝔏 and 𝔐=𝐿𝑋 are the trivial 𝛼-ideals. Let us study β„‘ and 𝔍.
Denote 𝐡=(0,1/2),𝐷=(1/2,0),𝐸=(1/2,1/2). Then, π΄βˆ—(β„‘)=π΅βˆ—(β„‘)=(0,0),π·βˆ—(β„‘)=πΈβˆ—(β„‘)=(1/2,1/2). So 𝐴,𝐡,πΈβˆˆπœ”βˆ—π›Ό(β„‘), But π·βˆ‰πœ”βˆ—π›Ό(β„‘). In addition, πœ”βˆ—π›Ό(β„‘,Ξ©)=πœ”π›Ό(Ξ©). However, π·βˆ—(𝔍)=(0,0), which means π·βˆˆπœ”βˆ—π›Ό(𝔍). It is easy to check πœ”βˆ—π›Ό(𝔍,Ξ©)=𝐿𝑋.
In this example, we see (πœ”π›Ό(𝐡))βˆ—(β„‘)[𝛼]=π΅βˆ—(β„‘)[𝛼]=βˆ…. However, πœ”βˆ—π›Ό(𝐡)[𝛼]=(π΅βˆ¨π΅βˆ—(β„‘))[𝛼]={𝑦}. Thus, (πœ”π›Ό(𝐡))βˆ—(β„‘)[𝛼]β‰ πœ”βˆ—π›Ό(𝐡)(β„‘)[𝛼]. Furthermore, (πœ”π›Ό(Ξ©))βˆ—(β„‘)={π΄βˆ—(β„‘)βˆΆπ΄βˆˆπœ”π›Ό(Ξ©)}={(0,0),(1/2,1/2)} is not an πœ”π›Ό-cotopology.

Theorem 19. Let (𝐿𝑋,πœ”π›Ό(Ω𝑖))(𝑖=1,2) be an πœ”π›Ό-opos. πœ”π›Ό(Ξ©1)βŠ‚πœ”π›Ό(Ξ©2). β„‘ is an 𝛼-ideal. π΄βˆˆπΏπ‘‹, then(1)π΄βˆ—(β„‘,πœ”π›Ό(Ξ©2))[𝛼]βŠ‚π΄βˆ—(β„‘,πœ”π›Ό(Ξ©1))[𝛼],(2)π΄βˆ—(β„‘,πœ”βˆ—π›Ό(β„‘,Ξ©1))[𝛼]βŠ‚π΄βˆ—(β„‘,πœ”π›Ό(Ξ©1))[𝛼],(3)πœ”βˆ—π›Ό(β„‘,Ξ©1)βŠ‚πœ”βˆ—π›Ό(β„‘,Ξ©2).

Theorem 20. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-π‘œπ‘π‘œπ‘ , an 𝛼-ideal β„‘βŠ‚πœ”π›Ό(Ξ©). Then, πœ”βˆ—π›Ό(β„‘)=πœ”π›Ό(Ξ©).

Proof. Obviously, πœ”βˆ—π›Ό(β„‘)βŠƒπœ”π›Ό(Ξ©). If π΄βˆ‰πœ”π›Ό(Ξ©), that is, 𝐴[𝛼]β‰ πœ”π›Ό(𝐴)[𝛼]. Take π‘¦βˆˆπœ”π›Ό(𝐴)[𝛼]βˆ’π΄[𝛼]. Let us prove π‘¦βˆˆπ΄βˆ—(β„‘)[𝛼]. In fact, if π‘¦βˆ‰π΄βˆ—(β„‘), there are π‘„βˆˆπœ‚πœ”π›Ό(𝑦𝛼),πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘„βˆ¨πΌ)[𝛼]. So π‘¦βˆˆπœ”π›Ό(𝐴)[𝛼]βŠ‚πœ”π›Ό(π‘„βˆ¨πΌ)[𝛼]=ξ€·πœ”π›Ό(𝑄)βˆ¨πœ”π›Όξ€Έ(𝐼)[𝛼]=(π‘„βˆ¨πΌ)[𝛼].(5)
It means π‘¦βˆˆπΌ[𝛼]. On the other hand, as 𝐴[𝛼]βŠ‚(π‘„βˆ¨πΌ)[𝛼], we have (𝐴∧𝐼)[𝛼]βŠ‚π‘„[𝛼]. Then, πœ”π›Ό(𝐴∧𝐼)[𝛼]=ξ€·πœ”π›Ό(𝐴)βˆ§πœ”π›Όξ€Έ(𝐼)[𝛼]=πœ”π›Ό(𝐴)[𝛼]∩𝐼[𝛼]βŠ‚πœ”π›Ό(𝑄)[𝛼]=𝑄[𝛼].(6) Notice that π‘¦βˆˆπœ”π›Ό(𝐴)[𝛼]∩𝐼[𝛼], we get π‘¦βˆˆπ‘„[𝛼], a contradiction. Hence, π‘¦βˆˆπ΄βˆ—(β„‘)[𝛼]. So π΄βˆ—(β„‘)[𝛼]βŠ„π΄[𝛼], which implies π΄βˆ‰πœ”βˆ—π›Ό(β„‘). Therefore, πœ”βˆ—π›Ό(β„‘)βŠ‚πœ”π›Ό(Ξ©). The proof is completed.

Theorem 21. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, and let β„‘ be an 𝛼-ideal. Take πœ”π›Ό(Ξ©)βˆ¨β„‘ as the supremum πœ”π›Ό- cotopology generated by πœ”π›Ό(Ξ©)βˆͺβ„‘. Then, πœ”βˆ—π›Ό(β„‘)=πœ”π›Ό(Ξ©)βˆ¨β„‘.

Proof. Put 𝔅={π‘ƒβˆ¨πΌβˆΆπ‘ƒβˆˆπœ”π›Ό(Ξ©),πΌβˆˆβ„‘}. It is easy to prove π”…βŠ‚πœ”βˆ—π›Ό(β„‘) is a base of πœ”βˆ—π›Ό(β„‘).

Lemma 22. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘,π”βŠ‚πΏπ‘‹ be two ideals. πΌβˆˆβ„‘, π΄βˆˆπΏπ‘‹. Then, (π΄βˆ’πΌ)βˆ—(𝔍,πœ”π›Ό(Ξ©))=π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘)).

Proof. By Theorem 21, π‘₯βˆ‰(π΄βˆ’πΌ)βˆ—ξ€·π”,πœ”π›Όξ€Έ(Ξ©)βŸΊβˆƒπ‘ƒβˆˆπœ‚πœ”π›Όξ€·π‘₯𝛼,π½βˆˆπ”,(π΄βˆ’πΌ)[𝛼]βŠ‚(π‘ƒβˆ¨π½)[𝛼]βŸΊβˆƒπ‘ƒβˆˆπœ‚πœ”π›Όξ€·π‘₯𝛼𝐴,π½βˆˆπ”,[𝛼]βŠ‚(π‘ƒβˆ¨π½βˆ¨πΌ)[𝛼]βŸΊβˆƒπ‘„βˆˆπœ‚πœ”βˆ—π›Ό(β„‘)ξ€·π‘₯𝛼𝐴,π½βˆˆπ”,[𝛼]βŠ‚(π‘„βˆ¨π½)[𝛼]⟺π‘₯βˆ‰π΄βˆ—ξ€·π”,πœ”βˆ—π›Όξ€Έ.(β„‘)(7)

Theorem 23. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘,π”βŠ‚πΏπ‘‹ be two 𝛼-ideals. π΄βˆˆπΏπ‘‹, then(1)π΄βˆ—(β„‘βˆ©π”)[𝛼]=π΄βˆ—(β„‘)[𝛼]βˆͺπ΄βˆ—(𝔍)[𝛼],(2)π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]=π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]βˆ©π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼].

Proof. (1) Suppose π‘₯βˆ‰π΄βˆ—(β„‘)[𝛼]βˆͺπ΄βˆ—(𝔍)[𝛼], there are 𝑃,π‘„βˆˆπœ‚πœ”π›Ό(π‘₯𝛼), and πΌβˆˆβ„‘,π½βˆˆπ”, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼], and 𝐴[𝛼]βŠ‚(π‘„βˆ¨π½)[𝛼]. Thus, 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)∧(π‘„βˆ¨π½)[𝛼]. As (π‘ƒβˆ§π‘„)∨(π‘ƒβˆ§π½)∨(πΌβˆ§π‘„)βˆˆπœ‚πœ”π›Ό(π‘₯𝛼), and πΌβˆ§π½βˆˆβ„‘βˆ©π”, it is clear that π‘₯βˆ‰π΄βˆ—(β„‘βˆ©π”)[𝛼]. So π΄βˆ—(β„‘βˆ©π”)[𝛼]βŠ‚π΄βˆ—(β„‘)[𝛼]βˆͺπ΄βˆ—(𝔍)[𝛼]. The reverse inclusion is obvious according to (3) in Theorem 16. Therefore, (1) holds.
(2) Suppose π‘₯βˆ‰π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]. There are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼), and πΌβˆˆβ„‘,π½βˆˆπ”,𝐾=𝐼∨𝐽, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΎ)[𝛼]. So (π΄βˆ’π½)[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼], (π΄βˆ’πΌ)[𝛼]βŠ‚(π‘ƒβˆ¨π½)[𝛼]. Hence, by Lemma 22, we have π‘₯βˆ‰(π΄βˆ’π½)βˆ—(β„‘,πœ”π›Ό(Ξ©))[𝛼]=π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼], and π‘₯βˆ‰(π΄βˆ’πΌ)βˆ—(𝔍,πœ”π›Ό(Ξ©))[𝛼]=π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼]. This shows that π‘₯βˆ‰π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]βˆ©π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼]. Therefore, π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]βŠƒπ΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]βˆ©π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼].
Conversely. suppose π‘₯βˆ‰π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]. Then, there exists π‘ƒβˆˆπœ‚πœ”βˆ—π›Ό(𝔍)(π‘₯𝛼), and πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Because of π‘ƒβˆˆπœ”βˆ—π›Ό(𝔍,Ξ©)=πœ”π›Ό(Ξ©)βˆ¨π”, there is π½βˆˆπ”, such that πœ”π›Ό(π‘ƒβˆ’π½)βˆˆπœ‚πœ”π›Ό(π‘₯𝛼). Hence, 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]=(π‘ƒβˆ’π½)[𝛼]βˆͺ(𝐼∨𝐽)[𝛼]. This shows π‘₯βˆ‰π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]. Therefore, π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]βŠ‚π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]. Similarly, we can prove π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]βŠ‚π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼]. So π΄βˆ—(β„‘βˆ¨π”,πœ”π›Ό(Ξ©))[𝛼]βŠ‚π΄βˆ—(β„‘,πœ”βˆ—π›Ό(𝔍))[𝛼]βˆ©π΄βˆ—(𝔍,πœ”βˆ—π›Ό(β„‘))[𝛼]. The proof is completed.

By Theorem 20, we get two important results.

Corollary 24. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘ be an 𝛼-ideal, then π΄βˆ—(β„‘,πœ”π›Ό(Ξ©))[𝛼]=π΄βˆ—(β„‘,πœ”βˆ—π›Ό(β„‘))[𝛼], and as a result πœ”βˆ—π›Ό(β„‘)=πœ”βˆ—π›Ό(πœ”βˆ—π›Ό(β„‘))(β„‘).

Proof. By (2) in Theorem 23, take β„‘=𝔍, clearly.

Corollary 25. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘,π”βŠ‚πΏπ‘‹ be two 𝛼-ideals. Then,(1)πœ”βˆ—π›Ό(β„‘βˆ¨π”)=πœ”βˆ—π›Ό(πœ”βˆ—π›Ό(𝔍))(β„‘)=πœ”βˆ—π›Ό(πœ”βˆ—π›Ό(β„‘))(𝔍),(2)πœ”βˆ—π›Ό(β„‘βˆ¨π”)=πœ”βˆ—π›Ό(𝔍)βˆ¨πœ”βˆ—π›Ό(β„‘),(3)πœ”βˆ—π›Ό(β„‘βˆ©π”)=πœ”βˆ—π›Ό(𝔍)βˆ©πœ”βˆ—π›Ό(β„‘).

Proof. (1) By (2) in Theorem 23, easy.
(2) Since πœ”π›Ό(Ξ©)βŠ‚πœ”βˆ—π›Ό(𝔍) for every 𝛼-ideal 𝔍, by (1) and Theorem 20, we have πœ”βˆ—π›Ό(β„‘βˆ¨π”)=πœ”βˆ—π›Ό(πœ”βˆ—π›Ό(𝔍))(β„‘)=πœ”βˆ—π›Ό(𝔍)βˆ¨β„‘=πœ”π›Ό(Ξ©)βˆ¨πœ”βˆ—π›Ό(𝔍)βˆ¨β„‘=πœ”βˆ—π›Ό(𝔍)βˆ¨πœ”βˆ—π›Ό(β„‘).
(3) Clearly, πœ”βˆ—π›Ό(β„‘βˆ©π”)βŠ‚πœ”βˆ—π›Ό(𝔍)βˆ©πœ”βˆ—π›Ό(β„‘). Conversely, π‘ƒβˆˆπœ”βˆ—π›Ό(𝔍)βˆ©πœ”βˆ—π›Ό(β„‘). Then, π‘ƒβˆ—(β„‘)[𝛼]βŠ‚π‘ƒ[𝛼], and π‘ƒβˆ—(𝔍)[𝛼]βŠ‚π‘ƒ[𝛼]. So by (1) in Theorem 23, π‘ƒβˆ—(β„‘βˆ©π”)[𝛼]=π‘ƒβˆ—(β„‘)[𝛼]βˆͺπ‘ƒβˆ—(𝔍)[𝛼]βŠ‚π‘ƒ[𝛼]. Therefore, π‘ƒβˆˆπœ”βˆ—π›Ό(β„‘βˆ©π”).

Theorem 26. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, and let β„‘ be an 𝛼-ideal. π΄βˆˆπΏπ‘‹. Then, πœ”βˆ—π›Ό(𝐴)[𝛼]=πœ”π›Ό(𝐴)[𝛼] if and only if β„‘βŠ‚πœ”π›Ό(Ξ©).

Proof. Sufficiency. according to (4) in Theorem 16, πœ”βˆ—π›Ό(𝐴)[𝛼]βŠ‚πœ”π›Ό(𝐴)[𝛼]. On the other hand, if π‘₯βˆ‰πœ”βˆ—π›Ό(𝐴)[𝛼], then π‘¦βˆ‰π΄βˆ—[𝛼] and π‘¦βˆ‰π΄[𝛼]. So there are π‘ƒβˆˆπœ‚πœ”π›Ό(𝑦𝛼), πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Here, we may assume π‘¦βˆ‰πΌ[𝛼], since β„‘ is an 𝛼-ideal. Therefore, πœ”π›Ό(𝐴)[𝛼]βŠ‚πœ”π›Ό(π‘ƒβˆ¨πΌ)[𝛼]=(π‘ƒβˆ¨πΌ)[𝛼]. This implies π‘¦βˆ‰πœ”π›Ό(𝐴)[𝛼]. πœ”βˆ—π›Ό(𝐴)[𝛼]βŠƒπœ”π›Ό(𝐴)[𝛼].
Necessary. if π΄βˆˆβ„‘, then π΄βˆ—[𝛼]=βˆ…. Since 𝐴[𝛼]=(π΄βˆ¨π΄βˆ—)[𝛼]=πœ”βˆ—π›Ό(𝐴)[𝛼]=πœ”π›Ό(𝐴)[𝛼], π΄βˆˆπœ”π›Ό(Ξ©).
Since (𝑋,πœ”π›Ό(Ξ©)[𝛼]) is an opos on 𝑋. β„‘[𝛼] is an ideal on 𝑋. We have the following results.

Theorem 27. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-π‘œπ‘π‘œπ‘ , and let β„‘ be an 𝛼-ideal. Then, for every π΄βˆˆπΏπ‘‹, (𝐴[𝛼])βˆ—=π΄βˆ—[𝛼], and therefore πœ”βˆ—π›Ό(𝐴)[𝛼]=π‘π‘™βˆ—(𝐴[𝛼]).

Proof. Since π‘₯βˆ‰π΄βˆ—[𝛼]βŸΊβˆƒπ‘ƒβˆˆπœ‚πœ”π›Όξ€·π‘₯𝛼,πΌβˆˆβ„‘,𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]βŸΊβˆƒπ‘ƒ[𝛼]βˆˆπœ”π›Ό(Ξ©)[𝛼]𝑃,π‘₯∈[𝛼]ξ€Έξ…ž,πΌβˆˆβ„‘,𝐴[𝛼]βˆ©ξ€·π‘ƒ[𝛼]ξ€Έξ…ž=𝐼[𝛼]ξ€·π΄βŸΊπ‘₯βˆ‰[𝛼]ξ€Έβˆ—,(8) then (𝐴[𝛼])βˆ—=π΄βˆ—[𝛼], πœ”βˆ—π›Ό(𝐴)[𝛼]=(π΄βˆ¨π΄βˆ—)[𝛼]=𝐴[𝛼]βˆͺ(𝐴[𝛼])βˆ—=π‘π‘™βˆ—(𝐴[𝛼]).

Corollary 28. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, and let β„‘ be an 𝛼-ideal. Then, for every π΄βˆˆπΏπ‘‹, π’³βˆ—π΄[𝛼]=π’³π΄βˆ—[𝛼]=𝒳(𝐴[𝛼])βˆ—.

Proof. π‘₯βˆ‰(π’³βˆ—π΄[𝛼])[𝛼]⇔thereexistsπ‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘,𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]⇔π‘₯βˆ‰π΄βˆ—[𝛼]⇔π‘₯βˆ‰(π’³π΄βˆ—[𝛼])[𝛼]. So π’³βˆ—π΄[𝛼]=π’³π΄βˆ—[𝛼]. Besides, by Theorem 27, the later equation is easy.

Corollary 29. Let (𝑋,Ξ”) be an opos on 𝑋, β„βŠ‚π‘‹ an ideal. The induced πœ”π›Ό-opos and 𝛼-ideal are denoted, respectively, by (𝐿𝑋,πœ”π›Ό(πœ”πΏ(Ξ”))) and β„‘={π’³πΌβˆΆπΌβˆˆβ„}. Then, πœ”βˆ—π›Ό(β„‘,πœ”π›Ό(πœ”πΏ(Ξ”)))[𝛼]=Ξ”βˆ—.

Proof. By Theorem 27 and Corollary 28, πœ”βˆ—π›Ό(β„‘,πœ”π›Ό(πœ”πΏ(Ξ”)))[𝛼]βŠ‚Ξ”βˆ—. On other hand, again by Corollary 28, for every π‘ˆβŠ‚π‘‹, πœ”βˆ—π›Ό(π’³π‘ˆ)[𝛼]=(π’³π‘ˆβˆ¨π’³βˆ—π‘ˆ)[𝛼]=π‘ˆβˆͺ(π’³π‘ˆβˆ—)[𝛼]=π‘ˆβˆͺπ‘ˆβˆ—=π‘π‘™βˆ—(π‘ˆ).

5. Compatibility of πœ”π›Ό(Ξ©) with β„‘

Definition 30. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. β„œβŠ‚πœ”π›Ό(Ξ©) is called an 𝛼-πœ”π›Ό-remote neighborhood family of 𝐴 (briefly, 𝛼-πœ”π›Ό-RF of 𝐴, if for all π‘₯∈𝐴[𝛼], there is π‘ƒβˆˆβ„œ, such that π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼)).

Definition 31. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos. Let β„‘ be an 𝛼-ideal. πœ”π›Ό(Ξ©) is said to be compatible with β„‘, denoted by πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘, if for any π΄βˆˆπΏπ‘‹ and π‘₯∈𝐴[𝛼], there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼], then π΄βˆˆβ„‘.

Example 32. In Example 18, we know πœ”π›Ό={(0,0),(0,1/2),(0,1),(1/2,1/2),(1/2,1),(1,1/2),(1,1)}, πœ‚πœ”π›Ό(π‘₯𝛼)={(0,0),(0,1/2),(0,1)},πœ‚πœ”π›Ό(𝑦𝛼)={(0,0)}. Take β„‘1=𝐿𝑋, β„‘2={(0,0)}, β„‘3={(0,0),(1/2,0),(1,0)}, β„‘4={(0,0),(0,1/2),(0,1)}.
It is easy to check ℑ𝑖,(𝑖≀4) are all 𝛼-ideals, πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘1 and πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘2. Take 𝐺=(1/2,1/2), then π‘₯∈𝐺[𝛼] and 𝑅=(0,1/2)βˆˆπœ‚πœ”π›Ό(π‘₯𝛼),𝐽=(1/2,0)βˆˆβ„‘3, such that 𝐺[𝛼]βŠ‚(π‘…βˆ¨π½)[𝛼]. But πΊβˆ‰β„‘3, therefore πœ”π›ΌΜΈβˆΌ(Ξ©)𝛼ℑ3. Furthermore, for all π΄βˆˆπΏπ‘‹, if 𝐴[𝛼]=βˆ… or 𝐴[𝛼]={𝑦}, then π΄βˆˆβ„‘4. If 𝐴[𝛼]={π‘₯} or 𝐴[𝛼]={π‘₯,𝑦}, there does not exist πΈβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘4, such that 𝐴[𝛼]βŠ‚(π‘…βˆ¨π½)[𝛼]. Therefore, πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘4.

Theorem 33. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, β„‘ an ideal, πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘. Then, 𝔅=πœ”βˆ—π›Ό(β„‘).

Proof. For every π΅βˆˆπœ”βˆ—π›Ό(β„‘), π΅βˆ—[𝛼]βŠ‚π΅[𝛼]. Put 𝐴=π΅βˆ’π΅βˆ—(β„‘). If π‘₯∈𝐴[𝛼], then π‘₯βˆ‰π΅βˆ—[𝛼]. So there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and πΌβˆˆβ„‘, such that 𝐡[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Thus, 𝐴[𝛼]βŠ‚π΅[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Since πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘, we get π΄βˆˆβ„‘. Besides, by (4) in Theorem 16, π΅βˆ—βˆˆπœ”π›Ό(Ξ©). Notice that 𝐡[𝛼]=(π΄βˆ¨π΅βˆ—)[𝛼]. Thus, π΅βˆˆπ”…, and πœ”βˆ—π›Ό(β„‘)βŠ‚π”…. The reverse inclusion is obvious. Therefore, 𝔅=πœ”βˆ—π›Ό(β„‘).

Lemma 34. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, β„‘ an 𝛼-ideal. Then, for every π΄βˆˆπΏπ‘‹, (π΄βˆ’π΄βˆ—(β„‘))[𝛼]∩(π΄βˆ’π΄βˆ—(β„‘))βˆ—(β„‘)[𝛼]=βˆ….

Proof. If π‘₯∈(π΄βˆ’π΄βˆ—(β„‘))[𝛼], so π‘₯βˆ‰π΄βˆ—(β„‘)[𝛼]. That is, there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼),πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. As (π΄βˆ’π΄βˆ—(β„‘))[𝛼]βŠ‚π΄[𝛼], we have π‘₯βˆ‰(π΄βˆ’π΄βˆ—(β„‘))βˆ—(β„‘)[𝛼]. Therefore, the conclusion holds.

Theorem 35. Let (𝐿𝑋,πœ”π›Ό(Ξ©)) be an πœ”π›Ό-opos, β„‘ an ideal. Then, the following statements are equivalent.(1)πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘.(2)If π΄βˆˆπΏπ‘‹ has an 𝛼-πœ”π›Ό-RF β„œ, satisfying for all π‘ƒβˆˆβ„œ, there is πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼], then π΄βˆˆβ„‘.(3)For all π΄βˆˆπΏπ‘‹, (π΄βˆ§π΄βˆ—(β„‘))[𝛼]=βˆ…β‡’π΄βˆˆβ„‘.(4)For all π΄βˆˆπΏπ‘‹, π΄βˆ’π΄βˆ—(β„‘)βˆˆβ„‘.(5)For all π΄βˆˆπœ”βˆ—π›Ό(β„‘), π΄βˆ’π΄βˆ—(β„‘)βˆˆβ„‘.(6)For all π΄βˆˆπΏπ‘‹, if βˆ„βˆ…β‰ π΅[𝛼]βŠ‚π΅βˆ—(β„‘)[𝛼], 𝐡[𝛼]βŠ‚π΄[𝛼], then π΄βˆˆβ„‘.

Proof. (1) β‡’ (2) If π΄βˆˆπΏπ‘‹ has a 𝛼-πœ”π›Ό-RF β„œ, satisfying the condition in (2), then for every π‘₯∈𝐴[𝛼], there are π‘ƒβˆˆβ„œ and πΌβˆˆβ„‘ such that π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. As πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘, we have π΄βˆˆβ„‘.
(2) β‡’ (1) By Definitions 30 and 31, clearly.
(1) β‡’ (3) For all π΄βˆˆπΏπ‘‹, (π΄βˆ§π΄βˆ—(β„‘))[𝛼]=βˆ…. If 𝐴[𝛼]=βˆ…, then π΄βˆˆβ„‘. If π‘₯∈𝐴[𝛼], then π‘₯βˆ‰π΄βˆ—(β„‘)[𝛼], there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Because πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘, we have π΄βˆˆβ„‘.
(3) β‡’ (1) if π΄βˆˆπΏπ‘‹ satisfying, for every π‘₯∈𝐴[𝛼], there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼]. Then, π‘₯βˆ‰π΄βˆ—(β„‘)[𝛼]. By (3), we have π΄βˆˆβ„‘. Hence, πœ”π›Ό(Ξ©)βˆΌπ›Όβ„‘.
(3) β‡’ (4) By Lemma 34, obviously.
(4) β‡’ (5) Directly.
(5) β‡’ (1) If for every π΄βˆˆπΏπ‘‹ and π‘₯∈𝐴[𝛼], there are π‘ƒβˆˆπœ‚πœ”π›Ό(π‘₯𝛼) and πΌβˆˆβ„‘, such that 𝐴[𝛼]βŠ‚(π‘ƒβˆ¨πΌ)[𝛼], this means (π΄βˆ§π΄βˆ—(β„‘))[𝛼]=βˆ…. Put 𝐡=π΄βˆ¨π΄βˆ—, then π΅βˆ—[𝛼]=(π΄βˆ¨π΄βˆ—)βˆ—[𝛼]=π΄βˆ—[𝛼]βˆͺ(π΄βˆ—)βˆ—[𝛼]=π΄βˆ—[𝛼]. So πœ”βˆ—π›Ό(𝐡)[𝛼]=(π΅βˆ¨π΅βˆ—(β„‘))[𝛼]=𝐡[𝛼]. This means π΅βˆˆπœ”βˆ—π›Ό(β„‘). By (5), we have π΅βˆ’π΅βˆ—βˆˆβ„‘. Since ξ€·π΅βˆ’π΅βˆ—ξ€Έ[𝛼]=ξ€·ξ€·π΄βˆ¨π΄βˆ—ξ€Έβˆ’ξ€·π΄βˆ¨π΄βˆ—ξ€Έβˆ—ξ€Έ[𝛼]=ξ€·π΄βˆ¨π΄βˆ—ξ€Έ[𝛼]βˆ’π΄βˆ—[𝛼]=𝐴[𝛼],(9) we draw a conclusion: for every π‘₯∈𝐴[𝛼], π‘₯βˆ‰π΅βˆ—[𝛼]=π΄βˆ—[𝛼]. Therefore, π΄βˆˆβ„‘.
(4) β‡’ (6) For any π΄βˆˆπΏπ‘‹, 𝐴=(π΄βˆ’π΄βˆ—)∨(π΄βˆ§π΄βˆ—). So π΄βˆ—[𝛼]=((π΄βˆ’π΄βˆ—)βˆ—[𝛼]βˆͺ