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Advances in Fuzzy Systems
VolumeΒ 2012Β (2012), Article IDΒ 582105, 8 pages
doi:10.1155/2012/582105
Research Article

On 𝜢 -Layer Ideal Topologies

Xiu-Yun Wu1Β and Li-Li Xie2

1Department of Mathematics, Institute of Computational Mathematics, Hunan University of Science and Engineering, Yongzhong 425100, China
2Department of English Teaching, Hunan University of Science and Engineering, Yongzhong 425100, China

Received 8 April 2012; Accepted 22 June 2012

Academic Editor: KatsuhiroΒ Honda

Copyright Β© 2012 Xiu-Yun Wu and Li-Li Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to study theory of two different kinds of 𝛼 -layer order-preserving operator space, namely, πœ” 𝛼 -opos and πœ” βˆ— 𝛼 ( 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” 𝛼 -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— 𝛼 ( 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 Janković 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” 𝛼 -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” 𝛼 -opos. We also prove it preserves many good properties. In the second part of the paper, we establish the theory of πœ” βˆ— 𝛼 ( 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” 𝛼 -opos and an 𝛼 -ideal. On the basis of its basic properties, we form πœ” βˆ— 𝛼 ( 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” 𝛼 -opos as well as the relations of the correspondent πœ” βˆ— 𝛼 -opos. We also obtain some equivalent conditions of πœ” βˆ— 𝛼 -opos under compatibility of 𝛼 -ideal and πœ” 𝛼 -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 ) , ℐ βŠ‚ 𝑋 is an ideal, π‘ˆ βŠ‚ 𝑋 . The concept of local function of π‘ˆ with respect to 𝜏 and ℐ is given by π‘ˆ βˆ— = { π‘₯ ∈ 𝑋 ∢ π‘ˆ ∩ 𝑉 βˆ‰ ℐ , 𝑉 ∈ 𝒩 ( 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 involution′. 𝑋 , π‘Œ will always denote nonempty crisp sets, A mapping 𝐴 ∢ 𝑋 β†’ 𝐿 is called an 𝐿 -fuzzy set. 𝐿 𝑋 is the set of all 𝐿 -fuzzy sets on 𝑋 . An element 𝑒 ∈ 𝐿 is called an irreducible element in 𝐿 , if 𝑝 ∨ π‘ž = 𝑒 implies 𝑝 = 𝑒 or π‘ž = 𝑒 , where 𝑝 , π‘ž ∈ 𝐿 . The set of all nonzero irreducible elements in 𝐿 will be denoted by 𝑀 ( 𝐿 ) (see [1]). If π‘₯ ∈ 𝑋 , 𝛼 ∈ 𝑀 ( 𝐿 ) , then π‘₯ 𝛼 is called a molecule in 𝐿 𝑋 . The set of all molecules in 𝐿 𝑋 is denoted by 𝑀 βˆ— ( 𝐿 𝑋 ) . If 𝐴 ∈ 𝐿 𝑋 , 𝛼 ∈ 𝑀 ( 𝐿 ) , take 𝐴 [ 𝛼 ] = { π‘₯ ∈ 𝑋 ∢ 𝐴 ( π‘₯ ) β‰₯ 𝛼 } . If 𝐸 βŠ‚ 𝑋 , the complement of 𝐸 , denoted by 𝐸 β€² , and 𝐸 β€² = 𝑋 βˆ’ 𝐸 = { 𝑦 ∈ 𝑋 ∢ 𝑦 βˆ‰ 𝐸 } [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” ∢ 𝐿 𝑋 β†’ 𝐿 𝑋 is called a 𝐿 -fuzzy order preserving operator in 𝐿 𝑋 , if it satisfies (1) πœ” ( 1 𝑋 ) = 1 𝑋 , (2) for all 𝐴 , 𝐡 ∈ 𝐿 𝑋 and 𝐴 ≀ 𝐡 implies πœ” ( 𝐴 ) ≀ πœ” ( 𝐡 ) . A set 𝐴 ∈ 𝐿 𝑋 is called an πœ” -set, if πœ” ( 𝐴 ) = 𝐴 . The set of all πœ” -sets in 𝐿 𝑋 is denoted by Ξ© . And ( 𝐿 𝑋 , Ξ© ) is called an order-preserving operator space (briefly, 𝐿 - 𝑓 opos). A molecule π‘₯ 𝛼 ∈ 𝑀 βˆ— ( 𝐿 𝑋 ) , 𝑃 ∈ Ξ© , 𝑃 is called an πœ” -remote neighborhood of π‘₯ 𝛼 , if π‘₯ 𝛼 βˆ‰ 𝑃 . The set of all πœ” -remote neighborhood of π‘₯ 𝛼 is denoted by πœ” πœ‚ ( π‘₯ 𝛼 ) . Let π‘₯ 𝛼 ∈ 𝑀 βˆ— ( 𝐿 𝑋 ) , 𝐴 ∈ 𝐿 𝑋 , π‘₯ 𝛼 is called an πœ” -adherent point of 𝐴 , if for all 𝑃 ∈ πœ” πœ‚ ( π‘₯ 𝛼 ) , 𝐴 βˆ‰ 𝑃 . The union of all πœ” -adherent points of 𝐴 is called the πœ” -closure of 𝐴 , denoted by 𝐴 βˆ’ πœ” . A set 𝐴 ∈ 𝐿 𝑋 is called πœ” -closed, if 𝐴 βˆ’ πœ” = 𝐴 . The set of all πœ” -closed sets in 𝐿 𝑋 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 𝜎 ∢ 𝑋 β†’ 𝑋 is called an order-preserving operator in 𝑋 , if it satisfies (1) 𝜎 ( 𝑋 ) = 𝑋 , (2) for all 𝐴 βŠ‚ 𝐡 βŠ‚ 𝑋 implies 𝜎 ( 𝐴 ) βŠ‚ 𝜎 ( 𝐡 ) . A set 𝐴 βŠ‚ 𝑋 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‘₯ ∈ 𝑋 , 𝑃 ∈ Ξ” , 𝑃 is called a 𝜎 -remote neighborhood of π‘₯ , if there is 𝑄 βŠ‚ 𝑋 , such that π‘₯ βˆ‰ 𝑄 , 𝑃 βŠ‚ 𝑄 . The set of all 𝜎 -remote neighborhood of π‘₯ is denoted by 𝜎 πœ‚ ( π‘₯ ) . Let π‘₯ ∈ 𝑋 , 𝐴 βŠ‚ 𝑋 , π‘₯ is called a 𝜎 -adherent point of 𝐴 , if for all 𝑃 ∈ 𝜎 πœ‚ ( π‘₯ ) , 𝐴 βŠ„ 𝑃 . The union of all 𝜎 -adherent points of 𝐴 is called the 𝜎 -closure of 𝐴 , denoted by 𝐴 βˆ’ 𝜎 . A set 𝐴 βŠ‚ 𝑋 is called 𝜎 -closed, if 𝐴 βˆ’ 𝜎 = 𝐴 . The set of all 𝜎 -closed sets in 𝑋 is denoted by Ξ” βˆ’ 𝜎 . Ξ” βˆ’ 𝜎 is finite union and infinite intersection preserving.

Let ( 𝑋 , Ξ” ) be an opos, and let 𝐿 be an fuzzy lattice, for all 𝛼 ∈ 𝐿 . A fuzzy set 𝐴 ∢ 𝐿 β†’ 𝑋 is called a lower continuous function, if { π‘₯ ∈ 𝑋 ∢ 𝐴 ( π‘₯ ) ≀ 𝛼 } ∈ Ξ” βˆ’ 𝜎 . Then, the set of all the lower continuous functions, denoted by πœ” 𝐿 ( Ξ” ) , consists an 𝐿 -fuzzy cotopology in 𝐿 𝑋 . The space ( 𝐿 𝑋 , πœ” 𝐿 ( Ξ” ) ) is called the induced 𝐿 - 𝑓 opos by ( 𝑋 , Ξ” ) .

A nonempty subfamily β„‘ of 𝐿 𝑋 is called an 𝛼 -ideal, if β„‘ satisfies the following conditions:(1) 𝐴 ∈ β„‘ , and 𝐡 [ 𝛼 ] βŠ‚ 𝐴 [ 𝛼 ] implies 𝐡 ∈ β„‘ ,(2) 𝐴 , 𝐡 ∈ β„‘ implies 𝐴 ∨ 𝐡 ∈ β„‘ .

It is easy to check if 𝐡 ∈ 𝐿 𝑋 , 𝐴 ∈ β„‘ , such that 𝐡 [ 𝛼 ] = 𝐴 [ 𝛼 ] , then 𝐡 ∈ β„‘ . Moreover, β„‘ of 𝐿 𝑋 is an 𝛼 -idea if and only if β„‘ [ 𝛼 ] = { 𝐴 [ 𝛼 ] ∢ 𝐴 ∈ β„‘ } is an ideal on 𝑋 . If 𝐡 ∈ 𝐿 𝑋 , 𝐴 ∈ β„‘ , such that 𝐡 [ 𝛼 ] = 𝐴 [ 𝛼 ] , then 𝐡 ∈ β„‘ . If β„‘ , 𝔍 are two 𝛼 -ideals, then β„‘ ∨ 𝔍 = { 𝐼 ∨ 𝐽 ∢ 𝐼 ∈ β„‘ , 𝐽 ∈ 𝔍 } and β„‘ ∩ 𝔍 are 𝛼 -ideals too.

In this paper, if 𝔉 βŠ‚ 𝐿 𝑋 , we denote 𝔉 [ 𝛼 ] = { 𝐴 ∢ 𝐴 ∈ 𝔉 } and 𝔍 ξ…ž = { 𝐴 ξ…ž ∢ 𝐴 ∈ 𝔉 } .

3. πœ” 𝛼 -Closed Set and πœ” 𝛼 -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 ( 𝐿 𝑋 , Ξ© ) be an 𝐿 - 𝑓 opos, 𝛼 ∈ 𝑀 ( 𝐿 ) . An operator πœ” 𝛼 ∢ 𝐿 𝑋 β†’ 𝐿 𝑋 is defined by, for all 𝐴 ∈ 𝐿 𝑋 , πœ” 𝛼  ξ€½ ( 𝐴 ) = 𝐺 ∈ Ξ© βˆ’ πœ” ∢ 𝐺 [ 𝛼 ] βŠƒ 𝐴 [ 𝛼 ] ξ€Ύ . ( 1 )

Theorem 2. Let ( 𝐿 𝑋 , Ξ© ) be an 𝐿 - 𝑓 opos, 𝛼 ∈ 𝑀 ( 𝐿 ) . 𝐴 , 𝐡 ∈ 𝐿 𝑋 , then the following statements hold:(1) 𝐴 [ 𝛼 ] βŠ‚ 𝐡 [ 𝛼 ] implies πœ” 𝛼 ( 𝐴 ) ≀ πœ” 𝛼 ( 𝐡 ) ;(2) 𝐴 [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] ;(3) πœ” 𝛼 ( 𝐴 ∨ 𝐡 ) = πœ” 𝛼 ( 𝐴 ) ∨ πœ” 𝛼 ( 𝐡 ) ;(4) πœ” 𝛼 ( πœ” 𝛼 ( 𝐴 ) ) = πœ” 𝛼 ( 𝐴 ) .

Definition 3. Let ( 𝐿 𝑋 , Ξ© ) be an 𝐿 - 𝑓 opos, 𝛼 ∈ 𝑀 ( 𝐿 ) . A set 𝐴 ∈ 𝐿 𝑋 is called πœ” 𝛼 -closed, if πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] = 𝐴 [ 𝛼 ] . The set of all πœ” 𝛼 -closed sets in 𝐿 𝑋 is denoted by πœ” 𝛼 ( Ξ© ) . ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) is called an πœ” 𝛼 -order-preserving operator space (briefly, πœ” 𝛼 -opos).

Theorem 4. Let ( 𝐿 𝑋 , Ξ© ) be an 𝐿 - 𝑓 opos, 𝛼 ∈ 𝑀 ( 𝐿 ) . Then,(1) 1 𝑋 ∈ πœ” 𝛼 ( Ξ© ) ;(2)if { 𝐴 𝑖 ∢ 𝑖 = 1 , 2 , … , 𝑛 } βŠ‚ πœ” 𝛼 ( Ξ© ) , then ⋁ 𝑛 𝑖 = 1 𝐴 𝑖 ∈ πœ” 𝛼 ( Ξ© ) ;(3)if { 𝐴 𝑖 ∢ 𝑖 ∈ 𝐼 } βŠ‚ πœ” 𝛼 ( Ξ© ) , then ∧ 𝑖 ∈ 𝐼 𝐴 𝑖 ∈ πœ” 𝛼 ( Ξ© ) .

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

Corollary 6. Let ( 𝐿 𝑋 , Ξ© ) be an 𝐿 - 𝑓 opos, 𝛼 ∈ 𝑀 ( 𝐿 ) . Then, πœ” 𝛼 ( Ξ© ) consists an 𝐿 -fuzzy cotopology on 𝐿 𝑋 .

Lemma 7. Let ( 𝑋 , Ξ” ) be an opos, ( 𝐿 𝑋 , πœ” 𝐿 ( Ξ” ) ) be an 𝐿 - 𝑓 opos induced by ( 𝑋 , Ξ” ) and let 𝐸 βŠ‚ 𝑋 . Then, 𝐸 ∈ Ξ” βˆ’ 𝜎 if and only if πœ’ 𝐸 ∈ ( πœ” 𝐿 ( Ξ” ) ) βˆ’ πœ” .

Theorem 8. Let ( 𝑋 , Ξ” ) be an opos, ( 𝐿 𝑋 , πœ” 𝐿 ( Ξ” ) ) an 𝐿 - 𝑓 opos induced by ( 𝑋 , Ξ” ) . 𝐴 ∈ 𝐿 𝑋 , 𝛼 ∈ 𝑀 ( 𝐿 ) . Then, 𝐴 ∈ πœ” 𝛼 ( πœ” 𝐿 ( Ξ” ) ) if and only if 𝐴 [ 𝛼 ] ∈ Ξ” βˆ’ 𝜎 .

Theorem 9. Let ( 𝑋 , Ξ” ) be an opos, ( 𝐿 𝑋 , πœ” 𝐿 ( Ξ” ) ) is an 𝐿 - 𝑓 opos induced by ( 𝑋 , Ξ” ) . Then, 𝐴 ∈ πœ” 𝐿 ( Ξ” ) if and only if for all 𝛼 ∈ 𝑀 ( 𝐿 ) , 𝐴 ∈ πœ” 𝛼 ( πœ” 𝐿 ( Ξ” ) ) .

Definition 10. Let ( 𝐿 𝑋 , πœ” 1 𝛼 ( Ξ© 1 ) ) , ( 𝐿 π‘Œ , πœ” 2 𝛼 ( Ξ© 2 ) ) be πœ” 1 𝛼 -opos, πœ” 2 𝛼 -opos, respectively. An 𝐿 -fuzzy homomorphism 𝑓 β†’ ∢ 𝐿 𝑋 β†’ 𝐿 π‘Œ is called ( πœ” 1 𝛼 ( Ξ© 1 ) , πœ” 2 𝛼 ( Ξ© 2 ) ) -continuous, if for all 𝐡 ∈ πœ” 2 𝛼 ( Ξ© 2 ) , then 𝑓 ← ( 𝐡 ) ∈ πœ” 1 𝛼 ( Ξ© 1 ) .

Theorem 11. Let ( 𝐿 𝑋 , πœ” 1 𝛼 ( Ξ© 1 ) ) , ( 𝐿 π‘Œ , πœ” 2 𝛼 ( Ξ© 2 ) ) be πœ” 1 𝛼 -opos, πœ” 2 𝛼 -opos, respectively. 𝑓 β†’ ∢ 𝐿 𝑋 β†’ 𝐿 π‘Œ is an 𝐿 -fuzzy mapping. Then, the following statements are equivalent:(1) 𝑓 β†’ is ( πœ” 1 𝛼 ( Ξ© 1 ) , πœ” 2 𝛼 ( Ξ© 2 ) ) -continuous;(2)for all 𝐡 ∈ 𝐿 π‘Œ , πœ” 1 𝛼 ( 𝑓 ← ( 𝐡 ) ) [ 𝛼 ] βŠ‚ 𝑓 ← ( πœ” 2 𝛼 ( 𝐡 ) ) [ 𝛼 ] ;(3)for all 𝐴 ∈ 𝐿 𝑋 , 𝑓 β†’ ( πœ” 1 𝛼 ( 𝐴 ) ) [ 𝛼 ] βŠ‚ πœ” 2 𝛼 ( 𝑓 β†’ ( 𝐴 ) ) [ 𝛼 ] .

Definition 12. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, π‘₯ 𝛼 ∈ 𝑀 βˆ— ( 𝐿 𝑋 ) , 𝐴 ∈ πœ” 𝛼 ( Ξ© ) . 𝐴 is called an πœ” 𝛼 -closed remote neighborhood of π‘Ž , if π‘₯ βˆ‰ 𝐴 [ 𝛼 ] . The set of all πœ” 𝛼 -closed remote neighborhood of π‘₯ 𝛼 will be denoted by πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) . An fuzzy point π‘₯ 𝛼 is called πœ” 𝛼 -adherent point of 𝐡 ∈ 𝐿 𝑋 , if for every 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐡 ( [ 𝛼 ] ) βŠ„ 𝑃 ( [ 𝛼 ] ) .

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

Theorem 14. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, π‘₯ 𝛼 ∈ 𝑀 βˆ— ( 𝐿 𝑋 ) , 𝐴 ∈ 𝐿 𝑋 , then(1) π‘₯ 𝛼 is an πœ” 𝛼 -adherent point of 𝐴 ∈ 𝐿 𝑋 if and only if π‘₯ ∈ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] ,(2) πœ” 𝛼 ( 𝐴 ) is the union of all πœ” 𝛼 -adherent points of 𝐴 .

4. πœ” 𝛼 - β„‘ -Closed Sets

Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. β„‘ βŠ‚ 𝐿 𝑋 be an 𝛼 -ideal. π‘₯ 𝛼 ∈ 𝑀 βˆ— ( 𝐿 𝑋 ) , πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) be the πœ” 𝛼 -remote family of π‘₯ 𝛼 . For any 𝐴 ∈ 𝐿 𝑋 , take 𝐴 βˆ— ξ€· β„‘ , πœ” 𝛼 ξ€Έ =  ξ€½ π‘₯ 𝛼 ∢ 𝐴 [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] , βˆ€ 𝑃 ∈ πœ‚ πœ” 𝛼 ξ€· π‘₯ 𝛼 ξ€Έ ξ€Ύ . , 𝐼 ∈ β„‘ ( 2 ) Then, 𝐴 βˆ— ( β„‘ , πœ” 𝛼 ) is called the local πœ” 𝛼 -remote function of 𝐴 with respect to β„‘ and πœ” 𝛼 ( Ξ© ) , simply denoted by 𝐴 βˆ— ( β„‘ ) or 𝐴 βˆ— .

Remark 15. If 𝐿 = [ 0 , 1 ] , πœ” is the closure operator, then ( 𝐴 βˆ— [ 𝛼 ] ) β€² is the local function of 𝐴 [ 𝛼 ] in ( 𝑋 , πœ” 𝛼 ( Ξ© ) [ 𝛼 ] ) , and π’œ βˆ— = ⋁ 𝛼 ∈ 𝐿 ( 𝐴 βˆ— [ 𝛼 ] ) β€² is the fuzzy local function of 𝐴 in [7]. Furthermore, we have ( 𝐴 βˆ— [ 𝛼 ] ) β€² = ( 𝐴 [ 𝛼 ] ) β€² βˆ— .

Theorem 16. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. β„‘ , 𝔍 βŠ‚ 𝐿 𝑋 be two 𝛼 -ideals. 𝐴 , 𝐡 ∈ 𝐿 𝑋 . Then, the following statements hold.(1) β„‘ [ 𝛼 ] = { βˆ… } β‡’ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] , β„‘ [ 𝛼 ] = 𝒫 ( 𝑋 ) β‡’ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] = βˆ… .(2) 𝐴 [ 𝛼 ] βŠ‚ 𝐡 [ 𝛼 ] β‡’ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βŠ‚ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] .(3) β„‘ [ 𝛼 ] βŠ‚ 𝔍 [ 𝛼 ] β‡’ 𝐴 βˆ— ( 𝔍 ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] .(4) 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] .(5) ( 𝐴 βˆ— ( β„‘ ) ) βˆ— ( β„‘ ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] .(6) 𝐴 ∈ β„‘ β‡’ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] = βˆ… .(7) ( 𝐴 ∨ 𝐡 ) βˆ— ( β„‘ ) [ 𝛼 ] = 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] .(8) ( 𝐴 βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] = ( ( 𝐴 βˆ’ 𝐡 ) βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] βŠ‚ ( 𝐴 βˆ’ 𝐡 ) βˆ— [ 𝛼 ] .(9) 𝐡 ∈ β„‘ β‡’ ( 𝐴 ∨ 𝐡 ) βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] = ( 𝐴 βˆ’ 𝐡 ) βˆ— [ 𝛼 ] .(10) 𝑃 ∈ πœ” 𝛼 ( Ξ© ) β‡’ 𝑃 βˆ— [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] .

Proof. (1) Suppose β„‘ [ 𝛼 ] = { βˆ… } , by Theorem 14, we have π‘₯ 𝛼 ≀ 𝐴 βˆ— ξ€· β„‘ , πœ” 𝛼 ξ€Έ ⟺ π‘₯ 𝛼 ≀  ξ€½ π‘₯ 𝛼 ∢ 𝐴 [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] , βˆ€ 𝑃 ∈ πœ‚ πœ” 𝛼 ξ€· π‘₯ 𝛼 ξ€Έ ξ€Ύ ⟺ π‘₯ 𝛼 ≀ πœ” 𝛼 ( 𝐴 ) . ( 3 )
If 𝑋 ∈ β„‘ [ 𝛼 ] , then there must be 𝐡 ∈ β„‘ , such that 𝐴 [ 𝛼 ] = 𝐡 [ 𝛼 ] = { βˆ… } . Therefore, 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] = βˆ… .
(2), (3), and (6) Easy.
(4) π‘₯ βˆ‰ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] implies πœ” 𝛼 ( 𝐴 ) ∈ πœ‚ 𝛼 ( π‘₯ 𝛼 ) . By 𝐴 [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] , we have π‘₯ βˆ‰ 𝐴 βˆ— [ 𝛼 ] . So 𝐴 βˆ— [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] . Thus, πœ” 𝛼 ( 𝐴 ) βˆ— [ 𝛼 ] βŠ‚ πœ” 𝛼 ( πœ” 𝛼 ( 𝐴 ) ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] . Besides, π‘₯ ∈ πœ” 𝛼 ( 𝐴 βˆ— ) [ 𝛼 ] implies for all 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐴 βˆ— [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] . Take 𝑦 ∈ 𝐴 βˆ— [ 𝛼 ] βˆ’ 𝑃 [ 𝛼 ] , so 𝑃 ∈ πœ‚ πœ” 𝛼 ( 𝑦 𝛼 ) and for all 𝑄 ∈ πœ‚ πœ” 𝛼 ( 𝑦 𝛼 ) , 𝐼 ∈ β„‘ , 𝐴 [ 𝛼 ] βŠ„ 𝑄 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . Particularly, let 𝑃 = 𝑄 . So 𝐴 [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . This means π‘₯ ∈ 𝐴 βˆ— [ 𝛼 ] . Hence, πœ” 𝛼 ( 𝐴 βˆ— ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— [ 𝛼 ] . Therefore, πœ” 𝛼 ( 𝐴 βˆ— ) [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] .
(5) By (4), ( 𝐴 βˆ— ( β„‘ ) ) βˆ— ( β„‘ ) [ 𝛼 ] = ( πœ” 𝛼 ( 𝐴 βˆ— ( β„‘ ) ) ) βˆ— ( β„‘ ) [ 𝛼 ] = πœ” 𝛼 ( πœ” 𝛼 ( 𝐴 βˆ— ( β„‘ ) ) ) [ 𝛼 ] = ( πœ” 𝛼 ( 𝐴 βˆ— ( β„‘ ) ) ) [ 𝛼 ] = 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] .
(7) By (2), ( 𝐴 ∨ 𝐡 ) βˆ— ( β„‘ ) [ 𝛼 ] βŠƒ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] . Conversely, if and only if π‘₯ ∈ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] , then for all 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ , 𝐴 [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] , and 𝐡 [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . Hence, ( 𝐴 ∨ 𝐡 ) [ 𝛼 ] βŠ„ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . This implies π‘₯ ∈ ( 𝐴 ∨ 𝐡 ) βˆ— ( β„‘ ) [ 𝛼 ] . So ( 𝐴 ∨ 𝐡 ) βˆ— ( β„‘ ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] .
(8) For each π‘₯ ∈ ( 𝐴 βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] , we may remark π‘₯ ∈ ( 𝐴 βˆ’ 𝐡 ) βˆ— [ 𝛼 ] . If not, there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ , such that ( 𝐴 βˆ’ 𝐡 ) [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . Thus, 𝐴 [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] βˆͺ 𝐡 [ 𝛼 ] . Since π‘₯ βˆ‰ 𝐡 βˆ— [ 𝛼 ] , there are 𝑄 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐽 ∈ β„‘ , 𝐡 [ 𝛼 ] βŠ‚ 𝑄 [ 𝛼 ] βˆͺ 𝐽 [ 𝛼 ] . This shows 𝐴 [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] βˆͺ 𝐡 [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] βˆͺ 𝑄 [ 𝛼 ] βˆͺ 𝐽 [ 𝛼 ] = ( 𝑃 ∨ 𝑄 ) [ 𝛼 ] βˆͺ ( 𝐼 ∨ 𝐽 ) [ 𝛼 ] . ( 4 ) Consequently, π‘₯ βˆ‰ 𝐴 βˆ— [ 𝛼 ] . But this contradicts with π‘₯ ∈ ( 𝐴 βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] . Therefore, ( 𝐴 βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] βŠ‚ ( ( 𝐴 βˆ’ 𝐡 ) βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] . The reverse inclusion is obvious. So ( 𝐴 βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] = ( ( 𝐴 βˆ’ 𝐡 ) βˆ— βˆ’ 𝐡 βˆ— ) [ 𝛼 ] .
(9) Obviously, ( 𝐴 ∨ 𝐡 ) βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] βˆͺ 𝐡 βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] . And ( 𝐴 βˆ’ 𝐡 ) βˆ— [ 𝛼 ] βŠ‚ 𝐴 βˆ— [ 𝛼 ] . Conversely, π‘₯ βˆ‰ ( 𝐴 βˆ’ 𝐡 ) βˆ— [ 𝛼 ] , there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ , ( 𝐴 βˆ’ 𝐡 ) [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] . So 𝐴 [ 𝛼 ] = ( 𝐴 βˆ’ 𝐡 ) [ 𝛼 ] βˆͺ 𝐡 [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] βˆͺ 𝐼 [ 𝛼 ] βˆͺ 𝐡 [ 𝛼 ] = 𝑃 [ 𝛼 ] βˆͺ ( 𝐼 ∨ 𝐡 ) [ 𝛼 ] . Thus, π‘₯ βˆ‰ 𝐴 βˆ— [ 𝛼 ] . Then, (9) holds.
(10) Suppose there is π‘₯ ∈ 𝑃 βˆ— [ 𝛼 ] βˆ’ 𝑃 [ 𝛼 ] . So 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) . By 𝑃 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] for each 𝐼 ∈ β„‘ , we have π‘₯ βˆ‰ 𝑃 [ 𝛼 ] , a contradiction.

Theorem 17. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. Let β„‘ βŠ‚ 𝐿 𝑋 be an ideal. 𝐴 ∈ 𝐿 𝑋 . Take πœ” βˆ— 𝛼 ( 𝐴 ) = 𝐴 ∨ 𝐴 βˆ— , then(1) πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = βˆ… , if and only if 𝐴 [ 𝛼 ] = βˆ… ,(2) πœ” βˆ— 𝛼 ( 𝐴 ∨ 𝐡 ) [ 𝛼 ] = πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] βˆͺ πœ” βˆ— 𝛼 ( 𝐡 ) [ 𝛼 ] , (3) πœ” βˆ— 𝛼 ( πœ” βˆ— 𝛼 ( 𝐴 ) ) [ 𝛼 ] = πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] ,(4) ( πœ” βˆ— 𝛼 ( 𝐴 βˆ— ) ) βˆ— [ 𝛼 ] = πœ” βˆ— 𝛼 ( 𝐴 βˆ— ) [ 𝛼 ] = ( πœ” βˆ— 𝛼 ( 𝐴 ) ) βˆ— [ 𝛼 ] = ( πœ” 𝛼 ( 𝐴 βˆ— ) ) βˆ— [ 𝛼 ] = 𝐴 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— 𝛼 ( β„‘ , Ξ© ) = { 𝐴 ∈ 𝐿 𝑋 ∢ πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = 𝐴 [ 𝛼 ] } , then πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) (simply denoted by πœ” βˆ— 𝛼 ( β„‘ ) ) consists an 𝐿 -fuzzy cotopology on 𝐿 𝑋 . So it is called πœ” 𝛼 -cotopology formed by the ideal β„‘ and πœ” 𝛼 ( Ξ© ) . The pair ( 𝐿 𝑋 , πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) ) , simply denoted by ( 𝐿 𝑋 , πœ” βˆ— 𝛼 ( β„‘ ) ) , is called πœ” βˆ— 𝛼 ( β„‘ ) - π‘œ 𝑝 π‘œ 𝑠 . By Theorem 16 (1), we have β„‘ [ 𝛼 ] = { βˆ… } β‡’ πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] , thus πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) = πœ” 𝛼 ( Ξ© ) , and β„‘ [ 𝛼 ] = 𝒫 ( 𝑋 ) β‡’ πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = 𝐴 [ 𝛼 ] , thus πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) = 𝐿 𝑋 . But in general, { βˆ… } βŠ‚ β„‘ βŠ‚ 𝒫 ( 𝑋 ) , 𝐴 [ 𝛼 ] βŠ‚ πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] , consequently, πœ” 𝛼 ( Ξ© ) βŠ‚ πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) βŠ‚ 𝐿 𝑋 . Moreover, if and only if πœ” 𝛼 ( Ξ© ) [ 𝛼 ] = { βˆ… , 𝑋 } , then πœ” βˆ— 𝛼 ( Ξ© ) = { 𝐴 ∢ 𝐴 [ 𝛼 ] ∈ β„‘ [ 𝛼 ] } .

Example 18. Let 𝑋 = { π‘₯ , 𝑦 } , 𝐿 = { 0 , 1 / 2 , 1 } . An 𝐿 -fuzzy set 𝐺 ∈ 𝐿 𝑋 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 𝐿 𝑋 . 𝛼 = 1 / 2 . Put πœ” 𝛼 β‹€ ( 𝐴 ) = { 𝐸 ∈ Ξ© ∢ 𝐸 [ 𝛼 ] βŠƒ 𝐴 [ 𝛼 ] } . Then, πœ” 𝛼 ( Ξ© ) = { 𝐴 ∢ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] = 𝐴 [ 𝛼 ] } consists an πœ” 𝛼 -cotopology, and πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) = { ( 0 , 0 ) , ( 0 , 1 / 2 ) , ( 0 , 1 ) } , πœ‚ πœ” 𝛼 ( 𝑦 𝛼 ) = { ( 0 , 0 ) } . Totally, there are four kinds of 𝛼 -ideals in 𝐿 𝑋 , namely, 𝔏 = { ( 0 , 0 ) } , 𝔐 = 𝐿 𝑋 , β„‘ = { ( 0 , 0 ) , ( 0 , 1 / 2 ) , ( 0 , 1 ) } and 𝔍 = { ( 0 , 0 ) , ( 1 / 2 , 0 ) , ( 1 , 0 ) } . 𝔏 and 𝔐 = 𝐿 𝑋 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, 𝐴 βˆ— ( β„‘ ) = 𝐡 βˆ— ( β„‘ ) = ( 0 , 0 ) , 𝐷 βˆ— ( β„‘ ) = 𝐸 βˆ— ( β„‘ ) = ( 1 / 2 , 1 / 2 ) . So 𝐴 , 𝐡 , 𝐸 ∈ πœ” βˆ— 𝛼 ( β„‘ ) , But 𝐷 βˆ‰ πœ” βˆ— 𝛼 ( β„‘ ) . In addition, πœ” βˆ— 𝛼 ( β„‘ , Ξ© ) = πœ” 𝛼 ( Ξ© ) . However, 𝐷 βˆ— ( 𝔍 ) = ( 0 , 0 ) , which means 𝐷 ∈ πœ” βˆ— 𝛼 ( 𝔍 ) . It is easy to check πœ” βˆ— 𝛼 ( 𝔍 , Ξ© ) = 𝐿 𝑋 .
In this example, we see ( πœ” 𝛼 ( 𝐡 ) ) βˆ— ( β„‘ ) [ 𝛼 ] = 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] = βˆ… . However, πœ” βˆ— 𝛼 ( 𝐡 ) [ 𝛼 ] = ( 𝐡 ∨ 𝐡 βˆ— ( β„‘ ) ) [ 𝛼 ] = { 𝑦 } . Thus, ( πœ” 𝛼 ( 𝐡 ) ) βˆ— ( β„‘ ) [ 𝛼 ] β‰  πœ” βˆ— 𝛼 ( 𝐡 ) ( β„‘ ) [ 𝛼 ] . Furthermore, ( πœ” 𝛼 ( Ξ© ) ) βˆ— ( β„‘ ) = { 𝐴 βˆ— ( β„‘ ) ∢ 𝐴 ∈ πœ” 𝛼 ( Ξ© ) } = { ( 0 , 0 ) , ( 1 / 2 , 1 / 2 ) } is not an πœ” 𝛼 -cotopology.

Theorem 19. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© 𝑖 ) ) ( 𝑖 = 1 , 2 ) be an πœ” 𝛼 -opos. πœ” 𝛼 ( Ξ© 1 ) βŠ‚ πœ” 𝛼 ( Ξ© 2 ) . β„‘ is an 𝛼 -ideal. 𝐴 ∈ 𝐿 𝑋 , then(1) 𝐴 βˆ— ( β„‘ , πœ” 𝛼 ( Ξ© 2 ) ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ , πœ” 𝛼 ( Ξ© 1 ) ) [ 𝛼 ] ,(2) 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( β„‘ , Ξ© 1 ) ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ , πœ” 𝛼 ( Ξ© 1 ) ) [ 𝛼 ] ,(3) πœ” βˆ— 𝛼 ( β„‘ , Ξ© 1 ) βŠ‚ πœ” βˆ— 𝛼 ( β„‘ , Ξ© 2 ) .

Theorem 20. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 - π‘œ 𝑝 π‘œ 𝑠 , an 𝛼 -ideal β„‘ βŠ‚ πœ” 𝛼 ( Ξ© ) . Then, πœ” βˆ— 𝛼 ( β„‘ ) = πœ” 𝛼 ( Ξ© ) .

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

Theorem 21. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, and let β„‘ be an 𝛼 -ideal. Take πœ” 𝛼 ( Ξ© ) ∨ β„‘ as the supremum πœ” 𝛼 - cotopology generated by πœ” 𝛼 ( Ξ© ) βˆͺ β„‘ . Then, πœ” βˆ— 𝛼 ( β„‘ ) = πœ” 𝛼 ( Ξ© ) ∨ β„‘ .

Proof. Put 𝔅 = { 𝑃 ∨ 𝐼 ∢ 𝑃 ∈ πœ” 𝛼 ( Ξ© ) , 𝐼 ∈ β„‘ } . It is easy to prove 𝔅 βŠ‚ πœ” βˆ— 𝛼 ( β„‘ ) is a base of πœ” βˆ— 𝛼 ( β„‘ ) .

Lemma 22. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. Let β„‘ , 𝔍 βŠ‚ 𝐿 𝑋 be two ideals. 𝐼 ∈ β„‘ , 𝐴 ∈ 𝐿 𝑋 . Then, ( 𝐴 βˆ’ 𝐼 ) βˆ— ( 𝔍 , πœ” 𝛼 ( Ξ© ) ) = 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) .

Proof. By Theorem 21, π‘₯ βˆ‰ ( 𝐴 βˆ’ 𝐼 ) βˆ— ξ€· 𝔍 , πœ” 𝛼 ξ€Έ ( Ξ© ) ⟺ βˆƒ 𝑃 ∈ πœ‚ πœ” 𝛼 ξ€· π‘₯ 𝛼 ξ€Έ , 𝐽 ∈ 𝔍 , ( 𝐴 βˆ’ 𝐼 ) [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐽 ) [ 𝛼 ] ⟺ βˆƒ 𝑃 ∈ πœ‚ πœ” 𝛼 ξ€· π‘₯ 𝛼 ξ€Έ 𝐴 , 𝐽 ∈ 𝔍 , [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐽 ∨ 𝐼 ) [ 𝛼 ] ⟺ βˆƒ 𝑄 ∈ πœ‚ πœ” βˆ— 𝛼 ( β„‘ ) ξ€· π‘₯ 𝛼 ξ€Έ 𝐴 , 𝐽 ∈ 𝔍 , [ 𝛼 ] βŠ‚ ( 𝑄 ∨ 𝐽 ) [ 𝛼 ] ⟺ π‘₯ βˆ‰ 𝐴 βˆ— ξ€· 𝔍 , πœ” βˆ— 𝛼 ξ€Έ . ( β„‘ ) ( 7 )

Theorem 23. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. Let β„‘ , 𝔍 βŠ‚ 𝐿 𝑋 be two 𝛼 -ideals. 𝐴 ∈ 𝐿 𝑋 , then(1) 𝐴 βˆ— ( β„‘ ∩ 𝔍 ) [ 𝛼 ] = 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐴 βˆ— ( 𝔍 ) [ 𝛼 ] ,(2) 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] = 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] ∩ 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] .

Proof. (1) Suppose π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐴 βˆ— ( 𝔍 ) [ 𝛼 ] , there are 𝑃 , 𝑄 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , and 𝐼 ∈ β„‘ , 𝐽 ∈ 𝔍 , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] , and 𝐴 [ 𝛼 ] βŠ‚ ( 𝑄 ∨ 𝐽 ) [ 𝛼 ] . Thus, 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) ∧ ( 𝑄 ∨ 𝐽 ) [ 𝛼 ] . As ( 𝑃 ∧ 𝑄 ) ∨ ( 𝑃 ∧ 𝐽 ) ∨ ( 𝐼 ∧ 𝑄 ) ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , and 𝐼 ∧ 𝐽 ∈ β„‘ ∩ 𝔍 , it is clear that π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ∩ 𝔍 ) [ 𝛼 ] . So 𝐴 βˆ— ( β„‘ ∩ 𝔍 ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝐴 βˆ— ( 𝔍 ) [ 𝛼 ] . The reverse inclusion is obvious according to (3) in Theorem 16. Therefore, (1) holds.
(2) Suppose π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] . There are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , and 𝐼 ∈ β„‘ , 𝐽 ∈ 𝔍 , 𝐾 = 𝐼 ∨ 𝐽 , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐾 ) [ 𝛼 ] . So ( 𝐴 βˆ’ 𝐽 ) [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] , ( 𝐴 βˆ’ 𝐼 ) [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐽 ) [ 𝛼 ] . Hence, by Lemma 22, we have π‘₯ βˆ‰ ( 𝐴 βˆ’ 𝐽 ) βˆ— ( β„‘ , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] = 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] , and π‘₯ βˆ‰ ( 𝐴 βˆ’ 𝐼 ) βˆ— ( 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] = 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] . This shows that π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] ∩ 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] . Therefore, 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] βŠƒ 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] ∩ 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] .
Conversely. suppose π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] . Then, there exists 𝑃 ∈ πœ‚ πœ” βˆ— 𝛼 ( 𝔍 ) ( π‘₯ 𝛼 ) , and 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Because of 𝑃 ∈ πœ” βˆ— 𝛼 ( 𝔍 , Ξ© ) = πœ” 𝛼 ( Ξ© ) ∨ 𝔍 , there is 𝐽 ∈ 𝔍 , such that πœ” 𝛼 ( 𝑃 βˆ’ 𝐽 ) ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) . Hence, 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] = ( 𝑃 βˆ’ 𝐽 ) [ 𝛼 ] βˆͺ ( 𝐼 ∨ 𝐽 ) [ 𝛼 ] . This shows π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] . Therefore, 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] . Similarly, we can prove 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] . So 𝐴 βˆ— ( β„‘ ∨ 𝔍 , πœ” 𝛼 ( Ξ© ) ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— ( β„‘ , πœ” βˆ— 𝛼 ( 𝔍 ) ) [ 𝛼 ] ∩ 𝐴 βˆ— ( 𝔍 , πœ” βˆ— 𝛼 ( β„‘ ) ) [ 𝛼 ] . The proof is completed.

By Theorem 20, we get two important results.

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

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

Corollary 25. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. Let β„‘ , 𝔍 βŠ‚ 𝐿 𝑋 be two 𝛼 -ideals. Then,(1) πœ” βˆ— 𝛼 ( β„‘ ∨ 𝔍 ) = πœ” βˆ— 𝛼 ( πœ” βˆ— 𝛼 ( 𝔍 ) ) ( β„‘ ) = πœ” βˆ— 𝛼 ( πœ” βˆ— 𝛼 ( β„‘ ) ) ( 𝔍 ) ,(2) πœ” βˆ— 𝛼 ( β„‘ ∨ 𝔍 ) = πœ” βˆ— 𝛼 ( 𝔍 ) ∨ πœ” βˆ— 𝛼 ( β„‘ ) ,(3) πœ” βˆ— 𝛼 ( β„‘ ∩ 𝔍 ) = πœ” βˆ— 𝛼 ( 𝔍 ) ∩ πœ” βˆ— 𝛼 ( β„‘ ) .

Proof. (1) By (2) in Theorem 23, easy.
(2) Since πœ” 𝛼 ( Ξ© ) βŠ‚ πœ” βˆ— 𝛼 ( 𝔍 ) for every 𝛼 -ideal 𝔍 , by (1) and Theorem 20, we have πœ” βˆ— 𝛼 ( β„‘ ∨ 𝔍 ) = πœ” βˆ— 𝛼 ( πœ” βˆ— 𝛼 ( 𝔍 ) ) ( β„‘ ) = πœ” βˆ— 𝛼 ( 𝔍 ) ∨ β„‘ = πœ” 𝛼 ( Ξ© ) ∨ πœ” βˆ— 𝛼 ( 𝔍 ) ∨ β„‘ = πœ” βˆ— 𝛼 ( 𝔍 ) ∨ πœ” βˆ— 𝛼 ( β„‘ ) .
(3) Clearly, πœ” βˆ— 𝛼 ( β„‘ ∩ 𝔍 ) βŠ‚ πœ” βˆ— 𝛼 ( 𝔍 ) ∩ πœ” βˆ— 𝛼 ( β„‘ ) . Conversely, 𝑃 ∈ πœ” βˆ— 𝛼 ( 𝔍 ) ∩ πœ” βˆ— 𝛼 ( β„‘ ) . Then, 𝑃 βˆ— ( β„‘ ) [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] , and 𝑃 βˆ— ( 𝔍 ) [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] . So by (1) in Theorem 23, 𝑃 βˆ— ( β„‘ ∩ 𝔍 ) [ 𝛼 ] = 𝑃 βˆ— ( β„‘ ) [ 𝛼 ] βˆͺ 𝑃 βˆ— ( 𝔍 ) [ 𝛼 ] βŠ‚ 𝑃 [ 𝛼 ] . Therefore, 𝑃 ∈ πœ” βˆ— 𝛼 ( β„‘ ∩ 𝔍 ) .

Theorem 26. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, and let β„‘ be an 𝛼 -ideal. 𝐴 ∈ 𝐿 𝑋 . Then, πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] if and only if β„‘ βŠ‚ πœ” 𝛼 ( Ξ© ) .

Proof. Sufficiency. according to (4) in Theorem 16, πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] . On the other hand, if π‘₯ βˆ‰ πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] , then 𝑦 βˆ‰ 𝐴 βˆ— [ 𝛼 ] and 𝑦 βˆ‰ 𝐴 [ 𝛼 ] . So there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( 𝑦 𝛼 ) , 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Here, we may assume 𝑦 βˆ‰ 𝐼 [ 𝛼 ] , since β„‘ is an 𝛼 -ideal. Therefore, πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] βŠ‚ πœ” 𝛼 ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] = ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . This implies 𝑦 βˆ‰ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] . πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] βŠƒ πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] .
Necessary. if 𝐴 ∈ β„‘ , then 𝐴 βˆ— [ 𝛼 ] = βˆ… . Since 𝐴 [ 𝛼 ] = ( 𝐴 ∨ 𝐴 βˆ— ) [ 𝛼 ] = πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = πœ” 𝛼 ( 𝐴 ) [ 𝛼 ] , 𝐴 ∈ πœ” 𝛼 ( Ξ© ) .
Since ( 𝑋 , πœ” 𝛼 ( Ξ© ) [ 𝛼 ] ) is an opos on 𝑋 . β„‘ [ 𝛼 ] is an ideal on 𝑋 . We have the following results.

Theorem 27. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 - π‘œ 𝑝 π‘œ 𝑠 , and let β„‘ be an 𝛼 -ideal. Then, for every 𝐴 ∈ 𝐿 𝑋 , ( 𝐴 [ 𝛼 ] ) βˆ— = 𝐴 βˆ— [ 𝛼 ] , and therefore πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = 𝑐 𝑙 βˆ— ( 𝐴 [ 𝛼 ] ) .

Proof. Since π‘₯ βˆ‰ 𝐴 βˆ— [ 𝛼 ] ⟺ βˆƒ 𝑃 ∈ πœ‚ πœ” 𝛼 ξ€· π‘₯ 𝛼 ξ€Έ , 𝐼 ∈ β„‘ , 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] ⟺ βˆƒ 𝑃 [ 𝛼 ] ∈ πœ” 𝛼 ( Ξ© ) [ 𝛼 ] ξ€· 𝑃 , π‘₯ ∈ [ 𝛼 ] ξ€Έ ξ…ž , 𝐼 ∈ β„‘ , 𝐴 [ 𝛼 ] ∩ ξ€· 𝑃 [ 𝛼 ] ξ€Έ ξ…ž = 𝐼 [ 𝛼 ] ξ€· 𝐴 ⟺ π‘₯ βˆ‰ [ 𝛼 ] ξ€Έ βˆ— , ( 8 ) then ( 𝐴 [ 𝛼 ] ) βˆ— = 𝐴 βˆ— [ 𝛼 ] , πœ” βˆ— 𝛼 ( 𝐴 ) [ 𝛼 ] = ( 𝐴 ∨ 𝐴 βˆ— ) [ 𝛼 ] = 𝐴 [ 𝛼 ] βˆͺ ( 𝐴 [ 𝛼 ] ) βˆ— = 𝑐 𝑙 βˆ— ( 𝐴 [ 𝛼 ] ) .

Corollary 28. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, and let β„‘ be an 𝛼 -ideal. Then, for every 𝐴 ∈ 𝐿 𝑋 , 𝒳 βˆ— 𝐴 [ 𝛼 ] = 𝒳 𝐴 βˆ— [ 𝛼 ] = 𝒳 ( 𝐴 [ 𝛼 ] ) βˆ— .

Proof. π‘₯ βˆ‰ ( 𝒳 βˆ— 𝐴 [ 𝛼 ] ) [ 𝛼 ] ⇔ t h e r e e x i s t s 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ , 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] ⇔ π‘₯ βˆ‰ 𝐴 βˆ— [ 𝛼 ] ⇔ π‘₯ βˆ‰ ( 𝒳 𝐴 βˆ— [ 𝛼 ] ) [ 𝛼 ] . So 𝒳 βˆ— 𝐴 [ 𝛼 ] = 𝒳 𝐴 βˆ— [ 𝛼 ] . Besides, by Theorem 27, the later equation is easy.

Corollary 29. Let ( 𝑋 , Ξ” ) be an opos on 𝑋 , ℐ βŠ‚ 𝑋 an ideal. The induced πœ” 𝛼 -opos and 𝛼 -ideal are denoted, respectively, by ( 𝐿 𝑋 , πœ” 𝛼 ( πœ” 𝐿 ( Ξ” ) ) ) and β„‘ = { 𝒳 𝐼 ∢ 𝐼 ∈ ℐ } . Then, πœ” βˆ— 𝛼 ( β„‘ , πœ” 𝛼 ( πœ” 𝐿 ( Ξ” ) ) ) [ 𝛼 ] = Ξ” βˆ— .

Proof. By Theorem 27 and Corollary 28, πœ” βˆ— 𝛼 ( β„‘ , πœ” 𝛼 ( πœ” 𝐿 ( Ξ” ) ) ) [ 𝛼 ] βŠ‚ Ξ” βˆ— . On other hand, again by Corollary 28, for every π‘ˆ βŠ‚ 𝑋 , πœ” βˆ— 𝛼 ( 𝒳 π‘ˆ ) [ 𝛼 ] = ( 𝒳 π‘ˆ ∨ 𝒳 βˆ— π‘ˆ ) [ 𝛼 ] = π‘ˆ βˆͺ ( 𝒳 π‘ˆ βˆ— ) [ 𝛼 ] = π‘ˆ βˆͺ π‘ˆ βˆ— = 𝑐 𝑙 βˆ— ( π‘ˆ ) .

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

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

Definition 31. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos. Let β„‘ be an 𝛼 -ideal. πœ” 𝛼 ( Ξ© ) is said to be compatible with β„‘ , denoted by πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ , if for any 𝐴 ∈ 𝐿 𝑋 and π‘₯ ∈ 𝐴 [ 𝛼 ] , there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] , then 𝐴 ∈ β„‘ .

Example 32. In Example 18, we know πœ” 𝛼 = { ( 0 , 0 ) , ( 0 , 1 / 2 ) , ( 0 , 1 ) , ( 1 / 2 , 1 / 2 ) , ( 1 / 2 , 1 ) , ( 1 , 1 / 2 ) , ( 1 , 1 ) } , πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) = { ( 0 , 0 ) , ( 0 , 1 / 2 ) , ( 0 , 1 ) } , πœ‚ πœ” 𝛼 ( 𝑦 𝛼 ) = { ( 0 , 0 ) } . Take β„‘ 1 = 𝐿 𝑋 , 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 β„‘ 𝑖 , ( 𝑖 ≀ 4 ) are all 𝛼 -ideals, πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ 1 and πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ 2 . Take 𝐺 = ( 1 / 2 , 1 / 2 ) , then π‘₯ ∈ 𝐺 [ 𝛼 ] and 𝑅 = ( 0 , 1 / 2 ) ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐽 = ( 1 / 2 , 0 ) ∈ β„‘ 3 , such that 𝐺 [ 𝛼 ] βŠ‚ ( 𝑅 ∨ 𝐽 ) [ 𝛼 ] . But 𝐺 βˆ‰ β„‘ 3 , therefore πœ” 𝛼 ΜΈ ∼ ( Ξ© ) 𝛼 β„‘ 3 . Furthermore, for all 𝐴 ∈ 𝐿 𝑋 , if 𝐴 [ 𝛼 ] = βˆ… or 𝐴 [ 𝛼 ] = { 𝑦 } , then 𝐴 ∈ β„‘ 4 . If 𝐴 [ 𝛼 ] = { π‘₯ } or 𝐴 [ 𝛼 ] = { π‘₯ , 𝑦 } , there does not exist 𝐸 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ 4 , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑅 ∨ 𝐽 ) [ 𝛼 ] . Therefore, πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ 4 .

Theorem 33. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, β„‘ an ideal, πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ . Then, 𝔅 = πœ” βˆ— 𝛼 ( β„‘ ) .

Proof. For every 𝐡 ∈ πœ” βˆ— 𝛼 ( β„‘ ) , 𝐡 βˆ— [ 𝛼 ] βŠ‚ 𝐡 [ 𝛼 ] . Put 𝐴 = 𝐡 βˆ’ 𝐡 βˆ— ( β„‘ ) . If π‘₯ ∈ 𝐴 [ 𝛼 ] , then π‘₯ βˆ‰ 𝐡 βˆ— [ 𝛼 ] . So there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐼 ∈ β„‘ , such that 𝐡 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Thus, 𝐴 [ 𝛼 ] βŠ‚ 𝐡 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Since πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ , we get 𝐴 ∈ β„‘ . Besides, by (4) in Theorem 16, 𝐡 βˆ— ∈ πœ” 𝛼 ( Ξ© ) . Notice that 𝐡 [ 𝛼 ] = ( 𝐴 ∨ 𝐡 βˆ— ) [ 𝛼 ] . Thus, 𝐡 ∈ 𝔅 , and πœ” βˆ— 𝛼 ( β„‘ ) βŠ‚ 𝔅 . The reverse inclusion is obvious. Therefore, 𝔅 = πœ” βˆ— 𝛼 ( β„‘ ) .

Lemma 34. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, β„‘ an 𝛼 -ideal. Then, for every 𝐴 ∈ 𝐿 𝑋 , ( 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] ∩ ( 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ) βˆ— ( β„‘ ) [ 𝛼 ] = βˆ… .

Proof. If π‘₯ ∈ ( 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] , so π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] . That is, there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) , 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . As ( 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] βŠ‚ 𝐴 [ 𝛼 ] , we have π‘₯ βˆ‰ ( 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ) βˆ— ( β„‘ ) [ 𝛼 ] . Therefore, the conclusion holds.

Theorem 35. Let ( 𝐿 𝑋 , πœ” 𝛼 ( Ξ© ) ) be an πœ” 𝛼 -opos, β„‘ an ideal. Then, the following statements are equivalent.(1) πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ .(2)If 𝐴 ∈ 𝐿 𝑋 has an 𝛼 - πœ” 𝛼 -RF β„œ , satisfying for all 𝑃 ∈ β„œ , there is 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] , then 𝐴 ∈ β„‘ .(3)For all 𝐴 ∈ 𝐿 𝑋 , ( 𝐴 ∧ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] = βˆ… β‡’ 𝐴 ∈ β„‘ .(4)For all 𝐴 ∈ 𝐿 𝑋 , 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ∈ β„‘ .(5)For all 𝐴 ∈ πœ” βˆ— 𝛼 ( β„‘ ) , 𝐴 βˆ’ 𝐴 βˆ— ( β„‘ ) ∈ β„‘ .(6)For all 𝐴 ∈ 𝐿 𝑋 , if βˆ„ βˆ… β‰  𝐡 [ 𝛼 ] βŠ‚ 𝐡 βˆ— ( β„‘ ) [ 𝛼 ] , 𝐡 [ 𝛼 ] βŠ‚ 𝐴 [ 𝛼 ] , then 𝐴 ∈ β„‘ .

Proof. (1) β‡’ (2) If 𝐴 ∈ 𝐿 𝑋 has a 𝛼 - πœ” 𝛼 -RF β„œ , satisfying the condition in (2), then for every π‘₯ ∈ 𝐴 [ 𝛼 ] , there are 𝑃 ∈ β„œ and 𝐼 ∈ β„‘ such that 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . As πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ , we have 𝐴 ∈ β„‘ .
(2) β‡’ (1) By Definitions 30 and 31, clearly.
(1) β‡’ (3) For all 𝐴 ∈ 𝐿 𝑋 , ( 𝐴 ∧ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] = βˆ… . If 𝐴 [ 𝛼 ] = βˆ… , then 𝐴 ∈ β„‘ . If π‘₯ ∈ 𝐴 [ 𝛼 ] , then π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] , there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Because πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ , we have 𝐴 ∈ β„‘ .
(3) β‡’ (1) if 𝐴 ∈ 𝐿 𝑋 satisfying, for every π‘₯ ∈ 𝐴 [ 𝛼 ] , there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] . Then, π‘₯ βˆ‰ 𝐴 βˆ— ( β„‘ ) [ 𝛼 ] . By (3), we have 𝐴 ∈ β„‘ . Hence, πœ” 𝛼 ( Ξ© ) ∼ 𝛼 β„‘ .
(3) β‡’ (4) By Lemma 34, obviously.
(4) β‡’ (5) Directly.
(5) β‡’ (1) If for every 𝐴 ∈ 𝐿 𝑋 and π‘₯ ∈ 𝐴 [ 𝛼 ] , there are 𝑃 ∈ πœ‚ πœ” 𝛼 ( π‘₯ 𝛼 ) and 𝐼 ∈ β„‘ , such that 𝐴 [ 𝛼 ] βŠ‚ ( 𝑃 ∨ 𝐼 ) [ 𝛼 ] , this means ( 𝐴 ∧ 𝐴 βˆ— ( β„‘ ) ) [ 𝛼 ] = βˆ… . Put 𝐡 = 𝐴 ∨ 𝐴 βˆ— , then 𝐡 βˆ— [ 𝛼 ] = ( 𝐴 ∨ 𝐴 βˆ— ) βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] βˆͺ ( 𝐴 βˆ— ) βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] . So πœ” βˆ— 𝛼 ( 𝐡 ) [ 𝛼 ] = ( 𝐡 ∨ 𝐡 βˆ— ( β„‘ ) ) [ 𝛼 ] = 𝐡 [ 𝛼 ] . This means 𝐡 ∈ πœ” βˆ— 𝛼 ( β„‘ ) . By (5), we have 𝐡 βˆ’ 𝐡 βˆ— ∈ β„‘ . Since ξ€· 𝐡 βˆ’ 𝐡 βˆ— ξ€Έ [ 𝛼 ] = ξ€· ξ€· 𝐴 ∨ 𝐴 βˆ— ξ€Έ βˆ’ ξ€· 𝐴 ∨ 𝐴 βˆ— ξ€Έ βˆ— ξ€Έ [ 𝛼 ] = ξ€· 𝐴 ∨ 𝐴 βˆ— ξ€Έ [ 𝛼 ] βˆ’ 𝐴 βˆ— [ 𝛼 ] = 𝐴 [ 𝛼 ] , ( 9 ) we draw a conclusion: for every π‘₯ ∈ 𝐴 [ 𝛼 ] , π‘₯ βˆ‰ 𝐡 βˆ— [ 𝛼 ] = 𝐴 βˆ— [ 𝛼 ] . Therefore, 𝐴 ∈ β„‘ .
(4) β‡’ (6) For any 𝐴 ∈ 𝐿 𝑋 , 𝐴 = ( 𝐴 βˆ’ 𝐴 βˆ— ) ∨ ( 𝐴 ∧ 𝐴 βˆ— ) . So 𝐴 βˆ— [ 𝛼 ] = ( ( 𝐴 βˆ’ 𝐴 βˆ— ) βˆ— [ 𝛼 ] βˆͺ ( 𝐴 ∧ 𝐴 βˆ— ) ) βˆ— [ 𝛼 ] . By (4), 𝐴 βˆ’ 𝐴 βˆ— ∈ β„‘ , so ( 𝐴 βˆ’ 𝐴 βˆ— ) βˆ— [ 𝛼 ] = βˆ… . Thus 𝐴 βˆ— [ 𝛼 ] = ( 𝐴 ∧ 𝐴 βˆ— ) βˆ— [ 𝛼 ] . Since ( 𝐴 ∧ 𝐴 βˆ— ) [ 𝛼 ] βŠ‚ 𝐴 βˆ— [ 𝛼 ] , we have ( 𝐴 ∧ 𝐴 βˆ— ) [ 𝛼 ] βŠ‚ ( 𝐴 ∧ 𝐴 βˆ— ) βˆ— [ 𝛼 ] . Here by the assumption in (6), ( 𝐴 ∧ 𝐴 βˆ— ) [ 𝛼 ] = βˆ… . This implies 𝐴 [ 𝛼 ] = ( 𝐴 βˆ’ 𝐴 βˆ— ) βˆ— [ 𝛼 ] . Again by (4), 𝐴 ∈ β„‘ .
(6) β‡’ (4) By Lemma 34, for any 𝐴 ∈ 𝐿 𝑋 , ( 𝐴 βˆ’ 𝐴 βˆ— ) [ 𝛼 ] ∩ ( 𝐴 βˆ’ 𝐴 βˆ— ) βˆ— [ 𝛼 ] = βˆ… . Let's prove 𝐴 βˆ’ 𝐴 βˆ— satisfies the condition in (6), In fact, if not, there is βˆ… β‰  𝐡 [ 𝛼 ] βŠ‚ 𝐡 βˆ— [ 𝛼 ] , 𝐡 [ 𝛼