Rough Filters in 𝐵𝐿-Algebras

Abstract

We apply the rough set theory to 𝐵𝐿-algebras. As a generalization of filters (subalgebras) of 𝐵𝐿-algebras, we introduce the notion of rough filters (subalgebras) of 𝐵𝐿-algebras and investigate some of their properties.

1. Introduction

The rough sets theory introduced by Pawlak [13] has often proved to be an excellent mathematical tool for the analysis of a vague description of objects (called actions in decision problems). Many different problems can be addressed by rough sets theory. During the last few years this formalism has been approached as a tool used in connection with many different areas of research. There have been investigations of the relations between rough sets theory and the Dempster-Shafer theory and between rough sets and fuzzy sets. Rough sets theory has also provided the necessary formalism and ideas for the development of some propositional machine learning systems. It has also been used for, among many others, knowledge representation; data mining; dealing with imperfect data; reducing knowledge representation and for analyzing attribute dependencies. The notions of rough relations and rough functions are based on rough sets theory and can be applied as a theoretical basis for rough controllers, among others. An algebraic approach to rough sets has been given by Iwinski [1]. Rough set theory is applied to semigroups and groups (see [2, 3]). In 1994, Biswas and Nanda [4] introduced and discussed the concept of rough groups and rough subgroups. Jun [5] applied rough set theory to BCK-algebras. Recently, Rasouli [6] introduced and studied the notion of roughness in 𝑀𝑉-algebras.

𝐵𝐿-algebras are the algebraic structures for Hàjek Basic Logic (𝐵𝐿-logic) [7], arising from the continuous triangular norms (𝑡-norms), familiar in the frameworks of fuzzy set theory. The language of propositional Hàjek basic logic [7] contains the binary connectives ∘, ⇒ and the constant 0.

Axioms of 𝐵𝐿 are as follows:(𝐴1)(𝜑⇒𝜓)⇒((𝜓⇒𝜔)⇒(𝜑⇒𝜔)),(𝐴2)(𝜑∘𝜓)⇒𝜑,(𝐴3)(𝜑∘𝜓)⇒(𝜓∘𝜑),(𝐴4)(𝜑∘(𝜑⇒𝜓))⇒(𝜓∘(𝜓⇒𝜑)),(𝐴5𝑎)(𝜑⇒(𝜓⇒𝜔))⇒((𝜑∘𝜓)⇒𝜔),(𝐴5𝑏)((𝜑∘𝜓)⇒𝜔)⇒(𝜑⇒(𝜓⇒𝜔)),(𝐴6)((𝜑⇒𝜓)⇒𝜔)⇒(((𝜓⇒𝜑)⇒𝜔)⇒𝜔),(𝐴7)0⇒𝜔.

𝐵𝐿-algebras rise as Lindenbaum algebras from the above logical axioms in a similar manner that Boolean algebras or 𝑀𝑉-algebras do from Classical logic or Lukasiewicz logic, respectively. 𝑀𝑉-algebras are 𝐵𝐿-algebras while the converse, in general, is not true. Indeed, 𝐵𝐿-algebras with involutory complement are 𝑀𝑉-algebras. Moreover, Boolean algebras are 𝑀𝑉-algebras and 𝑀𝑉-algebras with idempotent product are Boolean algebras. Filters theory plays an important role in studying these logical algebras. From logical point of view, various filters correspond to various sets of provable formula.

In this paper, we apply the rough set theory to 𝐵𝐿-algebras, and we introduce the notion of (lower) upper rough subalgebras and (lower) upper rough filters of 𝐵𝐿-algebras and obtain some related results.

2. Preliminaries

Definition 2.1. A 𝐵𝐿-algebra is an algebra (𝐿,∧,∨,∗,→,0,1) with four binary operations ∧,∨,∗,→ and two constants 0,1 such that:(𝐵𝐿1)(𝐿,∧,∨,0,1) is a bounded lattice,(𝐵𝐿2)(𝐿,∗,1) is a commutative monoid,(𝐵𝐿3)𝑐≤𝑎→𝑏 if and only if 𝑎∗𝑐≤𝑏, for all 𝑎,𝑏,𝑐∈𝐿, (i.e., ∗ and → form an adjoint pair),(𝐵𝐿4)𝑎∧𝑏=𝑎∗(𝑎→𝑏),(𝐵𝐿5)(𝑎→𝑏)∨(𝑏→𝑎)=1.

Examples of 𝐵𝐿-algebras [7] are 𝑡-algebras ([0,1],∧,∨,∗_𝑡,→_𝑡,0,1), where ([0,1],∧,∨,0,1) is the usual lattice on [0,1] and ∗_𝑡 is a continuous 𝑡-norm, whereas →_𝑡 is the corresponding residuum.

If (𝐿,∧,∨,⁻,0,1) is a Boolean algebra, then (𝐿,∧,∨,∗,→,0,1) is a 𝐵𝐿-algebra where the operation ∗ coincides with ∧ and 𝑥→𝑦=𝑥⁻∨𝑦, for all 𝑥,𝑦∈𝐿.

From now (𝐿,∧,∨,∗,→,0,1) or simply 𝐿 is a 𝐵𝐿-algebra.

A 𝐵𝐿-algebra is called an 𝑀𝑉-algebra if ¬¬𝑥=𝑥, for all 𝑥∈𝐿, where ¬𝑥=𝑥→0.

A 𝐵𝐿-algebra is nontrivial if 0≠1. For any 𝐵𝐿-algebra 𝐿, (𝐿,∧,∨,0,1) is a bounded distributive lattice. We denote the set of natural numbers by 𝑁 and define 𝑎⁰=1 and 𝑎^𝑛=𝑎^𝑛−1∗𝑎, for 𝑛∈𝑁⧵{0}. The order of 𝑎∈𝐿, 𝑎≠1, in symbols ord(𝑎) is the smallest 𝑛∈𝑁 such that 𝑎^𝑛=0; if no such 𝑛 exists, then ord(𝑎)=∞.

Lemma 2.2 (see [7–11]). In any 𝐵𝐿-algebra 𝐿, the following properties hold for all 𝑥,𝑦,𝑧∈𝐿:(1)𝑥≤𝑦 if and only if 𝑥→𝑦=1,(2)𝑥→(𝑦→𝑧)=(𝑥∗𝑦)→𝑧=𝑦→(𝑥→𝑧),(3)if 𝑥≤𝑦, then 𝑦→𝑧≤𝑥→𝑧 and 𝑧→𝑥≤𝑧→𝑦,(4)𝑦→𝑥≤(𝑧→𝑦)→(𝑧→𝑥) and 𝑥→𝑦≤(𝑦→𝑧)→(𝑥→𝑧),(5)𝑥≤𝑦 implies 𝑥∗𝑧≤𝑦∗𝑧,(6)1→𝑥=𝑥, 𝑥→𝑥=1, 𝑥≤𝑦→𝑥, 𝑥→1=1, 0→𝑥=1,(7)(𝑥→𝑦)→(𝑥→𝑧)=(𝑥∧𝑦)→𝑧,(8)𝑥→𝑦≤(𝑥∗𝑧)→(𝑦∗𝑧),(9)𝑥≤𝑦→𝑥 and 𝑥∗𝑦≤𝑥,𝑦,(10)𝑥∨𝑦=[(𝑥→𝑦)→𝑦]∧[(𝑦→𝑥)→𝑥].

For any 𝐵𝐿-algebra 𝐿, 𝐵(𝐿) denotes the Boolean algebra of all complemented elements in lattice of 𝐿.

Proposition 2.3 (see [7, 11]). For 𝑒∈𝐿, the following are equivalent:(i)𝑒∈𝐵(𝐿),(ii)𝑒∗𝑒=𝑒 and 𝑒=𝑒⁻⁻,(iii)𝑒∗𝑒=𝑒 and 𝑒⁻→𝑒=𝑒,(iv)𝑒∨𝑒⁻=1,(v)(𝑒→𝑥)→𝑒=𝑒, for every 𝑥∈𝐿.

Hàjek [7] defined a filter of a 𝐵𝐿-algebra 𝐿 to be a nonempty subset 𝐹 of 𝐿 such that (i) if 𝑎,𝑏∈𝐹 implies 𝑎∗𝑏∈𝐹 and (ii) if 𝑎∈𝐹, 𝑎≤𝑏, then 𝑏∈𝐹. Turunen [8] defined a deductive system of a 𝐵𝐿-algebra 𝐿 to be a nonempty subset 𝐷 of 𝐿 such that (i) 1∈𝐷 and (ii) 𝑥∈𝐷 and 𝑥→𝑦∈𝐷 imply 𝑦∈𝐷. Note that a subset 𝐹 of a 𝐵𝐿-algebra 𝐿 is a deductive system of 𝐿 if and only if 𝐹 is a filter of 𝐿 [8].

Let 𝑈 denote a nonempty set of objects called the univers, and let 𝜃⊆𝑈×𝑈 be an equivalence relation on 𝑈. The pair (𝑈;𝜃) is called a Pawlak approximation space. The equivalence relation 𝜃 partitions the set 𝑈 into disjoint subsets. Let 𝑈/𝜃 denote the quotient set consisting of all the equivalence classes of 𝜃. The empty set ∅ and the elements of 𝑈/𝜃 are called elementary sets. A finite union of elementary sets, that is, the union of one or more elementary sets, is called a composed set [12]. The family of all composed sets is denoted by Com(Apr). It is a subalgebra of the Boolean algebra 2^𝑈 formed by the power set of 𝑈. A set which is a union of elementary sets is called a definable set [12]. The family of all definable sets is denoted by Def(Apr). For a finite universe, the family of definable sets is the same as the family of composed sets. A Pawlak approximation space defines uniquely a topological space (𝑈;Def(Apr)), in which Def(Apr) is the family of all open and closed sets [13]. In connection to rough set theory there exist two views. The operator-oriented view interprets rough set theory as an extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are related to the interior and closure operators in topological spaces, the necessity and possibility operators in modal logic, and lower and upper approximations in interval structures. The set-oriented view focuses on the interpretation and characterization of members of rough sets. Both operator-oriented and set-oriented views are useful in the understanding and application of the theory of rough sets.

Definition 2.4. For an approximation space (𝑈;𝜃), by a rough approximation in (𝑈;𝜃) we mean a mapping Apr∶𝑃(𝑈)→𝑃(𝑈)×𝑃(𝑈) defined for every 𝑋∈𝑃(𝑈) by Apr(𝑋)=(Apr(𝑋);Apr(𝑋)), where Apr(𝑋)=𝑥∈𝑈∶[𝑥]_𝜃∩𝑋≠∅,Apr(𝑋)=𝑥∈𝑈∶[𝑥]_𝜃⊆𝑋. (2.1)Apr(𝑋) is called a lower rough approximation of 𝑋 in (𝑈;𝜃), whereas Apr(𝑋) is called an upper rough approximation of 𝑋 in (𝑈;𝜃).

Let 𝐹 be a filter of a 𝐵𝐿-algebra 𝐿. Define relation ≡_𝐹 on 𝐿 as follows: 𝑥≡_𝐹𝑦iff𝑥⟶𝑦∈𝐹,𝑦⟶𝑥∈𝐹. (2.2) Then ≡_𝐹 is a congruence relation on 𝐿. 𝐿/𝐹 denotes the set of all congruence classes of ≡_𝐹, that is, 𝐿/𝐹={[𝑥]_𝐹∶𝑥∈𝐿}, thus 𝐿/𝐹 is a 𝐵𝐿-algebra.

3. Lower and Upper Approximations in 𝐵𝐿-Algebras

Definition 3.1. Let 𝐿 be a 𝐵𝐿-algebra and 𝐹 a filter of 𝐿. For any nonempty subset 𝑋 of 𝐿, the sets Apr_𝐹(𝑋)=𝑥∈𝐿∶[𝑥]_𝐹⊆𝑋,Apr_𝐹(𝑋)=𝑥∈𝐿∶[𝑥]_𝐹∩𝑋≠∅ (3.1) are called, respectively, the lower and upper approximations of the set 𝑋 with respect to the filter 𝐹. Therefore, when 𝑈=𝐿 and 𝜃 is the induced congruence relation by filter 𝐹, then we use the pair (𝐿,𝐹) instead of the approximation space (𝑈,𝜃). Also, in this case we use the symbols Apr_𝐹(𝑋) and Apr_𝐹(𝑋) instead of Apr(𝑋) and Apr(𝑋).

Proposition 3.2. Let (𝐿,𝐹) be an approximation space and 𝑋,𝑌⊆𝐿. Then the following hold:(1)Apr_𝐹(𝑋)⊆𝑋⊆Apr_𝐹(𝑋),(2)Apr_𝐹(𝑋∩𝑌)=Apr_𝐹(𝑋)∩Apr_𝐹(𝑌),(3)Apr_𝐹(𝑋)∪Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋∪𝑌),(4)Apr_𝐹(𝑋∩𝑌)⊆Apr_𝐹(𝑋)∩Apr_𝐹(𝑌),(5)Apr_𝐹(𝑋)∪Apr_𝐹(𝑌)=Apr_𝐹(𝑋∪𝑌),(6)Apr_𝐹(Apr_𝐹(𝑋))⊆Apr_𝐹(Apr_𝐹(𝑋)),(7)Apr_𝐹(Apr_𝐹(𝑋))⊆Apr_𝐹(Apr_𝐹(𝑋)),(8)Apr_𝐹(𝑋^𝑐)=(Apr_𝐹(𝑋))^𝑐,Apr_𝐹(𝑋^𝑐)=(Apr_𝐹(𝑋))^𝑐,(9) if 𝑋≠∅, then Apr_𝐿(𝑋)=𝐿,(10)if 𝑋≠𝐿, then Apr_𝐿(𝑋)=∅.

Proof. The proof is similar to the proof of Theorem  2.1 of [14].

Proposition 3.3. Let (𝐿,𝐹) be an approximation space. Then, Apr_𝐹 is a closure operator and Apr_𝐹 is an interior operator.

Proof. Let 𝑋 be an arbitrary subset of 𝐿.(i)By Proposition 3.2 part (1), we have 𝑋⊆Apr_𝐹(𝑋).(ii)We will show that Apr_𝐹(Apr_𝐹(𝑋))=Apr_𝐹(𝑋). Suppose that 𝑥∈Apr_𝐹(Apr_𝐹(𝑋)). By Definition 3.1, we have [𝑥]_𝐹∩Apr_𝐹(𝑋)≠∅. Hence there exists 𝑦∈Apr_𝐹(𝑋) such that [𝑥]_𝐹=[𝑦]_𝐹. By Definition 3.1, [𝑦]_𝐹∩𝑋≠∅, and so we get that [𝑥]_𝐹∩𝑋≠∅. Hence 𝑥∈Apr_𝐹(𝑋) and then Apr_𝐹(Apr_𝐹(𝑋))⊆Apr_𝐹(𝑋). By part (1) of Proposition 3.2, we have Apr_𝐹(𝑋)⊆Apr_𝐹(Apr_𝐹(𝑋)).(iii)Suppose that 𝑋⊆𝑌. We will show that Apr_𝐹(𝑋)⊆Apr_𝐹(𝑌). Let 𝑥∈Apr_𝐹(𝑋). Then [𝑥]_𝐹∩𝑋≠∅. Since 𝑋⊆𝑌, we get that [𝑥]_𝐹∩𝑌≠∅, that is, 𝑥∈Apr_𝐹(𝑌). Analogously, we can prove that Apr_𝐹 is an interior operator.

Definition 3.4. (𝐿,𝐹) is an approximation space and 𝑋⊆𝐿. 𝑋 is called definable with respect to 𝐹, if Apr_𝐹(𝑋)=Apr_𝐹(𝑋).

Proposition 3.5. Let (𝐿,𝐹) be an approximation space. Then ∅, 𝐿, and [𝑥]_𝐹 are definable respect to 𝐹.

Proof. The proof is straightforward.

Proposition 3.6. Let (𝐿,𝐹) be an approximation space. If 𝐹={1}, then every subset of 𝐿 is definable.

Proof. Let 𝑋 be an arbitrary subset of 𝐿. We have [𝑥]_𝐹=𝑦∈𝐿∶𝑥⟶𝑦=1,𝑦⟶𝑥=1=𝑦∈𝐿∶𝑥=𝑦={𝑥}, (3.2) for all 𝑥∈𝐿. We get that Apr_𝐹(𝑋)=𝑥∈𝐿∶[𝑥]_𝐹⊆𝑋={𝑥∈𝐿∶{𝑥}⊆𝑋}=𝑋,Apr_𝐹(𝑋)=𝑥∈𝐿∶[𝑥]_𝐹∩𝑋≠∅={𝑥∈𝐿∶{𝑥}∩𝑋≠∅}=𝑋. (3.3) Hence 𝑋 is definable.

Let 𝑋 and 𝑌 be nonempty subsets of 𝐿. Then we define two sets,.𝑋∗𝑌=𝑥∗𝑦∶𝑥∈𝑋,𝑦∈𝑌𝑋⟶𝑌=𝑥⟶𝑦∶𝑥∈𝑋,𝑦∈𝑌 (3.4) If either 𝑋 or 𝑌 is empty, then we define 𝑋∗𝑌=∅ and 𝑋→𝑌=∅. Clearly, 𝑋∗𝑌=𝑌∗𝑋, for every 𝑋,𝑌∈𝐿. 𝑋∗𝑌 and 𝑋→𝑌 are called, respectively, Minkowski product and Minkowski arrow. However, Minkowski arrow is not the residuum of Minkowski product.

Lemma 3.7. Let (𝐿,𝐹) be an approximation space and 𝑋,𝑌⊆𝐿. Then the following hold:(1)Apr_𝐹(𝑋)∗Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋∗𝑌),(2)Apr_𝐹(𝑋)→Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋→𝑌).

Proof. (1) Let 𝑧∈Apr_𝐹(𝑋)∗Apr_𝐹(𝑌), and so there exist 𝑥∈Apr_𝐹(𝑋) and 𝑦∈Apr_𝐹(𝑌) such that 𝑧=𝑥∗𝑦. We have [𝑥]_𝐹∩𝑋≠∅ and [𝑦]_𝐹∩𝑌≠∅. There exist 𝑎∈𝑋 and 𝑏∈𝑌 such that [𝑥]_𝐹=[𝑎]_𝐹 and [𝑦]_𝐹=[𝑏]_𝐹. We get that 𝑎∗𝑏∈𝑋∗𝑌 such that [𝑎∗𝑏]_𝐹=[𝑥∗𝑦]_𝐹. Hence [𝑥∗𝑦]_𝐹∩(𝑋∗𝑌)≠∅, that is, Apr_𝐹(𝑋)∗Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋∗𝑌).
Similarly, we can prove (2).

Definition 3.8. Let (𝐿,𝐹) be an approximation space. A nonempty subset 𝑆 of 𝐿 is called an upper (resp., a lower) rough subalgebra (or filter) of 𝐿, if the upper (resp., the lower) approximation of 𝑆 is a subalgebra (or filter) of 𝐿. If 𝑆 is both an upper and a lower rough subalgebra (or filter) of 𝐿, we say 𝑆 is a rough subalgebra (or filter) of 𝐿.

Proposition 3.9. Let (𝐿,𝐹) be an approximation space. If 𝑆 is a subalgebra of 𝐿, then 𝑆 is an upper rough subalgebra of 𝐿.

Proof. We will show that Apr_𝐹(𝑆) is a subalgebra of 𝐿. Since 0,1∈𝑆, 0∈[0]_𝐹 and 1∈[1]_𝐹, then [0]_𝐹∩𝑆≠∅ and [1]_𝐹∩𝑆≠∅. Hence 0,1∈Apr_𝐹(𝑆). Taking 𝑋=𝑌=𝑆 in Lemma 3.7, by 𝑆 is a subalgebra of 𝐿, we obtain Apr_𝐹(𝑆)∗Apr_𝐹(𝑆)⊆Apr_𝐹(𝑆∗𝑆)⊆Apr_𝐹(𝑆),Apr_𝐹(𝑆)⟶Apr_𝐹(𝑆)⊆Apr_𝐹(𝑆⟶𝑆)⊆Apr_𝐹(𝑆⟶𝑆). (3.5) Hence 𝑆 is an upper rough subalgebra.

Let (𝐿,𝐹) be an approximation space and 𝑆 a subalgebra of 𝐿. The following example shows that 𝑆 may not be a lower rough subalgebra of 𝐿 in general.

Example 3.10. Let 𝐿={0,𝑎,𝑏,𝑐,𝑑,1}, where 0<𝑑<𝑐<𝑎,𝑏<1. Define ∗ and → as follow: →0𝑎𝑏𝑐𝑑10𝑎111111𝑏01𝑏𝑏𝑑1𝑐0𝑎1𝑎𝑑1𝑑0111𝑑11𝑑11111∗0𝑎𝑏𝑐𝑑10𝑎𝑏𝑐𝑑10𝑎000000𝑏0𝑎𝑐𝑐𝑑𝑎𝑐0𝑐𝑏𝑐𝑑𝑏𝑑0𝑐𝑐𝑐𝑑𝑐10𝑑𝑑𝑑0𝑑0𝑎𝑏𝑐𝑑1 (3.6) Then (𝐿,∧,∨,∗,→,0,1) is a 𝐵𝐿-algebra. It is easy to check that 𝐹={1,𝑏} is a filter of 𝐿 and 𝑆={0,𝑎,1} is a subalgebra of 𝐿. Since [1]_𝐹=𝐹⊈𝑆, then 𝑆 is not be a lower rough subalgebra of 𝐿.

Theorem 3.11. Let (𝐿,𝐹) be an approximation space and 𝑋 a nonempty subset of 𝐿. Then the following hold:(i)𝑋⊆𝐹 if and only if Apr_𝐹(𝑋)=𝐹,(ii)𝐹⊆𝑋 if and only if 𝐹⊆Apr_𝐹(𝑋).

Proof. (i) Let 𝑋⊆𝐹 and 𝑧∈Apr_𝐹(𝑋). Then [𝑧]_𝐹∩𝑋≠∅, and so there exists 𝑥∈𝑋 such that [𝑥]_𝐹=[𝑧]_𝐹. Since 𝑋⊆𝐹, then [𝑥]_𝐹=𝐹. So 𝑧∈𝐹, that is, Apr_𝐹(𝑋)⊆𝐹. Now let 𝑧∈𝐹. Then [𝑧]_𝐹=𝐹 and so [𝑧]_𝐹∩𝑋=𝐹∩𝑋=𝑋≠∅. Thus 𝑧∈Apr_𝐹(𝑋), and hence Apr_𝐹(𝑋)=𝐹.
The converse follows from Proposition 3.2 part (1).
(ii) Let 𝐹⊆𝑋 and 𝑥∈𝐹. Then [𝑥]_𝐹=𝐹⊆𝑋, and so 𝑥∈Apr_𝐹(𝑋). Therefore 𝐹⊆Apr_𝐹(𝑋).

Theorem 3.12. Let (𝐿,𝐹) be an approximation space and 𝐽 a filter of 𝐿. Then the following hold:(1)𝐹⊆𝐽 if and only if Apr_𝐹(𝐽)=𝐽=Apr_𝐹(𝐽),(2)Apr_𝐹(𝐹)=𝐹=Apr_𝐹(𝐹),(3)𝐹⊆Apr_𝐹(𝐽).

Proof. (1) Suppose that 𝐹⊆𝐽. By Proposition 3.2 part (1), 𝐽⊆Apr_𝐹(𝐽) and Apr_𝐹(𝐽)⊆𝐽. Now let 𝑥∈Apr_𝐹(𝐽). Then [𝑥]_𝐹∩𝐽≠∅ and so there exists 𝑦∈𝐽 such that [𝑥]_𝐹=[𝑦]_𝐹. We get that 𝑥→𝑦,𝑦→𝑥∈𝐹. Since 𝐹⊆𝐽, 𝐽 is a filter and 𝑦∈𝐽, then 𝑥∈𝐽, that is, Apr_𝐹(𝐽)⊆𝐽. Hence Apr_𝐹(𝐽)=𝐽. Similarly, we can obtain 𝐽⊆Apr_𝐹(𝐽). Conversely, suppose that Apr_𝐹(𝐽)=𝐽=Apr_𝐹(𝐽) and 𝑥∈𝐹. We have 1∈[𝑥]_𝐹∩𝐽. Hence 𝑥∈Apr_𝐹(𝐽). Thus 𝐹⊆Apr_𝐹(𝐽). We get that 𝐹⊆𝐽.
(2) The result follows from (1).
(3) Let 𝑥∈𝐹. Then 1∈𝐹∩𝐽=[𝑥]_𝐹∩𝐽 and so 𝑥∈Apr_𝐹(𝐽). Therefore 𝐹⊆Apr_𝐹(𝐽).

Lemma 3.13. Let 𝐿 be linearly ordered and 𝐹 a filter of 𝐿. If 𝑥≤𝑦 and [x]_𝐹≠[𝑦]_𝐹, then for each 𝑡∈[𝑥]_𝐹 and 𝑠∈[𝑦]_𝐹, 𝑡≤𝑠.

Proof. Let there exist 𝑡∈[𝑥]_𝐹 and 𝑠∈[𝑦]_𝐹 such that 𝑠<𝑡. Then 𝑡→𝑥≤𝑠→𝑥, and also 𝑡→𝑥∈𝐹. So 𝑠→𝑥∈𝐹. By 𝑥≤𝑦 we get that 𝑦→𝑠≤𝑥→𝑠, also we have 𝑦→𝑠∈𝐹, and so 𝑥→𝑠∈𝐹. Thus 𝑠∈[𝑥]_𝐹∩[𝑦]_𝐹, that is, [𝑥]_𝐹=[𝑦]_𝐹, it is a contradiction. Thus for each 𝑡∈[𝑥]_𝐹 and 𝑠∈[𝑦]_𝐹, 𝑡≤𝑠.

Theorem 3.14. Let (𝐿,𝐹) be an approximation space and 𝐽 a filter of 𝐿. Then the following hold.(1) If 𝐹⊆𝐽, then 𝐽 is a rough filter of 𝐿.(2) If 𝐽⊆𝐹, then 𝐽 is an upper rough filter of 𝐿.(3) If 𝐿 is linearly ordered, then 𝐽 is an upper rough filter of 𝐿.

Proof. (1) The proof follows from Theorem 3.12 part (1).
(2) The proof is easy by Theorem 3.11 part (i).
(3) Let 𝑥,𝑦∈Apr_𝐹(𝐽). Then it is easy to see that 𝑥∗𝑦∈Apr_𝐹(𝐽). If 𝑥≤𝑦 and 𝑥∈Apr_𝐹(𝐽), then [𝑥]_𝐹∩𝐽≠∅ and so there is 𝑡∈𝐽 such that [𝑥]_𝐹=[𝑡]_𝐹. If [𝑥]_𝐹=[𝑦]_𝐹, we get that 𝑦∈Apr_𝐹(𝐽). If [𝑥]_𝐹≠[𝑦]_𝐹, then by Lemma 3.13 we obtain 𝑡≤𝑦. So by 𝑡∈𝐽 we get that 𝑦∈𝐽, that is, 𝑦∈Apr_𝐹(𝐽).

If 𝑋 is a nonempty subset of a 𝐵𝐿-algebra 𝐿, we let ¬𝑋={¬𝑥∣𝑥∈𝑋}. It is easy to see that for every nonempty subset 𝑋,𝑌 of 𝐿, 𝑋⊆𝑌 implies that ¬𝑋⊆¬𝑌.

Proposition 3.15. Let 𝐹 be a filter of 𝐿 and 𝑋 a nonempty subset of 𝐿. Then ¬Apr_𝐹(𝑋)⊆Apr_𝐹(¬𝑋).

Proof. Let 𝑧∈¬Apr_𝐹(𝑋). Then there is 𝑡∈Apr_𝐹(𝑋) such that 𝑧=¬𝑡 and so [𝑡]_𝐹∩𝑋≠∅. Thus there exists ℎ∈𝑋 such that [𝑡]_𝐹=[ℎ]_𝐹, hence [𝑧]_𝐹=[¬𝑡]_𝐹=[¬ℎ]_𝐹. Also ℎ∈𝑋 implies that ¬ℎ∈¬𝑋 and so [𝑧]_𝐹∩¬𝑋=[¬ℎ]_𝐹∩¬X≠∅. Therefore 𝑧∈Apr_𝐹(¬𝑋) and hence ¬Apr_𝐹(𝑋)⊆Apr_𝐹(¬𝑋).

Remark 3.16. (1) We cannot replace the inclusion symbol ⊆ by an equal sign in Proposition 3.15. Consider filter 𝐹={1,𝑏} in Example 3.10. Let subset 𝑋={0,𝑐} of 𝐿; we have ¬𝑋={1,0}, ¬Apr_𝐹(𝑋)=¬{0,𝑎,𝑐}={0,1} and Apr_𝐹(¬𝑋)={0,𝑏,1}. Therefore, Apr_𝐹(¬𝑋)⊈¬Apr_𝐹(𝑋).
(2) We can show that Proposition 3.15 may not be true for Apr_𝐹. Consider Example 3.10, filter 𝐹={1,𝑏} and subset 𝑋={0,𝑎,𝑐} of 𝐿. We can get that ¬𝑋={0,1}, Apr_𝐹(¬𝑋)={0} and ¬Apr_𝐹(𝑋)=¬{0,𝑎,𝑐}={0,1}. Therefore, ¬Apr_𝐹(𝑋)⊈Apr_𝐹(¬𝑋). Also by considering 𝑋={1} we can check that Apr_𝐹(¬𝑋)={0} and ¬Apr_𝐹(𝑋)=∅. Thus Apr_𝐹(¬𝑋)⊈¬Apr_𝐹(𝑋).

Let 𝐿 be a 𝐵𝐿-algebra. An element 𝑎 of 𝐿 is said to be regular if and only if ¬¬𝑎=𝑎. The set of all regular elements of 𝐿 is denoted by Reg(𝐿). The set of regular elements is also denoted by 𝑀𝑉(𝐿) in [10] where it is proved that it is the largest sub 𝑀𝑉-algebra of 𝐿.

Lemma 3.17. Let 𝐹 be a filter of 𝐿 and ∅≠𝑋⊆𝐿. Then,(i)Reg(𝐿)∩Apr_𝐹(¬𝑋)⊆¬Apr_𝐹(¬¬𝑋),(ii)Reg(𝐿)∩Apr_𝐹¬(𝑋∩Reg(𝐿))⊆¬Apr_𝐹(𝑋).

Proof. (i) Let 𝑧∈Reg(𝐿)∩Apr_𝐹(¬𝑋). Then [𝑧]_𝐹∩¬𝑋≠∅ and ¬¬𝑧=𝑧 and so there exists 𝑥∈𝑋 such that [¬𝑥]_𝐹=[𝑧]_𝐹. Thus we have 𝑧=¬¬𝑧=¬(¬𝑧) and [¬𝑧]_𝐹∩¬¬𝑋=[¬¬𝑥]_𝐹∩¬¬𝑋≠∅ and hence ¬𝑧∈Apr_𝐹(¬¬𝑋). Therefore, 𝑧∈¬Apr_𝐹(¬¬𝑋), and this implies that Reg(𝐿)∩Apr_𝐹(¬𝑋)⊆¬Apr_𝐹(¬¬𝑋).
(ii) Let 𝑧∈Reg(𝐿)∩Apr_𝐹¬(𝑋∩Reg(𝐿)). Then ¬¬𝑧=𝑧 and [𝑧]_𝐹∩¬(𝑋∩Reg(𝐿))≠∅. So there exists 𝑥∈𝑋∩Reg(𝐿) such that [¬𝑥]_𝐹=[𝑧]_𝐹 and ¬¬𝑥=𝑥. We get that 𝑧=¬(¬𝑧) and [¬𝑧]_𝐹∩𝑋=[𝑥]_𝐹∩𝑋≠∅ and so 𝑧∈¬Apr_𝐹(𝑋). Therefore, Reg(𝐿)∩Apr_𝐹¬(𝑋∩Reg(𝐿))⊆¬Apr_𝐹(𝑋).

An element 𝑎 of 𝐿 is said to be dense if and only if ¬𝑎=0. We denote by 𝐷𝑠(𝐿) the set of the dense elements of 𝐿. 𝐷𝑠(𝐿) is a filter of 𝐿 [15].

Lemma 3.18. Let 𝐹 be a filter of 𝐿. Then 𝐹⊆Apr_𝐹(𝐷𝑠(𝐿))⊆{𝑧∈𝐿∣¬¬𝑧∈𝐹}.

Proof. Let 𝑧∈Apr_𝐹(𝐷𝑠(𝐿)). Then [𝑧]_𝐹∩𝐷𝑠(𝐿)≠∅ and so there is 𝑡∈[𝑧]_𝐹 such that ¬𝑡=0. Thus [0]_𝐹=[¬𝑡]_𝐹=[¬𝑧]_𝐹 and hence ¬¬𝑧∈𝐹. Also since 𝐷𝑠(𝐿) is a filter of 𝐿, then by Theorem 3.12 part (3), 𝐹⊆Apr_𝐹(𝐷𝑠(𝐿)).

Lemma 3.19. Let 𝐹 be a filter of 𝐿 and 𝑋 a nonempty set of 𝐿. Then 𝑋 is definable respect to 𝐹 if and only if Apr_𝐹(𝑋)=𝑋 or Apr_𝐹(𝑋)=𝑋.

Proof. Suppose that 𝑋 is definable. Then 𝑋⊆Apr_𝐹(𝑋)=Apr_𝐹(𝑋)⊆𝑋 and so Apr_𝐹(𝑋)=𝑋=Apr_𝐹(𝑋). Conversely, let Apr_𝐹(𝑋)=𝑋. We show that Apr_𝐹(𝑋)=𝑋. We have Apr_𝐹(𝑋)⊆𝑋. Now let 𝑥∈𝑋. and 𝑧∈[𝑥]_𝐹. Then [𝑧]_𝐹∩𝑋=[𝑥]_𝐹∩𝑋≠∅ and so 𝑧∈Apr_𝐹(𝑋)=𝑋. Therefore Apr_𝐹(𝑋)=𝑋. If Apr_𝐹(𝑋)=𝑋, then we show that Apr_𝐹(𝑋)=𝑋. Let 𝑥∈Apr_𝐹(𝑋). Then [𝑥]_𝐹∩𝑋≠∅ and so there is 𝑧∈𝑋 such that [𝑥]_𝐹=[𝑧]_𝐹. By hypothesis we get that [𝑧]_𝐹⊆𝑋, hence 𝑥∈𝑋. Therefore Apr_𝐹(𝑋)⊆𝑋. Since 𝑋⊆Apr_𝐹(𝑋), then Apr_𝐹(𝑋)=𝑋.

By Theorem 3.12 part (1) and Lemma 3.19 we have the following:

Corollary 3.20. Let 𝐹 and 𝐽 be two filters of 𝐿. Then 𝐽 is definable respect to 𝐹 if and only if 𝐹⊆𝐽.

Let 𝑋 and 𝑌 be two nonempty subsets of 𝐿. Then we define𝑋⊙𝑌=𝑡∈𝐿∣𝑥∗𝑦≤𝑡,forsome𝑥∈𝑋,𝑦∈𝑌. (3.7) If either 𝑋 or 𝑌 are empty, then we define 𝑋⊙𝑌=∅. If 𝐴 and 𝐵 are two filters of 𝐿, then it is clear that 𝐴⊙𝐵 is the smallest filter containing of 𝐴 and 𝐵. For any subsets 𝑋,𝑌,𝑍 of 𝐿 we have 𝑋⊙𝑌=𝑌⊙𝑋 and (𝑋⊙𝑌)⊙𝑍=𝑋⊙(𝑌⊙𝑍). It is easy to see that Apr_𝐹(𝑋)∪Apr_𝐸(𝑋)⊆Apr_𝐹⊙𝐸(𝑋), for any filter 𝐸 and 𝐹 of 𝐿 and nonempty subset 𝑋 of 𝐿. Since 1∈𝑋⊙𝑌⊆Apr_𝐹(𝑋⊙𝑌), for nonempty subsets 𝑋,𝑌 of 𝐿 and filter 𝐹 of 𝐿, then we can conclude that 𝐹⊆Apr_𝐹(𝑋⊙𝑌).

Proposition 3.21. Let 𝐹 be a filter of 𝐿 and 𝑋,𝑌 be nonempty subsets of 𝐿. Then(i)Apr_𝐹(𝑋⊙𝑌)⊆Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌),(ii) If 𝑋,𝑌⊆𝐹, then Apr_𝐹(𝑋⊙𝑌)=Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌),(iii) If 𝐿 is linearly ordered, then Apr_𝐹(𝑋⊙𝑌)=Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌).

Proof. (i) Let 𝑧∈Apr_𝐹(𝑋⊙𝑌). Then [𝑧]_𝐹∩(𝑋⊙𝑌)≠∅ and so there is 𝑡∈𝑋⊙𝑌 such that [𝑧]_𝐹=[𝑡]_𝐹. Hence there are 𝑥∈𝑋 and 𝑦∈𝑌 such that 𝑥∗𝑦≤𝑡. On the other hand, [𝑧]_𝐹=[𝑡]_𝐹 implies that 𝑡→𝑧∈𝐹, then there is 𝑓∈𝐹 such that 𝑡≤𝑓→𝑧. By hypothesis we can conclude that 𝑥∗𝑦≤𝑡≤𝑓→𝑧, and hence 𝑥∗(𝑦∗𝑓)≤𝑧. Also by Lemma 2.2, we have 𝑓≤𝑦∗𝑓→𝑦 and 𝑓≤𝑦→𝑦∗𝑓, thus 𝑓∈𝐹 implies that (𝑦∗𝑓→𝑦)∈𝐹 and (𝑦→𝑦∗𝑓)∈𝐹. Therefore [𝑦∗𝑓]_𝐹=[𝑦]_𝐹, hence 𝑦∗𝑓∈Apr_𝐹(𝑌), and also we have 𝑥∈Apr_𝐹(𝑋). So 𝑧∈Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌).
(ii) If 𝑋,𝑌⊆𝐹, then 𝑋⊙𝑌⊆𝐹. So by Theorem 3.11Apr_𝐹(𝑋)=Apr_𝐹(𝑌)=Apr_𝐹(𝑋⊙𝑌)=𝐹. Therefore Apr_𝐹(𝑋⊙𝑌)=𝐹=𝐹⊙𝐹=Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌).
(iii) Let 𝐿 be linearly ordered and 𝑧∈Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌). Then there are ℎ∈Apr_𝐹(𝑋) and 𝑘∈Apr_𝐹(𝑌) such that ℎ∗𝑘≤𝑧. So [ℎ]_𝐹∩𝑋≠∅ and [𝑘]_𝐹∩𝑌≠∅ imply that there are 𝑥∈𝑋 and 𝑦∈𝑌 such that [𝑥]_𝐹=[ℎ]_𝐹 and [𝑦]_𝐹=[𝑘]_𝐹. Hence [𝑥∗𝑦]_𝐹=[ℎ∗𝑘]_𝐹 and we have 𝑥∗𝑦∈𝑋⊙𝑌. If [𝑧]_𝐹=[ℎ∗𝑘]_𝐹, then by hypothesis we get that [𝑧]_𝐹∩𝑋⊙𝑌≠∅ and hence 𝑧∈Apr_𝐹(𝑋⊙𝑌). If [𝑧]_𝐹≠[ℎ∗𝑘]_𝐹, then by Lemma 3.13𝑥∗𝑦≤𝑧 and so 𝑧∈𝑋⊙𝑌. Therefore [𝑧]_𝐹∩(𝑋⊙𝑌)≠∅, that is, 𝑧∈Apr_𝐹(𝑋⊙𝑌).

Proposition 3.22. Let 𝐹 be a filter of 𝐿 and 𝑋,𝑌 be nonempty subsets of 𝐿. Then(i)Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋⊙𝑌).(ii)If 𝑋 and 𝑌 are definable respect to 𝐹, then Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌)=Apr_𝐹(𝑋⊙𝑌).

Proof. (i) Let 𝑧∈Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌). Then 𝑥∗𝑦≤𝑧, for some 𝑥∈Apr_𝐹(𝑋) and 𝑦∈Apr_𝐹(𝑌). Consider 𝑏∈[𝑧]_𝐹, so there is 𝑓∈𝐹 such that 𝑧≤𝑓→𝑏. Hence 𝑥∗𝑦≤𝑓→𝑏, we can obtain that 𝑥∗(𝑦∗𝑓)≤𝑏. Thus by hypothesis we have [𝑦∗𝑓]_𝐹=[𝑦]_𝐹⊆𝑌, [𝑥]_𝐹⊆𝑋 and 𝑥∗(𝑦∗𝑓)≤𝑏, hence 𝑏∈𝑋⊙𝑌. Therefore 𝑧∈Apr_𝐹(𝑋⊙𝑌), it implies that Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌)⊆Apr_𝐹(𝑋⊙𝑌).
(ii) Since 𝑋 and 𝑌 are definable respect to 𝐹, then Apr_𝐹(𝑋)=𝑋 and Apr_𝐹(𝑌)=𝑌. By part (i), we get that 𝑋⊙𝑌⊆Apr_𝐹(𝑋⊙𝑌)⊆𝑋⊙𝑌. Therefore Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌)=𝑋⊙𝑌=Apr_𝐹(𝑋⊙𝑌).

By the following example, we show that we cannot replace the inclusion symbol ⊆ by an equal sign in general in the above proposition part (i).

Example 3.23. Let 𝐿={0,𝑎,𝑏,𝑐,𝑑,1}, where 0<𝑑,𝑏<𝑎<1, 0<𝑑<𝑐<1. Define ∗ and → as follow: →0𝑎𝑏𝑐𝑑10𝑎111111𝑏𝑑1𝑎𝑐𝑐1𝑐𝑐11𝑐𝑐1𝑑𝑏𝑎𝑏1𝑎11𝑎1𝑎111∗0𝑎𝑏𝑐𝑑10𝑎𝑏𝑐𝑑10𝑎000000𝑏0𝑏𝑏𝑑0𝑎𝑐0𝑏𝑏00𝑏𝑑0𝑑0𝑐𝑑𝑐1000𝑑0𝑑.0𝑎𝑏𝑐𝑑1 (3.8) Then (𝐿,∧,∨,∗,→,0,1) is a 𝐵𝐿-algebra. It is easy to check that 𝐹={1,𝑐} is a filter of 𝐿. By considering subsets 𝑋={𝑎} and 𝑌={𝑐} of 𝐿, we have Apr_𝐹(𝑋⊙𝑌)={1,𝑎,𝑑,𝑐} and Apr_𝐹(𝑋)=Apr_𝐹(𝑌)=∅. Therefore Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌)≠Apr_𝐹(𝑋⊙𝑌).

Proposition 3.24. Let 𝐹 and 𝐽 be two filters of 𝐿 and 𝑋 be a nonempty subset of 𝐿. Then(i)If 𝑋⊆𝐵(𝐿), then Apr_𝐹⊙𝐽(𝑋)⊆Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋),(ii)Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋)⊆Apr_𝐹⊙𝐽(𝑋)⊙Apr_𝐹⊙𝐽(𝑋), (iii)Apr_𝐹⊙𝐽(𝑋)⊆Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋).

Proof. (i) Let 𝑧∈Apr_𝐹⊙𝐽(𝑋). Then [𝑧]_𝐹⊙𝐽∩𝑋≠∅ and so there is 𝑥∈𝑋 such that 𝑥→𝑧∈𝐹⊙𝐽. Thus there are 𝑓∈𝐹 and 𝑒∈𝐽 such that 𝑓∗𝑒∗𝑥≤𝑧, by 𝑋⊆𝐵(𝐴), we get that (𝑓∗𝑥)∗(𝑒∗𝑥)≤𝑧. Since [𝑓∗𝑥]_𝐹=[𝑥]_𝐹 and [𝑒∗𝑥]_𝐽=[𝑥]_𝐽, then we have 𝑧∈Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋).
(ii) Let ℎ∈Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋). Then 𝑡∗𝑠≤ℎ, for some 𝑡,𝑠∈𝐿 such that [𝑡]_𝐹∩𝑋≠∅ and [𝑠]_𝐽∩𝑋≠∅. So [𝑡]_𝐹⊙𝐽∩𝑋≠∅ and [𝑠]_𝐹⊙𝐽∩𝑋≠∅ and hence 𝑡,𝑠∈Apr_𝐹⊙𝐽(𝑋). Thus ℎ∈Apr_𝐹⊙𝐽(𝑋)⊙Apr_𝐹⊙𝐽(𝑋).
(iii) Let 𝑧∈Apr_𝐹⊙𝐽(𝑋). Then [𝑧]_𝐹⊙𝐽⊆𝑋. Since 𝑧∗𝑧≤𝑧 and [𝑧]_𝐹,[𝑧]_𝐽⊆[𝑧]_𝐹⊙𝐸⊆𝑋, hence 𝑧∈Apr_𝐹(𝑋)⊙Apr_𝐽(𝑋).

By Proposition 3.21 part (i) and Proposition 3.24 part (i) we can obtain the following corollary.

Corollary 3.25. Let 𝐹 and 𝐽 be two filters of 𝐿 and 𝑋,𝑌 be nonempty subsets of 𝐵(𝐿). Then Apr_𝐹⊙𝐽(𝑋⊙𝑌)⊆(Apr_𝐹(𝑋)⊙Apr_𝐹(𝑌))⊙(Apr_𝐽(𝑋)⊙Apr_𝐽(𝑌)).

Let 𝐽 and 𝐹 be two filters of 𝐿 such that 𝐽⊆𝐹 and let 𝑋 be a nonempty subset of 𝐿. Then it is easy to see that Apr_𝐽(𝑋)⊆Apr_𝐹(𝑋) and Apr_𝐹(𝑋)⊆Apr_𝐽(𝑋). So we have Apr_𝐹∩𝐽(𝑋)⊆Apr_𝐹(𝑋)∩Apr_𝐽(𝑋) and Apr_𝐹(𝑋)∩Apr_𝐽(𝑋)⊆Apr_𝐹∩𝐽(𝑋).

Consider 𝐵𝐿-algebra 𝐿 in Example 3.10. We can see that 𝐽={1,𝑎} and 𝐹={1,𝑏} are two filters of 𝐿. Put 𝑋={𝑎,𝑏}. We have Apr_𝐹∩𝐽(𝑋)={𝑎,𝑏} and Apr_𝐹(𝑋)=Apr_𝐽(𝑋)={1,𝑏,𝑐,𝑎}, so Apr_𝐹∩𝐽(𝑋)≠Apr_𝐹(𝑋)∩Apr_𝐽(𝑋). Also Apr_𝐹∩𝐽(𝑋)=𝑋 and Apr_𝐹(𝑋)=Apr_𝐽(𝑋)=∅, thus Apr_𝐹(𝑋)∩Apr_𝐽(𝑋)≠Apr_𝐹∩𝐽(𝑋).

By the following proposition we can obtain some conditions that Apr_𝐹∩𝐽(𝑋)=Apr_𝐹(𝑋)∩Apr_𝐽(𝑋) or Apr_𝐹(𝑋)∩Apr_𝐽(𝑋)=Apr_𝐹∩𝐽(𝑋).

Proposition 3.26. Let 𝐹 and 𝐽 be two filters of 𝐿 and 𝑋 be a nonempty subset of 𝐿. Then(i)If 𝑋⊆𝐹∩𝐽 or 𝑋 is definable respect to 𝐽 or 𝐹, then Apr_𝐹∩𝐽(𝑋)=Apr_𝐹(𝑋)∩Apr_𝐽(𝑋),(ii)If 𝑋 is a filter of 𝐿 containing 𝐽 and 𝐹, then Apr_𝐹(𝑋)∩Apr_𝐽(𝑋)=Apr_𝐹∩𝐽(𝑋).

Proof. (i) Assume that 𝑋⊆𝐹∩𝐽. Then by Theorem 3.11 part (i), Apr_𝐹∩𝐽(𝑋)=𝐹∩𝐽=Apr_𝐹(𝑋)∩Apr_𝐽(𝑋). If 𝑋 is definable respect to 𝐽, then Apr_𝐹(𝑋)∩Apr_𝐽(𝑋)=𝑋⊆Apr_𝐹∩𝐽(𝑋), and it proves theorem.
(ii) If 𝑋 is a filter of 𝐿 containing 𝐽 and 𝐹, then by Theorem 3.12 part (1) Apr_𝐹∩𝐽(𝑋)=𝑋=𝑋∩𝑋=Apr_𝐹(𝑋)∩Apr_𝐽(𝑋).

Proposition 3.27. Let 𝐿 and 𝐿^′ be two 𝐵𝐿-algebras and 𝑓∶𝐿→𝐿^′ be a homomorphism. Then(i)If 𝑋 be a nonempty subset of 𝐿 and 𝐹 be a filter of 𝐿^′, then 𝑓⁻¹(Apr_𝐹(𝑓(𝑋))=Apr_𝑓⁻¹_(𝐹)(𝑋),(ii)𝑓(Apr_ker𝑓(𝑋))=𝑓(𝑋), for any nonempty subset 𝑋 of 𝐿.(iii)Let 𝑓 be onto, 𝑌 be a nonempty subset of 𝐿 and 𝐹 a filter of 𝐿. If ker𝑓⊆𝐹, then 𝑓(Apr_𝐹(𝑌))=Apr_𝑓(𝐹)(𝑓(𝑌)).

Proof. (i) By hypothesis we have

𝑧∈𝑓⁻¹(Apr_𝐹(𝑓(𝑋))⇔𝑓(𝑧)∈(Apr_𝐹(𝑓(𝑋))⇔[𝑓(𝑧)]_𝐹∩𝑓(𝑋)≠∅⇔[𝑓(𝑧)]_𝐹=[𝑓(𝑥)]_𝐹, for some 𝑥∈𝑋⇔[𝑧]_𝑓⁻¹_(𝐹)=[𝑥]_𝑓⁻¹_(𝐹), for some 𝑥∈𝑋⇔[𝑧]_𝑓⁻¹_(𝐹)∩𝑋≠∅⇔𝑧∈Apr_𝑓⁻¹_(𝐹)(𝑋).

(ii) The proof is easy by part (i).
(iii) Let 𝑧∈𝑓(Apr_𝐹(𝑌)). Then there is 𝑡∈Apr_𝐹(𝑌) such that 𝑧=𝑓(𝑡), and so there exists 𝑦∈𝑌 such that [𝑡]_𝐹=[𝑦]_𝐹. Thus by [𝑓(𝑡)]_𝑓(𝐹)=[𝑓(𝑦)]_𝑓(𝐹) we get that [𝑓(𝑡)]_𝑓(𝐹)∩𝑓(𝑌)≠∅. Therefore 𝑧∈Apr_𝑓(𝐹)(𝑓(𝑌)). Now let 𝑧∈Apr_𝑓(𝐹)(𝑓(𝑌)). Then [𝑧]_𝑓(𝐹)∩𝑓(𝑌)≠∅. By hypothesis we have ℎ∈𝐿 such that 𝑧=𝑓(ℎ), and so there is 𝑦∈𝑌 such that [𝑓(ℎ)]_𝑓(𝐹)=[𝑓(𝑦)]_𝑓(𝐹). Since 𝐹 is a filter of 𝐿 and ker𝑓⊆𝐹, then we obtain that [𝑦]_𝐹=[ℎ]_𝐹. So ℎ∈Apr_𝐹(𝑌), hence 𝑧=𝑓(ℎ)∈𝑓(Apr_𝐹(𝑌)).