Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 648983 | 11 pages | https://doi.org/10.1155/2012/648983

Generalized Lower and Upper Approximations in Quantales

Academic Editor: Jin L. Kuang
Received08 Aug 2011
Revised21 Oct 2011
Accepted30 Oct 2011
Published01 Feb 2012

Abstract

We introduce the concepts of set-valued homomorphism and strong set-valued homomorphism of a quantale which are the extended notions of congruence and complete congruence, respectively. The properties of generalized lower and upper approximations, constructed by a set-valued mapping, are discussed.

1. Introduction

The concept of Rough set was introduced by Pawlak [1] as a mathematical tool for dealing with vagueness or uncertainty. In Pawlak’s rough sets, the equivalence classes are the building blocks for the construction of the lower and upper approximations. It soon invoked a natural question concerning a possible connection between rough sets and algebraic systems. Biswas and Nanda [2] introduced the notion of rough subgroups. Kuroki [3] and Qimei [4] introduced the notions of a rough ideal and a rough prime ideal in a semigroup, respectively. Davvaz in [5] introduced the notion of rough subring with respect to an ideal of a ring. Rough modules have been investigated by Davvaz and Mahdavipour [6]. Rasouli and Davvaz studied the roughness in MV-algebra [7]. In [8–12], the roughness of various hyperstructures are discussed. Further, some authors consider the rough set in a fuzzy algebraic system, see [13–16]. The concept of quantale was introduced by Mulvey [17] in 1986 with the purpose of studying the spectrum of C*-algebra, as well as constructive foundations for quantum mechanics. There are abundant contents in the structure of quantales, because quantale can be regarded as the generalization of frame. Since quantale theory provides a powerful tool in studying noncommutative structures, it has wide applications, especially in studying noncommutative C*-algebra theory, the ideal theory of commutative ring, linear logic, and so on. The quantale theory has aroused great interests of many researchers, and a great deal of new ideas and applications of quantale have been proposed in twenty years [18–24].

The majority of studies on rough sets for algebraic structures such as semigroups, groups, rings, and modules have been concentrated on a congruence relation. An equivalence relation is sometimes difficult to be obtained in real-world problems due to the vagueness and incompleteness of human knowledge. From this point of view, Davvaz [25] introduced the concept of set-valued homomorphism for groups. And then, Yamak et al. [26, 27] introduced the concepts of set-valued homomorphism and strong set-valued homomorphism of a ring and a module. In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism in quantales are introduced. We discuss the properties of generalized lower and upper approximations in quantales.

2. Preliminaries

In this section, we give some basic notions and results about quantales and rough set theory (see [19, 22, 25, 28]), which will be necessary in the next sections.

Definition 2.1. A quantale is a complete lattice 𝑄 with an associative binary operation ∘ satisfying ğ‘Žâˆ˜âŽ›âŽœâŽî˜ğ‘–âˆˆğ¼ğ‘ğ‘–âŽžâŽŸâŽ =î˜ğ‘–âˆˆğ¼î€·ğ‘Žâˆ˜ğ‘ğ‘–î€¸,âŽ›âŽœâŽî˜ğ‘–âˆˆğ¼ğ‘Žğ‘–âŽžâŽŸâŽ âˆ˜ğ‘=î˜ğ‘–âˆˆğ¼î€·ğ‘Žğ‘–âˆ˜ğ‘î€¸,(2.1) for all ğ‘Ž,𝑏,ğ‘Žğ‘–,𝑏𝑖∈𝑄(𝑖∈𝐼).
An element 𝑒∈𝑄 is called a left (right) unit if and only if ğ‘’âˆ˜ğ‘Ž=ğ‘Ž (ğ‘Žâˆ˜ğ‘’=ğ‘Ž), 𝑒 is called a unit if it is both a right and left unit.
A quantale 𝑄 is called a commutative quantale if ğ‘Žâˆ˜ğ‘=ğ‘âˆ˜ğ‘Ž for all ğ‘Ž,𝑏∈𝑄.
A quantale 𝑄 is called an idempotent quantale if ğ‘Žâˆ˜ğ‘Ž=ğ‘Ž for all ğ‘Žâˆˆğ‘„.
A subset 𝑆 of 𝑄 is called a subquantale of 𝑄 if it is closed under ∘ and arbitrary sups.
In a quantale 𝑄, we denote the top element of 𝑄 by 1 and the bottom by 0. For 𝐴,𝐵⊆𝑄, we write 𝐴∘𝐵 to denote the set {ğ‘Žâˆ˜ğ‘âˆ£ğ‘Žâˆˆğ´,𝑏∈𝐵}, 𝐴∨𝐵 to denote {ğ‘Žâˆ¨ğ‘âˆ£ğ‘Žâˆˆğ´,𝑏∈𝐵} and ⋁𝑖∈𝐼𝐴𝑖={â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–âˆ£ğ‘Žğ‘–âˆˆğ´ğ‘–}.

Definition 2.2. Let 𝑄 be a quantale, a subset ∅≠𝐼⊆𝑄 is called a left (right) ideal of 𝑄 if(1)ğ‘Ž,𝑏∈𝐼 implies ğ‘Žâˆ¨ğ‘âˆˆğ¼,(2)ğ‘Žâˆˆğ¼,𝑏∈𝑄 and ğ‘â‰¤ğ‘Ž imply 𝑏∈𝐼 for all,(3)ğ‘Žâˆˆğ‘„ and 𝑥∈𝐼 imply ğ‘Žâˆ˜ğ‘¥âˆˆğ¼ (ğ‘¥âˆ˜ğ‘Žâˆˆğ¼).
A subset 𝐼⊆𝑄 is called an ideal if it is both a left and a right ideal.
Let 𝑋 be a subset of 𝑄, we write ↓𝑋={𝑦∈𝑄∣𝑦≤𝑥 for some 𝑥∈𝑋}, 𝑋 is a lower set if and only if 𝑋=↓𝑋. It is obvious that an ideal 𝐼 is a directed lower set. For every (left, right) ideal 𝐼 of 𝑄, it is easy to see that 0∈𝐼.
An ideal of 𝑄 is called a prime ideal if ğ‘Žâˆ˜ğ‘âˆˆğ¼ implies ğ‘Žâˆˆğ¼ or 𝑏∈𝐼 for all ğ‘Ž,𝑏∈𝑄.
An ideal 𝐼 of 𝑄 is called a semi-prime ideal if ğ‘Žâˆ˜ğ‘Žâˆˆğ¼ implies ğ‘Žâˆˆğ¼ for all ğ‘Žâˆˆğ‘„.
An ideal 𝐼 of 𝑄 (𝐼≠𝑄) is called a primary ideal if for all ğ‘Ž,𝑏∈𝑄, ğ‘Žâˆ˜ğ‘âˆˆğ¼ and ğ‘Žâˆ‰ğ¼ imply 𝑏𝑛∈𝐼 for some 𝑛>0. (𝑏𝑛=𝑏∘𝑏∘⋯∘𝑏𝑛).

Definition 2.3. A nonempty subset 𝑀⊆𝑄 is called an 𝑚-system of 𝑄, if for all ğ‘Ž,𝑏∈𝑀, ↓(ğ‘Žâˆ˜1∘𝑏)∩𝑀≠∅.
A nonempty subset 𝑆⊆𝑄 is called a multiplicative set of 𝑄, if ğ‘Žâˆ˜ğ‘âˆˆğ‘† for all ğ‘Ž,𝑏∈𝑆.
Every ideal of 𝑄 is both an 𝑚-system and a multiplicative set.

Definition 2.4. Let 𝑄 be a quantale, an equivalence relation 𝜃 on 𝐿 is called a congruence on 𝑄 if for all ğ‘Ž,𝑏,𝑐,𝑑,ğ‘Žğ‘–,𝑏𝑖∈𝑄(𝑖∈𝐼), we have(1)ğ‘Žğœƒğ‘,𝑐𝜃𝑑⇒(ğ‘Žâˆ˜ğ‘)𝜃(𝑏∘𝑑),(2)ğ‘Žğ‘–ğœƒğ‘ğ‘–(𝑖∈𝐼)⇒(â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–)𝜃(⋁𝑖∈𝐼𝑏𝑖).It is obvious that [ğ‘Ž]𝜃∘[𝑏]𝜃⊆[ğ‘Žâˆ˜ğ‘]𝜃, ⋁𝑖∈𝐼[ğ‘Žğ‘–]𝜃⊆[â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–]𝜃 for all ğ‘Ž,𝑏,ğ‘Žğ‘–âˆˆğ‘„(𝑖∈𝐼).

Definition 2.5. Let 𝑄 be a quantale, a congruence 𝜃 on 𝑄 is called a complete congruence, if(1)[ğ‘Ž]𝜃∘[𝑏]𝜃=[ğ‘Žâˆ˜ğ‘]𝜃 for all ğ‘Ž,𝑏∈𝑄,(2)⋁𝑖∈𝐼[ğ‘Žğ‘–]𝜃=[â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–]𝜃 for all ğ‘Žğ‘–âˆˆğ‘„(𝑖∈𝐼).

Definition 2.6. Let (𝑄1,∘1) and (𝑄2,∘2) be two quantales. A map 𝑓∶𝑄1→𝑄2 is said to be a homomorphism if(1)𝑓(ğ‘Žâˆ˜1𝑏)=𝑓(ğ‘Ž)∘2𝑓(𝑏) for all ğ‘Ž,𝑏∈𝑄1;(2)𝑓(â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–)=⋁𝑖∈𝐼𝑓(ğ‘Žğ‘–) for all ğ‘Žğ‘–âˆˆğ‘„1(𝑖∈𝐼).

Definition 2.7. Let 𝑈 and 𝑊 be two nonempty universes. Let 𝑇 be a set-valued mapping given by 𝑇∶𝑈→𝑃(𝑊), where 𝑃(𝑊) denotes the set of all subsets of 𝑊. Then the triple (𝑈,𝑊,𝑇) is referred to as a generalized approximation space. For any set 𝐴⊆𝑊, the generalized lower and upper approximations, 𝑇−(𝐴) and 𝑇−(𝐴), are defined by 𝑇−(𝐴)={𝑥∈𝑈∣𝑇(𝑥)⊆𝐴},𝑇−(𝐴)={𝑥∈𝑈∣𝑇(𝑥)∩𝐴≠∅}.(2.2)
The pair (𝑇−(𝐴),𝑇−(𝐴)) is referred to as a generalized rough set.

From the definition, the following theorems can be easily derived.

Theorem 2.8. Let 𝑈,𝑊 be nonempty universes and 𝑇∶𝑈→𝑃∗(𝑊) be a set-valued mapping, where 𝑃∗(𝑊) denotes the set of all nonempty subsets of 𝑊. If 𝐴⊆𝑊, then 𝑇−(𝐴)⊆𝑇−(𝐴).

If 𝑊=𝑈 and 𝑅𝑇={(𝑥,𝑦)∣𝑦∈𝑇(𝑥)} is an equivalence relation on 𝑈, then the pair (𝑈,𝑅𝑇) is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlak’s rough set.

Theorem 2.9. Let (𝑈,𝑊,𝑇) be a generalized approximation space, its lower and upper approximation operators satisfy the following properties. For all 𝐴,𝐵∈𝑃(𝑊), (1)𝑇−(𝐴)=(𝑇−(𝐴𝑐))𝑐, 𝑇−(𝐴)=(𝑇−(𝐴𝑐))𝑐,(2)𝑇−(𝑊)=𝑈, 𝑇−(∅)=∅,(3)𝑇−(𝐴∩𝐵)=𝑇−(𝐴)∩𝑇−(𝐵), 𝑇−(𝐴∪𝐵)=𝑇−(𝐴)∪𝑇−(𝐵),(4)𝐴⊆𝐵⇒𝑇−(𝐴)⊆𝑇−(𝐵), 𝑇−(𝐴)⊆𝑇−(𝐵),(5)𝑇−(𝐴∪𝐵)⊇𝑇−(𝐴)∪𝑇−(𝐵), 𝑇−(𝐴∩𝐵)⊆𝑇−(𝐴)∩𝑇−(𝐵),where 𝐴𝑐 is the complement of the set 𝐴.

3. Generalized Rough Subsets in Quantales

In this paper, (𝑄1,∘1) and (𝑄2,∘2) are two quantales.

Theorem 3.1. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued mapping and ∅≠𝐴,𝐵⊆𝑄2. Then(1)𝑇−(𝐴)∪𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵), if 0∈𝐴∩𝐵,(2)𝑇−(𝐴)∪𝑇−(𝐵)⊆𝑇−(𝐴∘2𝐵), if 𝑒∈𝐴∩𝐵,(3)𝑇−(𝐴)∩𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵),(4)𝑇−(𝐴)∩𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵), if 𝑄2 is an idempotent quantale,(5)𝑇−(𝐴)∪𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵), if 0∈𝐴∩𝐵,(6)𝑇−(𝐴)∪𝑇−(𝐵)⊆𝑇−(𝐴∘2𝐵), if 𝑒∈𝐴∩𝐵.

Proof. (1) Suppose that ğ‘Žâˆˆğ´, we have ğ‘Ž=ğ‘Žâˆ¨0∈𝐴∨𝐵 for 0∈𝐵. So 𝐴⊆𝐴∨𝐵. Similarly, 𝐵⊆𝐴∨𝐵. So 𝐴∪𝐵⊆𝐴∨𝐵. By Theorem 2.9, we have 𝑇−(𝐴)∪𝑇−(𝐵)=𝑇−(𝐴∪𝐵)⊆𝑇−(𝐴∨𝐵).
(2) Suppose that ğ‘Žâˆˆğ´, we have ğ‘Ž=ğ‘Žâˆ˜2𝑒∈𝐴∘2𝐵 for 𝑒∈𝐵. So 𝐴⊆𝐴∘2𝐵. Similarly, 𝐵⊆𝐴∘2𝐵. So 𝐴∪𝐵⊆𝐴∘2𝐵. By Theorem 2.9, we have 𝑇−(𝐴)∪𝑇−(𝐵)=𝑇−(𝐴∪𝐵)⊆𝑇−(𝐴∘2𝐵).
(3) It is obvious that 𝐴∩𝐵⊆𝐴∨𝐵. By Theorem 2.9, we have 𝑇−(𝐴)∩𝑇−(𝐵)=𝑇−(𝐴∩𝐵)⊆𝑇−(𝐴∨𝐵).
(4) Since 𝑄2 is an idempotent quantale, we have 𝐴∩𝐵⊆𝐴∘𝐵. By Theorem 2.9, we have 𝑇−(𝐴)∩𝑇−(𝐵)=𝑇−(𝐴∩𝐵)⊆𝑇−(𝐴∘2𝐵).
(5) and (6) The proofs are similar to (1) and (2), respectively.

Definition 3.2. A set-valued mapping 𝑇∶𝑄1→𝑃(𝑄2) is called a set-valued homomorphism if(1)𝑇(ğ‘Ž)∘2𝑇(𝑏)⊆𝑇(ğ‘Žâˆ˜1𝑏) for all ğ‘Ž,𝑏∈𝑄1,(2)⋁𝑖∈𝐼𝑇(ğ‘Žğ‘–)⊆𝑇(â‹ğ‘–âˆˆğ¼ğ‘Žğ‘–) for all ğ‘Žğ‘–âˆˆğ‘„1(𝑖∈𝐼).
𝑇 is called a strong set-valued homomorphism if the equalities in (1), (2) hold.

Example 3.3. (1) Let 𝜃 be a congruence on 𝑄2. Then the set-valued mapping 𝑇∶𝑄1→𝑃(𝑄2) defined by 𝑇(𝑥)=[𝑥]𝜃 is a set-valued homomorphism but not necessarily a strong set-valued homomorphism. If 𝜃 is complete, then 𝑇 is a strong set-valued homomorphism.
(2) Let 𝑓 be a quantale homomorphism from 𝑄1 to 𝑄2. Then the set-valued mapping 𝑇∶𝑄1→𝑃(𝑄2) defined by 𝑇(ğ‘Ž)={𝑓(ğ‘Ž)} is a strong set-valued homomorphism.

Theorem 3.4. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued homomorphism and ∅≠𝐴,𝐵⊆𝑄2. Then(1)𝑇−(𝐴)∨𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵),(2)𝑇−(𝐴)∘1𝑇−(𝐵)⊆𝑇−(𝐴∘2𝐵),(3)𝑇−(𝐴)∩𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵),(4)𝑇−(𝐴)∩𝑇−(𝐵)⊆𝑇−(𝐴∘2𝐵), if 𝑄1 is an idempotent quantale.

Proof. (1) Suppose that 𝑐∈𝑇−(𝐴)∨𝑇−(𝐵), there exist ğ‘Žâˆˆğ‘‡âˆ’(𝐴),𝑏∈𝑇−(𝐵) such that 𝑐=ğ‘Žâˆ¨ğ‘. So there exist 𝑥∈𝐴∩𝑇(ğ‘Ž) and 𝑦∈𝐵∩𝑇(𝑏). Hence 𝑥∨𝑦∈𝐴∨𝐵 and 𝑥∨𝑦∈𝑇(ğ‘Ž)∨𝑇(𝑏). Since 𝑇 is a set-valued homomorphism, we have 𝑥∨𝑦∈𝑇(ğ‘Žâˆ¨ğ‘). Therefore, 𝑇(ğ‘Žâˆ¨ğ‘)∩(𝐴∨𝐵)≠∅ which implies that 𝑐=ğ‘Žâˆ¨ğ‘âˆˆğ‘‡âˆ’(𝐴∨𝐵).
(2) The proof is similar to (1).
(3) Suppose that 𝑥∈𝑇−(𝐴)∩𝑇−(𝐵), there exist ğ‘Žâˆˆğ´âˆ©ğ‘‡(𝑥) and 𝑏∈𝐵∩𝑇(𝑥). Since 𝑇 is a set-valued homomorphism, we have ğ‘Žâˆ¨ğ‘âˆˆğ‘‡(𝑥)∨𝑇(𝑥)⊆𝑇(𝑥). So ğ‘Žâˆ¨ğ‘âˆˆğ‘‡(𝑥)∩(𝐴∨𝐵) which implies that 𝑥∈𝑇−(𝐴∨𝐵).
(4) Suppose that 𝑥∈𝑇−(𝐴)∩𝑇−(𝐵), there exist ğ‘Žâˆˆğ´âˆ©ğ‘‡(𝑥) and 𝑏∈𝐵∩𝑇(𝑥). Since 𝑇 is a set-valued homomorphism and 𝑄1 is idempotent, we have ğ‘Žâˆ¨ğ‘âˆˆğ‘‡(𝑥)∘2𝑇(𝑥)⊆𝑇(𝑥∘1𝑥)=𝑇(𝑥). So ğ‘Žâˆ¨ğ‘âˆˆğ‘‡(𝑥)∩(𝐴∨𝐵) which implies that 𝑥∈𝑇−(𝐴∘2𝐵).

Theorem 3.5. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and ∅≠𝐴,𝐵⊆𝑄2. Then(1)𝑇−(𝐴)∨𝑇−(𝐵)⊆𝑇−(𝐴∨𝐵),(2)𝑇−(𝐴)∘1𝑇−(𝐵)⊆𝑇−(𝐴∘2𝐵).

Proof. (1) Suppose that 𝑐∈𝑇−(𝐴)∨𝑇−(𝐵), there exist ğ‘Žâˆˆğ‘‡âˆ’(𝐴),𝑏∈𝑇−(𝐵) such that 𝑐=ğ‘Žâˆ¨ğ‘. Hence 𝑇(ğ‘Ž)⊆𝐴 and 𝑇(𝑏)⊆𝐵. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(ğ‘Žâˆ¨ğ‘)=𝑇(ğ‘Ž)∨𝑇(𝑏)⊆𝐴∨𝐵 which implies that 𝑐=ğ‘Žâˆ¨ğ‘âˆˆğ‘‡âˆ’(𝐴∨𝐵).
(2) The proof is similar to (1).

4. Generalized Rough Ideals in Quantales

Theorem 4.1. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued mapping. If 𝐼 and 𝐽 are, respectively, a right and a left ideal of 𝑄2, then(1)𝑇−(𝐼∘2𝐽)⊆𝑇−(𝐼)∩𝑇−(𝐽),(2)𝑇−(𝐼∘2𝐽)⊆𝑇−(𝐼)∩𝑇−(𝐽),(3)𝑇−(𝐼∧𝐽)⊆𝑇−(𝐼)∩𝑇−(𝐽),(4)𝑇−(𝐼)∧𝑇−(𝐽)=𝑇−(𝐼∧𝐽),(5)𝑇−(𝐼)∪𝑇−(𝐽)⊆𝑇−(𝐼∨𝐽),(6)𝑇−(𝐼)∪𝑇−(𝐽)⊆𝑇−(𝐼∨𝐽).
If 𝑄1 is an idempotent quantale and 𝑇 is a set-valued homomorphism, then the equalities in (1)–(3) hold.

Proof. Since 𝐼 and 𝐽 are, respectively, a right and a left ideal of 𝑄2, we have 𝐼∘2𝐽⊆𝐼∩𝐽, 𝐼∧𝐽=𝐼∩𝐽 and 𝑜∈𝐼∩𝐽. By Theorem 2.9, we get the conclusion (1)–(4). By Theorem 3.1, we get (5) and (6).
If 𝑄1 is idempotent, we first show that 𝑇−(𝐼)∩𝑇−(𝐽)⊆𝑇−(𝐼∘2𝐽). Suppose that 𝑥∈𝑇−(𝐼)∩𝑇−(𝐽), there exist 𝑦∈𝐼,𝑧∈𝐽 such that 𝑦,𝑧∈𝑇(𝑥). So, 𝑦∘2𝑧∈𝐼∘2𝐽 and 𝑦∘2𝑧∈𝑇(𝑥)∘2𝑇(𝑥). Since T is a set-valued homomorphism and 𝑄1 is idempotent, we have 𝑦∘2𝑧∈𝑇(𝑥∘1𝑥)=𝑇(𝑥). Therefore, 𝑥∈𝑇−(𝐼∘2𝐽). So the equality in (1) holds. Since 𝑄1 is idempotent, we have 𝐼∩𝐽⊆𝐼∘2𝐽. So 𝐼∩𝐽=𝐼∘2𝐽=𝐼∩𝐽. By Theorem 2.9, we get 𝑇−(𝐼∘2𝐽)=𝑇−(𝐼∩𝐽)=𝑇−(𝐼)∩𝑇−(𝐽). Since the equality in (1) holds, we have 𝑇−(𝐼∧𝐽)=𝑇−(𝐼∘2𝐽)=𝑇−(𝐼)∩𝑇−(𝐽).

Lemma 4.2. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued homomorphism. If 𝐴 is a lower set of 𝑄2, then 𝑇−(𝐴) is a lower set of 𝑄1.

Proof. Suppose 𝑥≤𝑦∈𝑇−(𝐴), then 𝑇(𝑦)⊆𝐴. Let 𝑧∈𝑇(𝑥),ğ‘Žâˆˆğ‘‡(𝑦), we have ğ‘§âˆ¨ğ‘Žâˆˆğ‘‡(𝑥)∨𝑇(𝑦)⊆𝑇(𝑥∨𝑦)=𝑇(𝑦)⊆𝐴. Since 𝐴 is a lower set, we have 𝑧∈𝐴. Therefore, 𝑇(𝑥)⊆𝐴 which implies that 𝑥∈𝑇−(𝐴).

Lemma 4.3. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism. If 𝐴 is a lower set of 𝑄2, then 𝑇−(𝐴) is a lower set of 𝑄1.

Proof. Suppose that 𝑥≤𝑦∈𝑇−(𝐴), there exists 𝑧∈𝑇(𝑦)∩𝐴. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(𝑥)∨𝑇(𝑦)=𝑇(𝑥∨𝑦)=𝑇(𝑦). So there exist ğ‘Žâˆˆğ‘‡(𝑥),𝑏∈𝑇(𝑦) such that 𝑧=ğ‘Žâˆ¨ğ‘. Since 𝐴 is a lower set, we have ğ‘Žâˆˆğ´. Thus 𝑇(𝑥)∩𝐴≠∅ which follows that 𝑥∈𝑇−(𝐴).

Lemma 4.4. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued homomorphism and 𝐴 a nonempty subset of 𝑄2. If 𝐴 is closed under arbitrary (resp., finite) sups, then 𝑇−(𝐴) is closed under arbitrary (resp., finite) sups.

Proof. Let 𝐵⊆𝑇−(𝐴). For each 𝑏∈𝐵, we have 𝑏∈𝑇−(𝐴), then there exist 𝑥𝑏∈𝑇(𝑏)∩𝐴. Since 𝑇 is a set-valued homomorphism, we have ⋁𝑏∈𝐵𝑥𝑏∈⋁𝑏∈𝐵𝑇(𝑏)⊆𝑇(⋁𝐵). And we have ⋁𝑏∈𝐵𝑥𝑏∈𝐴 for A is closed under arbitrary sups. So 𝑇(⋁𝐵)∩𝐴≠∅ which implies that ⋁𝐵∈𝑇−(𝐴).

Lemma 4.5. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐴 a nonempty subset of 𝑄2. If 𝐴 is closed under arbitrary (resp., finite) sups, then 𝑇−(𝐴) is closed under arbitrary (resp., finite) sups.

Proof. Let 𝐵⊆𝑇−(𝐴). For each 𝑏∈𝐵, we have 𝑇(𝑏)⊆𝐴. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(⋁𝐵)=⋁𝑇(𝐵). Suppose 𝑧∈𝑇(⋁𝐵)=⋁𝑇(𝐵), there exist 𝑥𝑏∈𝑇(𝑏)⊆𝐴 (𝑏∈𝐵) such that 𝑧=⋁𝑏∈𝐵𝑥𝑏. Since 𝐴 is closed under arbitrary sups, we have 𝑧=⋁𝑏∈𝐵𝑥𝑏∈𝐴. So 𝑇(⋁𝐵)⊆𝐴 which implies that ⋁𝐵∈𝑇−(𝐴).

Theorem 4.6. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued homomorphism. If 𝐴 is a subquantale of 𝑄2, then 𝑇−(𝐴) is a subquantale of 𝑄1.

Proof. Since 𝑇 is a set-valued homomorphism and 𝐴 is a subquantale, by Theorems 3.4 and 2.9, we have 𝑇−(𝐴)∘1𝑇−(𝐴)⊆𝑇−(𝐴∘2𝐴)⊆𝑇−(𝐴). So, 𝑇−(𝐴) is closed under ∘1.
Since 𝐴 is closed under arbitrary sups, by Lemma 4.4, we get 𝑇−(𝐴) is closed under arbitrary sups.

Theorem 4.7. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism. If 𝐴 is a subquantale of 𝑄2, then 𝑇−(𝐴) is a subquantale of 𝑄1.

Proof. The proof is similar to Theorem 4.6.

Theorem 4.8. Let 𝑇∶𝑄1→𝑃∗(𝑄2) be a strong set-valued homomorphism and 𝐼 a right (left) ideal of 𝑄2. Then 𝑇−(𝐼) is, if it is nonempty, a right (left) ideal of 𝑄1.

Proof. Suppose that ğ‘Ž,𝑏∈𝑇−(𝐼). Since 𝐼 is closed under finite sups, by Lemma 4.4, we have 𝑇−(𝐼) is closed under finite sups. So ğ‘Žâˆ¨ğ‘âˆˆğ‘‡âˆ’(𝐼).
Since 𝐼 is a lower set, by Lemma 4.3, we have 𝑇−(𝐼) is a lower set.
Suppose ğ‘Žâˆˆğ‘„1,𝑥∈𝑇−(𝐼), there exists 𝑦∈𝐼∩𝑇(𝑥). Since 𝐼 is a right ideal of 𝑄2, we have 𝑦∘2𝑏∈𝐼 for each 𝑏∈𝑇(ğ‘Ž)⊆𝑄2. So 𝑦∘2𝑏∈𝑇(𝑥)∘2𝑇(ğ‘Ž)⊆𝑇(𝑥∘1ğ‘Ž). Therefore, 𝑇(𝑥∘1ğ‘Ž)∩𝐼≠∅ which implies that 𝑥∘1ğ‘Žâˆˆğ‘‡âˆ’(𝐼).

Theorem 4.9. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a right (left) ideal of 𝑄2. Then 𝑇−(𝐼) is, if it is nonempty, a right (left) ideal of 𝑄1.

Proof. Suppose ğ‘Ž,𝑏∈𝑇−(𝐼). Since 𝐼 is closed under finite sups, by Lemma 4.5, we have 𝑇−(𝐼) is closed under finite sups. So, ğ‘Žâˆ¨ğ‘âˆˆğ‘‡âˆ’(𝐼).
Since 𝐼 is a lower set, by Lemma 4.2, we have 𝑇−(𝐼) is a lower set.
Suppose ğ‘Žâˆˆğ‘„1,𝑥∈𝑇−(𝐼), we have 𝑇(𝑥)⊆𝐼. Let 𝑦∈𝑇(𝑥∘1ğ‘Ž). Since 𝑇 is a strong set-valued homomorphism, we have 𝑦∈𝑇(𝑥)∘2𝑇(ğ‘Ž), then there exist 𝑦1∈𝑇(𝑥)⊆𝐼, 𝑦2∈𝑇(ğ‘Ž) such that 𝑦=𝑦1∘2𝑦2. Since 𝐼 is a right ideal, we have 𝑦=𝑦1∘2𝑦2∈𝐼. Therefore, 𝑇(𝑥∘1ğ‘Ž)⊆𝐼 which implies that 𝑥∘1ğ‘Žâˆˆğ‘‡âˆ’(𝐼).

Definition 4.10. A subset 𝐴⊆𝑄2 is called a generalized rough ideal (subquantale) of 𝑄1 if 𝑇−(𝐴) and 𝑇−(𝐴) are ideals (subquantales) of 𝑄1.
The following corollary follows from Theorems 4.6–4.9.

Corollary 4.11. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 an ideal (a subquantale) of 𝑄2. If 𝑇−(𝐼) and 𝑇−(𝐼) are nonempty, then 𝐼 is a generalized rough ideal (subquantale) of 𝑄1.

From the above, we know that an ideal is a generalized rough ideal with respect to a strong set-valued homomorphism. The following example shows that the converse does not hold in general.

Example 4.12. Let 𝑄1={0,ğ‘Ž,1} and 𝑄2={0,𝑏,𝑐,1} be quantales shown in Figures 1 and 2 and Tables 1 and 2.
Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism as defined by 𝑇(0)={0′},𝑇(ğ‘Ž)={𝑏,𝑐},𝑇(1)={1′}. Let 𝐴={0′,𝑏,𝑐}⊆𝑄2, 𝐵={𝑏,𝑐}⊆𝑄2, then 𝑇−(𝐴)={0,ğ‘Ž}=𝑇−(𝐴) and 𝑇−(𝐵)={ğ‘Ž}=𝑇−(𝐵). It is obvious that 𝐴 is a generalized rough ideal of 𝑄1 but 𝐴 is not an ideal of 𝑄2 and 𝐵 is a generalized rough subquantale of 𝑄1 but 𝐵 is not a subquantale of 𝑄2.


∘ 1 0 ğ‘Ž 1

00 ğ‘Ž 1
ğ‘Ž 0 ğ‘Ž 1
10 ğ‘Ž 1


∘ 2 0′ 𝑏 𝑐 1′

0′ 0′ 𝑏 𝑐 1′
𝑏 0′ 𝑏 𝑐 1′
𝑐 0′ 𝑏 𝑐 1′
1′ 0′ 𝑏 𝑐 1′

Theorem 4.13. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a prime ideal of 𝑄2. Then 𝑇−(𝐼) is, if it is nonempty, a prime ideal of 𝑄1.

Proof. By Theorem 4.8, we get 𝑇−(𝐼) is an ideal of 𝑄1.
Let ğ‘Žâˆ˜1𝑏∈𝑇−(𝐼), there exists 𝑥∈𝐼∩𝑇(ğ‘Žâˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have 𝑥∈𝑇(ğ‘Ž)∘2𝑇(𝑏), then there exist 𝑦∈𝑇(ğ‘Ž),𝑧∈𝑇(𝑏) such that 𝑥=𝑦∘2𝑧∈𝐼. Since 𝐼 is a prime ideal of 𝑄2, we have 𝑦∈𝐼 or 𝑧∈𝐼. So 𝑇(ğ‘Ž)∩𝐼≠∅ or 𝑇(𝑏)∩𝐼≠∅ which implies that ğ‘Žâˆˆğ‘‡âˆ’(𝐼) or 𝑏∈𝑇−(𝐼).

Theorem 4.14. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a prime ideal of 𝑄2. Then 𝑇−(𝐼) is, if it is nonempty, a prime ideal of 𝑄1.

Proof. By Theorem 4.9, we get 𝑇−(𝐼) is an ideal of 𝑄1.
Let ğ‘Žâˆ˜1𝑏∈𝑇−(𝐼), we have 𝑇(ğ‘Žâˆ˜1𝑏)⊆𝐼. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(ğ‘Ž)∘2𝑇(𝑏)⊆𝐼. We assume that ğ‘Žâˆ‰ğ‘‡âˆ’(𝐼), then 𝑇(ğ‘Ž)⊆𝐼, there exists 𝑥∈𝑇(ğ‘Ž) but 𝑥∉𝐼. If 𝑦∈𝑇(𝑏), then 𝑥∘2𝑦∈𝑇(ğ‘Ž)∘2𝑇(𝑏)⊆𝐼. Since 𝐼 is a prime ideal of 𝑄2, we have 𝑦∈𝐼. Therefore, 𝑇(𝑏)⊆𝐼 which implies that 𝑏∈𝑇−(𝐼).

We call 𝐼⊆𝑄2 is a generalized rough prime ideal of 𝑄1 if 𝑇−(𝐼) and 𝑇−(𝐼) are ideals of 𝑄1.

Theorem 4.15. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a semiprime ideal of 𝑄2. Then 𝑇−(𝐼) is, if it is nonempty, a semiprime ideal of 𝑄1.

Proof. By Theorem 4.9, we get 𝑇−(𝐼) is an ideal of 𝑄1.
Suppose that ğ‘Žâˆ˜1ğ‘Žâˆˆğ‘‡âˆ’(𝐼), we have 𝑇(ğ‘Žâˆ˜1ğ‘Ž)⊆𝐼. Let 𝑥∈𝑇(ğ‘Ž), we have 𝑥∘2𝑥∈𝑇(ğ‘Ž)∘2𝑇(ğ‘Ž)⊆𝑇(ğ‘Žâˆ˜1ğ‘Ž)⊆𝐼. Since 𝐼 is a semi-prime ideal, we have 𝑥∈𝐼. So 𝑇(ğ‘Ž)⊆𝐼 which implies that ğ‘Žâˆˆğ‘‡âˆ’(𝐼).

Theorem 4.16. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a primary ideal of 𝑄2. Then 𝑇−(𝐼) is, if 𝑇−(𝐼)≠∅ and 𝑇−(𝐼)≠𝑄1, a primary ideal of 𝑄1.

Proof. By Theorem 4.8, we get 𝑇−(𝐼) is an ideal of 𝑄1.
Suppose that ğ‘Ž,𝑏∈𝑄,ğ‘Žâˆ˜1𝑏∈𝑇−(𝐼) and ğ‘Žâˆ‰ğ‘‡âˆ’(𝐼), there exists 𝑥∈𝐼∩𝑇(ğ‘Žâˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have 𝑥∈𝑇(ğ‘Ž)∘2𝑇(𝑏), there exist 𝑦∈𝑇(ğ‘Ž), 𝑧∈𝑇(𝑏) such that 𝑥=𝑦∘2𝑧∈𝐼. Since ğ‘Žâˆ‰ğ‘‡âˆ’(𝐼), we get 𝑦∉𝐼. Since 𝐼 is a primary ideal, we have 𝑧𝑛∈𝐼 for some 𝑛>0 Since 𝑇 is a strong set-valued homomorphism, we have 𝑧𝑛∈𝑇(𝑏𝑛). So 𝑇(𝑏𝑛)∩𝐼≠∅ which implies that 𝑏𝑛∈𝑇−(𝐼).

Theorem 4.17. Let 𝑇∶𝑄1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a primary ideal of 𝑄2. Let 𝑄2 be a commutative quantale and 𝑇(𝑥) a finite set for each 𝑥∈𝑄1. Then 𝑇−(𝐼) is, if 𝑇−(𝐼)≠∅ and 𝑇−(𝐼)≠𝑄1, a primary ideal of 𝑄1.

Proof. By Theorem 4.9, we get 𝑇−(𝐼) is an ideal of 𝑄1.
Suppose ğ‘Ž,𝑏∈𝑄,ğ‘Žâˆ˜1𝑏∈𝑇−(𝐼) and ğ‘Žâˆ‰ğ‘‡âˆ’(𝐼), then 𝑇(ğ‘Žâˆ˜1𝑏)⊆𝐼 and 𝑇(ğ‘Ž)⊆𝐼. So there exists 𝑥∈𝑇(ğ‘Ž) with 𝑥∉𝐼. We assume that 𝑇(𝑏)={𝑦1,𝑦2,…,𝑦𝑚}, then 𝑥∘2𝑦𝑖∈𝑇(ğ‘Ž)∘2𝑇(𝑏)⊆𝑇(ğ‘Žâˆ˜1𝑏)⊆𝐼(𝑖=1,2,…,𝑚). Since 𝐼 is a primary ideal, there exists 𝑦𝑛𝑖𝑖∈𝐼 for some 𝑛𝑖>0(𝑖=1,2,…,𝑚). Let 𝑛=max{𝑛𝑖∣𝑖=1,2,…,𝑚}. Since 𝐼 is an ideal, we have 𝑦𝑙𝑖∈𝐼(𝑙≥𝑛). Let 𝑧∈𝑇(𝑏𝑚𝑏𝑛)=𝑇(𝑏)𝑚𝑛. Since 𝑄2 is commutative, there exists {𝑖1,𝑖2,…,𝑖𝑠}⊆{1,2,…,𝑚} such that 𝑧=𝑦𝑡1𝑖1∘2𝑦𝑡2𝑖2∘2⋯∘2𝑦𝑡𝑠𝑖𝑠 with 𝑡1+𝑡2+⋯+𝑡𝑠=𝑚𝑛, where 𝑡𝑗>0(𝑗=1,2,…,𝑠). Assume that 𝑡𝑗<𝑛,(𝑗=1,2,…,𝑠), then 𝑡1+𝑡2+⋯+𝑡𝑠<𝑠𝑛≤𝑚𝑛. It contradicts with 𝑡1+𝑡2+⋯+𝑡𝑠=𝑚𝑛. Therefore, there is 1≤𝑘≤𝑠 such that 𝑡𝑘≥𝑛,(𝑗=1,2,…,𝑠), we have 𝑦𝑡𝑘𝑖𝑘∈𝐼. Since 𝐼 is an ideal, we get 𝑧∈𝐼. Hence 𝑇(𝑏𝑚𝑛)⊆𝐼 which implies that 𝑏𝑚𝑛∈𝑇−(𝐼).

Theorem 4.18. Let 𝑇∶𝑄1→𝑃(𝑄2) be a set-valued homomorphism. If 𝑆 is a multiplicative set of 𝑄2, then 𝑇−(𝑆) is, if it is nonempty, a multiplicative of 𝑄1. If 𝑇 is a strong set-valued homomorphism, then 𝑇−(𝑆) is, if it is nonempty, a multiplicative of 𝑄1.

Proof. Suppose that ğ‘Ž,𝑏∈𝑇−(𝑆), there exist 𝑥∈𝑇(ğ‘Ž)∩𝑆, 𝑦∈𝑇(𝑏)∩𝑆. Hence 𝑥∘𝑦∈𝑇(ğ‘Ž)∘2𝑇(𝑏)⊆𝑇(ğ‘Žâˆ˜1𝑏). Since 𝑆 is a multiplicative set, we have 𝑥∘2𝑦∈𝑆. Therefore, 𝑥∘2𝑦∈𝑇(ğ‘Žâˆ˜1𝑏)∩𝑆 which implies that ğ‘Žâˆ˜1𝑏∈𝑇−(𝑆).
Suppose ğ‘Ž,𝑏∈𝑇−(𝑆), then 𝑇(ğ‘Ž)⊆𝑆,𝑇(𝑏)⊆𝑆. Let 𝑥∈𝑇(ğ‘Žâˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have 𝑥∈𝑇(ğ‘Ž)∘2𝑇(𝑏). There exist 𝑦∈𝑇(ğ‘Ž)⊆𝑆,𝑧∈𝑇(𝑏)⊆𝑆 such that 𝑥=𝑦∘2𝑧. Since 𝑆 is a multiplicative set, we have 𝑥=𝑦∘2𝑧∈𝑆. So 𝑇(ğ‘Žâˆ˜1𝑏)⊆𝑆 which implies that ğ‘Žâˆ˜1𝑏∈𝑇−(𝑆).

We call 𝐴⊆𝑄2 a generalized rough multiplicative set (𝑚-system) of 𝑄1, if 𝑇−(𝐴),𝑇−(𝐴) are multiplicative sets (𝑚-systems) of 𝑄1.

Theorem 4.19. Let 𝑇∶𝑄1→𝑃∗(𝑄2) be a set-valued homomorphism. If 𝑄1 is commutative and 𝐴⊆𝑄2 with 1∉𝑇−(𝐴), then the following statements are equivalent:(1)𝐴 is a generalized rough prime ideal of 𝑄1,(2)𝐴 is a generalized rough ideal and 𝐴𝑐 is a generalized rough multiplicative set of 𝑄1,(3)𝐴 is a generalized rough ideal and 𝐴𝑐 is a generalized rough 𝑚-system of 𝑄1.

Proof. (1)⇒(2): Since 𝐴 is a generalized rough prime ideal of 𝑄1, we get 𝐴 is a generalized rough ideal of 𝑄1 and 𝑇−(𝐴), 𝑇−(𝐴) are prime ideals of 𝑄1. Now we show that 𝑇−(𝐴𝑐) is a multiplicative set. Let ğ‘Ž, 𝑏∈𝑇−(𝐴𝑐). By Theorem 2.9(1), we have ğ‘Ž,𝑏∈(𝑇−(𝐴))𝑐. Since 𝑇−(𝐴) is a prime ideal, we have ğ‘Žâˆ˜1𝑏∉𝑇−(𝐴) which implies that ğ‘Žâˆ˜1𝑏∈(𝑇−(𝐴))𝑐=𝑇−(𝐴𝑐). So 𝑇−(𝐴𝑐) is a multiplicative set. Similarly, 𝑇−(𝐴𝑐) is a multiplicative set.
(2)⇒(3): Let ğ‘Ž, 𝑏∈𝑇−(𝐴𝑐). Since 1∉𝑇−(𝐴) and 𝑇(1)≠∅, we have 1∈𝑇−(𝐴𝑐). Since 𝑇−(𝐴𝑐) is a multiplicative set, we have ğ‘Žâˆ˜11∘1𝑏∈𝑇−(𝐴𝑐). So ↓(ğ‘Žâˆ˜11∘1𝑏)∩𝑇−(𝐴𝑐)≠∅. Therefore, 𝑇−(𝐴𝑐) is an 𝑚-system. Similarly, we have 𝑇−(𝐴𝑐) is a 𝑚-system.
(3)⇒(1): Let ğ‘Žâˆ˜1𝑏∈𝑇−(𝐴). Suppose ğ‘Žâˆ‰ğ‘‡âˆ’(𝐴) and 𝑏∉𝑇−(𝐴), then ğ‘Ž, 𝑏∈(𝑇−(𝐴))𝑐=𝑇−(𝐴𝑐). Since 𝑇−(𝐴𝑐) is an 𝑚-system, we have ↓(ğ‘Žâˆ˜11∘1𝑏)∩𝑇−(𝐴𝑐)≠∅. Thus ↓(ğ‘Žâˆ˜11∘1𝑏)∩(𝑇−(𝐴))𝑐≠∅. Since 𝑇−(𝐴) is an ideal and 𝑄1 is commutative, we have ğ‘Žâˆ˜11∘1𝑏∈𝑇−(𝐴) and ↓(ğ‘Žâˆ˜11∘1𝑏)⊆𝑇−(𝐴). It contradicts with ↓(ğ‘Žâˆ˜11∘1𝑏)∩(𝑇−(𝐴))𝑐≠∅. So, ğ‘Žâˆˆğ‘‡âˆ’(𝐴) or 𝑏∈𝑇−(𝐴). Therefore, 𝑇−(𝐴) is a prime ideal. Similarly, we have 𝑇−(𝐴) as a prime ideal.

5. Conclusion

The Pawlak rough sets on the algebraic sets such as semigroups, groups, rings, modules, and lattices were mainly studied by a congruence relation. However, the generalized Pawlak rough set was defined for two universes and proposed on generalized binary relations. Can we extended congruence relations to two universes for algebraic sets? Therefore, Davvaz [25] introduced the concept of set-valued homomorphism for groups which is a generalization of the concept of congruence. In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism of quantales are introduced. We construct generalized lower and upper approximations by means of a set-valued mapping and discuss the properties of them. We obtain that the concept of generalized rough ideal (subquantale) is the extended notion of ideal (subquantale). Many results in [29] are the special case of the results in this paper. It is an interesting research topic of rough set, we will further study it in the future.

Acknowledgments

The authors are grateful to the reviewers for their valuable suggestions to improve the paper. This work was supported by the National Science Foundation of China (no. 11071061).

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Copyright © 2012 Qimei Xiao and Qingguo Li. 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.


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