The Homomorphisms and Operations of Rough Groups

Abstract

Based on the light relation between a normal subgroup and a complete congruence relation of a group, we consider the homomorphism problem of rough groups and rough quotient groups and investigate their operational properties. Some new results are obtained.

1. Introduction

Rough set theory, proposed by Pawlak [1], is an extension of set theory for study of information systems characterized by inexact and uncertain information. It has been demonstrated to be useful in fields such as knowledge discovery, data mining, decision analysis, pattern recognition, and algebra.

Rough set theory includes three basic elements: the universe set, the binary relations, and a subset described by a pair of ordinary sets. In the past few years, most studies have been focusing on the binary relations and the subsets; many interesting and constructive extensions to binary relations and the subsets have been proposed [1–8]. But the researchers have paid little attention to another basic element: the universe set. In real world, some universe has been given operations, such as the set of natural numbers and the set of real numbers. So, it is very natural to ask what would happen if we substitute an algebraic structure for the universe set. Biswas and Nanda [9] generalized the universe of rough sets to groups and introduced the notion of rough subgroups and some new properties of rough approximations have been obtained. Jiang et al. [10] investigated the product structure of fuzzy rough sets on a group and provided some new algebraic structures. Yin et al. [11] studied fuzzy roughness of -ary hypergroups based on a complete residuated lattice. Xiao and Zhang [12] studied the rough sets on a semigroup and proposed two new algebraic structures—rough prime ideals and rough fuzzy prime ideals. In [6], a new algebraic definition for pseudo-Cayley graphs containing Cayley graphs has been proposed, a rough approximation was expanded to pseudo-Cayley graphs, and some new properties have been obtained. For more other papers on this line please refer to [12–24], which have greatly enriched the theoretical research of rough sets.

The aim of this paper is to investigate the homomorphism problem of rough group and rough quotient groups. The rest of the paper is organized as follows. In Section 2, we recall some basic notions and results which will be used throughout the paper. In Section 3, the homomorphism problems of rough groups and rough quotient groups are studied and some related properties are discussed. In Section 4, congruence relation and the operation of rough groups are investigated.

2. Preliminaries

Let [25] be a group with unit element and let be an equivalence relation on . If we have , then is called a congruence relation and indicates congruence class of about . Furthermore, if , then is called complete.

Lemma 1 (see [1]). Let be a congruence relation of . Then, is a normal subgroup of and ; contrarily, if is a normal subgroup of , one can define a congruence relation , where ; then is a congruence relation of , and .

Lemma 1 indicates the congruence relation of and the normal subgroup of is one to one correspondence.

Theorem 2. Let be a normal subgroup of ; then is a complete congruence relation.

Proof. Consider . Therefore, is a complete congruence relation.

According to Theorem 2, we know the complete congruence relation of and the normal subgroup of is also one to one correspondence.

Definition 3 (see [1]). Let be an equivalence relation on and a nonempty subset of . Then, the sets are called, respectively, the -lower and -upper approximations of the set . And is called a rough set with respect to .

Definition 4 (see [1]). Let be an equivalence relation on and a nonempty subset of . Then are called, respectively, the lower and upper rough quotient of .

Lemma 5 (see [1]). Let be an equivalence relation on and, for all , one has the following.(1). (2); . (3); . (4)If , then , .

Lemma 6. Let be an equivalence relation on and one has the following.(1). (2); . (3); . (4)If , then , .

3. The Homomorphism Problem of Rough Groups and Rough Quotient Groups

Definition 7. Let be a complete congruence relation on and ; if is a subgroup (a normal subgroup) of , then is called the lower rough group (lower rough normal subgroup) of and is called the lower rough quotient group (lower rough normal quotient group); if is a subgroup (a normal subgroup) of , then is called the upper rough group (upper rough normal subgroup) of and is called the upper rough quotient group (upper rough normal quotient group); if are all the subgroup (normal subgroup) of , then is called rough group (rough normal subgroup) of .

Theorem 8. Let be a complete congruence relation on , , and . If is a subgroup (normal subgroup) of , then is rough group (rough normal subgroup) of and are, respectively, lower rough quotient group (lower rough normal quotient group) and upper rough quotient group (upper rough normal quotient group).

Corollary 9. Let be a complete congruence relation on , , and . Then, is a subgroup (normal subgroup) of is a subgroup (normal subgroup) of .

Corollary 10. Let be a complete congruence relation on , , and . If is a subgroup (normal subgroup) of , then(1);  , (2);  .

Corollary 11. Let be a complete congruence relation on and let be a subgroup of and . Then,

Proof. Based on the third isomorphism theorem of group, it is easy to prove this corollary.

Lemma 12. Let and be two groups; is a homomorphism. If is a congruence relation on and , then is a congruence relation on . Further, if is an injective, then (1);  (2), where .

Proof. It is easy to prove is a congruence relation on . (1)Because is a normal subgroup on , then is the normal subgroup on . So , so .(2)It follows immediately from (1).

Theorem 13. Let be an injective homomorphism. If is a complete congruence relation on and , then is a complete congruence relation on .

Proof. According to Lemma 12, it is easy to prove is a congruence relation on . Now we prove it is complete; , we have = = = = = .
Therefore, is complete.

Lemma 14. Let be a surjective homomorphism. If is a congruence relation on , then is a congruence relation on . Further, if is an injective, then(1); (2), where .

Proof. It is easy to prove is a congruence relation on . (1)Because is a normal subgroup on , then is the normal subgroup on . So , and because is injective, then ; that is, . On the contrary ; that is, ; hence, .(2)It follows immediately from (1).

Theorem 15. Let be an injective homomorphism. If is a complete congruence relation on , then is a complete congruence relation on .

Proof. According to Lemma 14, it is easy to prove is a congruence relation on . Now we prove it is complete; , we have .
Therefore, is complete.

Lemma 16. Let be a surjective homomorphism. If is a congruence relation on , , and , then(1); (2)if is injective, then .

Proof. (1) Consider , , , , , , , that is, ; on the contrary, , , , , , , , , , , that is, , so .
(2) Consider , = = = = , so = .

Theorem 17. Let be a surjective homomorphism, let be a congruence relation on , and let be a subgroup of and . Then,(1); (2)if is injective, then .

Proof. (1) Suppose that is a subgroup of ; then is a subgroup on and is a subgroup on ; according to Lemma 16  ; therefore, .
(2) If is injective, according to Lemma 16, we have ; therefore, ≅.

Corollary 18. Let be a surjective homomorphism, let be a complete congruence relation of based on , and let be a subgroup of and . Then,(1); (2)if is injective, then .

Lemma 19. Let be a surjective homomorphism and let be a congruence relation on and . Then(1); (2)if is an injective, then .

Proof. (1) Consider , , , , and, hence, .
(2) Consider = , so .

Theorem 20. Let be a surjective homomorphism and let be a congruence relation on and is a subgroup of . Then,(1); (2)if is an injective, then .

Proof. By Lemma 19 and the first isomorphism theorem of group, we have ;  .

4. Congruence Relation and the Operation of Rough Group

Lemma 21. Let be the congruence relations on . Then .

Proof. Consider , , , and, therefore, .

Lemma 22. Let be two congruence relations on and . Then .

Proof. It is easy to prove that . On the contrary, , ; that is, ; therefore, .

Lemma 23. Let be two congruence relations on and . Then .

Proof. Consider ; hence .

Theorem 24. Let be two complete congruence relations on . Then is a complete congruence relation on .

Proof. Because are both the normal subgroups of , then is a normal subgroup of , so is a complete congruence relation.

Theorem 25. Let be two congruence relations on and . Then(1); (2).

Proof. (1) Consider . Because , then , , ; therefore, .
(2) Consider or or , and, therefore .

Lemma 26 (see [25]). Let be two congruence relations on . Then is a congruence relation on .

Theorem 27. Let be two complete congruence relations on and . Then is a complete congruence relation on .

Proof. By Lemma 26, is the congruence relation and ; hence, is the complete congruence relation.

Lemma 28. Let be two congruence relations on . Then .

Proof. Consider , , ; that is, . On the contrary, , , , , ; that is, ; hence, .

Lemma 29. Let be two congruence relations on and . Then .

Proof. Consider .

Theorem 30. Let be two congruence relations on and . If is a subgroup of and . Then(1); (2).

Proof. (1) Consider , , , , , , , , , so .
(2) Consider , because , and , ; that is, . On the contrary, , , , , = = ; that is, , so .

Theorem 31. Let be complete congruence relation on and . Then(1); (2).

Proof. (1) Consider , , , , and , , , because , , , = ; that is, . On the contrary , , , , , , , ; that is, . Therefore .
(2) , if , then , Contradiction, that is, , , so, , that is, . On the contrary, , , , , , that is, , so, .

Corollary 32. Let be a complete congruence relation on and . Then(1); (2).

Corollary 33. Let be a complete congruence relation on and let be a subgroup of and . Then(1); (2).

Proof. Because is a subgroup, then are both the subgroups of , so , , , and is an equivalence relation; then ; hence, we can get (1) and (2).

Corollary 34. Let be a complete congruence relation on and let be two subgroups of . Then(1); (2).

Proof. (1) Because , and , so ; hence, and ; therefore, (1) holds. (2) Because , so .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Fundamental Research Funds for the Central Universities (no. YX2014-08) and the NSFC of China (Grant no. 61370193).