International Journal of Mathematics and Mathematical Sciences

Volume 2010, Article ID 303412, 9 pages

http://dx.doi.org/10.1155/2010/303412

## -Subalgebras in BCK/BCI-Algebras Based on Point -Structures

^{1}Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea^{2}Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea^{3}Department of Mathematics, University of Ulsan, Ulsan 680-749, Republic of Korea

Received 16 March 2010; Revised 10 August 2010; Accepted 2 September 2010

Academic Editor: Andrzej Skowron

Copyright © 2010 Young Bae Jun et al. 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 notion of -subalgebras of several types is introduced, and related properties are investigated. Conditions for an -structure to be an -subalgebra of type are provided, and a characterization of an -subalgebra of type is considered.

#### 1. Introduction

A (crisp) set in a universe can be defined in the form of its characteristic function yielding the value for elements belonging to the set and the value for elements excluded from the set So far, most of the generalization of the crisp set have been conducted on the unit interval and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point into the interval Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [1] introduced a new function which is called negative-valued function, and constructed -structures. They applied -structures to BCK/BCI-algebras, and discussed -subalgebras and -ideals in BCK/BCI-algebras. Jun et al. [2] considered closed ideals in BCH-algebras based on -structures. To obtain more general form of an -subalgebra in BCK/BCI-algebras, we define the notions of -subalgebras of types , and and investigate related properties. We provide a characterization of an -subalgebra of type We give conditions for an -structure to be an -subalgebra of type

#### 2. Preliminaries

Let be the class of all algebras with type . By a * BCI-algebra* we mean a system in which the following axioms hold: (i)(ii)(iii)(iv)

for all If a BCI-algebra satisfies for all then we say that is a * BCK-algebra*. We can define a partial ordering by
In a BCK/BCI-algebra , the following hold:

for all

A nonempty subset of a BCK/BCI-algebra is called a * subalgebra* of if for all For our convenience, the empty set is regarded as a subalgebra of

We refer the reader to the books [3, 4] for further information regarding BCK/BCI-algebras.

For any family of real numbers, we define

Denote by the collection of functions from a set to We say that an element of is a * negative-valued function* from to (briefly, *-function* on ). By an *-structure* we mean an ordered pair of and an -function on In what follows, let denote a BCK/BCI-algebra and an -function on unless otherwise specified.

*Definition 2.1 (see [1]). *By a * subalgebra* of based on -function (briefly, *-subalgebra* of ), we mean an -structure in which satisfies the following assertion:

For any -structure and the set
is called a * closed **-support* of and the set
is called an * open **-support* of

Using the similar method to the transfer principle in fuzzy theory (see [5, 6]), Jun et al. [2] considered transfer principle in -structures as follows.

Theorem 2.2 (-transfer principle [2]). *An -structure satisfies the property if and only if for all *

Lemma 2.3 (see [1]). *An -structure is an -subalgebra of if and only if every open -support of is a subalgebra of for all *

#### 3. Generalized -Subalgebras

Let be an -structure in which is given by
where In this case, is denoted by and we call a * point **-structure*. For any -structure we say that a point -structure is an *-subset* (resp., *-subset*) of if (resp., ). If a point -structure is an -subset of or an -subset of we say is an *-subset* of

Theorem 3.1. *For any -structure the following are equivalent: *(1)* is an -subalgebra of *(2)*for any and if two point -structures and are -subsets of then the point -structure is an -subset of *

*Proof. *(1) (2). Let and be such that and are -subsets of Then and It follows from (2.3) that
so that the point -structure is an -subset of

(2) (1). For any note that and are point -structures which are -subsets of Using (2), we know that the point -structure is an -subset of Thus and so is an -subalgebra of

*Definition 3.2. *An -structure is called an *-subalgebra* of type (i) (resp., and ) if whenever two point -structures and are -subsets of then the point -structure is an -subset (resp., -subset and -subset) of (ii) (resp., and ) if whenever two point -structures and are -subsets of then the point -structure is an -subset (resp., -subset and -subset) of

Note that every -subalgebra of type is an -subalgebra of (see Theorem 3.1). Note also that every -subalgebra of types and is an -subalgebra of type

*Example 3.3. *Let be a set with a -operation table which is given by Table 1. Then is a BCK-algebra (see [4]). Consider an -structure in which is defined by
It is routine to verify that is an -subalgebra of types and But it is not of type

*Example 3.4. *Let be a BCI-algebra with a -operation table which is given by Table 2. Consider an -structure in which is defined by
Then is an -subalgebra of type But(1) is not of type since two point -structures and are -subsets of but the point -structure
is not an -subset of since (2) is not of type since two point -structures and are -subsets of but the point -structure
is not an -subset of (3)is not of type since two point -structures and are -subsets of but the point -structure
is not an -subset of

*Example 3.5. *Let be a set with a -operation table which is given by Table 3. Then is a BCK-algebra (see [4]). Consider an -structure in which is defined by
Then is an -subalgebra of type

Theorem 3.6. *If is an -subalgebra of type then the open -support of is a subalgebra of *

*Proof. *Let be an -subalgebra of type If is zero, that is, for all then which is a subalgebra of Assume that is nonzero and let Then and Suppose that Note that and are point -structures which are -subsets of But the point -structure is not an -subset of because This is a contradiction, and so that is, Hence is a subalgebra of

Theorem 3.7. *If is an -subalgebra of type then the open -support of is a subalgebra of *

*Proof. *Let Then and If then
Thus the point -structure is not an -subset of which is impossible since and are point -structures which are -subsets of Therefore, that is, This shows that the open -support of is a subalgebra of

Theorem 3.8. *If is an -subalgebra of type then the open -support of is a subalgebra of *

*Proof. *Let Then and which imply that and are point -structures which are -subsets of If then the point -structure is not an -subset of a contradiction. Therefore, that is, and so the open -support of is a subalgebra of

Theorem 3.9. *If is an -subalgebra of type then is constant on the open -support of *

*Proof. *Assume that is not constant on the open -support of Then there exists such that Then either or Suppose that and choose such that Then and and so and are point -structures which are -subsets of Since
the point -structure is not an -subset of which is a contradiction. Next assume that Then and so is an -subset of Note that
and thus is not an -subset of This is impossible, and therefore is constant on the open -support of

Theorem 3.10. *An -structure is an -subalgebra of type if and only if it satisfies
*

*Proof. *Suppose that is an -subalgebra of type For any assume that If for some then there exists such that Thus, point -structures and are -subsets of but the point -structure is not an -subset of a contradiction. Hence whenever for all Now suppose that Then point -structures and are -subsets of which imply that the point -structure is an -subset of Hence Otherwise, that is, is not an -subset of This is a contradiction. Consequently, for all

Conversely, assume that (3.12) is valid. Let and be such that two point -structures and are -subsets of If then is an -subset of Suppose that Then Otherwise, we have
a contradiction. It follows that
and so is an -subset of Consequently, is an -subset of and thus is an -subalgebra of type

We provide conditions for an -structure to be an -subalgebra of type

Theorem 3.11. *Let be a subalgebra of and let be an -structure such that *(1)(2)*Then is an -subalgebra of type *

*Proof. *Let and be such that two point -structures and are -subsets of Then and Thus because if it is impossible, then or Thus or and so or This is a contradiction. Hence If then and thus the point -structure is an -subset of If then and so the point -structure is an -subset of Therefore, the point -structure is an -subset of This completes the proof.

Theorem 3.12. *Let be an -subalgebra of type If is not constant on the open -support of then for some In particular, *

*Proof. *Assume that for all Since is not constant on the open -support of there exists such that Then either or For the case choose such that Then the point -structure is an -subset of Since is an -subset of It follows from (a1) that the point -structure is an -subset of But, implies that the point -structure is not an -subset of Also, implies that the point -structure is not an -subset of This is a contradiction. Now, if then we can take such that Then is an -subset of and induces that is not an -subset of Since
is not an -subset of Hence is not an -subset of which is a contradiction. Therefore for some We now prove that Assume that Note that there exists such that and so Choose such that Then and thus the point -structure is an -subset of Now we have
and Hence is not an -subset of a contradiction. Therefore

Corollary 3.13. *If is an -subalgebra of types or in which is not constant on the open -support of then for some In particular, *

Theorem 3.14. *Let be a BCK-algebra and let be an -subalgebra of type such that is not constant on the open -support of If
**
then for all *

*Proof. *Assume that for all Since is not constant on the open -support of there exists such that Then Choose such that Then is an -subset of Note that the point -structure is an -subset of It follows that is an -subset of But induces that is not an -subset of and induces that is not an -subset of This is a contradiction, and so for some Now, if possible, let Then there exists such that Thus Take such that Then two point -structures and are -subsets of but is not an -subset of a contradiction. Hence Finally let for some Taking such that then two point -structures and are -subsets of But
implies that the point -structure is not an -subset of Hence the point -structure is not an -subset of a contradiction. Therefore for all

#### Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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