FUZZY BCI-SUBALGEBRAS WITH INTERVAL-VALUED MEMBERSHIP FUNCTIONS

The purpose of this paper is to define the notion of an interval-valued fuzzy BCI-subalgebra (briefly, an i-v fuzzy BCI-subalgebra) of a BCI-algebra. Necessary and sufficient conditions for an i-v fuzzy set to be an i-v fuzzy BCI-subalgebra are stated. A way to make a new i-v fuzzy BCI-subalgebra from old one is given. The images and inverse images of i-v fuzzy BCI-subalgebras are defined, and how the images or inverse images of i-v fuzzy BCI-subalgebras become i-v fuzzy BCI-subalgebras is studied. 2000 Mathematics Subject Classification. Primary 06F35, 03B52.


Introduction.
The notion of BCK-algebras was proposed by Iami and Iséki in 1966.In the same year, Iséki [2] introduced the notion of a BCI-algebra which is a generalization of a BCK-algebra.Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK/BCI-algebras and their relationship with other universal structures including lattices and Boolean algebras.Fuzzy sets were initiated by Zadeh [3].In [4], Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set (i.e., a fuzzy set with an interval-valued membership function).This interval-valued fuzzy set is referred to as an i-v fuzzy set.In [4], Zadeh also constructed a method of approximate inference using his i-v fuzzy sets.In [1], Biswas defined interval-valued fuzzy subgroups (i.e., i-v fuzzy subgroups) of Rosenfeld's nature, and investigated some elementary properties.In this paper, using the notion of interval-valued fuzzy set by Zadeh, we introduce the concept of an interval-valued fuzzy BCI-subalgebra (briefly, i-v fuzzy BCI-subalgebra) of a BCI-algebra, and study some of their properties.Using an i-v level set of an i-v fuzzy set, we state a characterization of an i-v fuzzy BCI-subalgebra.We prove that every BCI-subalgebra of a BCI-algebra X can be realized as an i-v level BCI-subalgebra of an i-v fuzzy BCI-subalgebra of X.In connection with the notion of homomorphism, we study how the images and inverse images of i-v fuzzy BCI-subalgebras become i-v fuzzy BCI-subalgebras.

Preliminaries.
In this section, we include some elementary aspects that are necessary for this paper.
Note that the equality 0 We now review some fuzzy logic concepts.Let X be a set.A fuzzy set in X is a function µ : X → [0, 1].Let f be a mapping from a set X into a set Y .Let ν be a fuzzy set in Y .Then the inverse image of ν, denoted by for all y ∈ Y , where An interval-valued fuzzy set (briefly, i-v fuzzy set) A defined on X is given by where µ L A and µ U A are two fuzzy sets in X such that µ L A (x) ≤ µ U A (x) for all x ∈ X. (2.3) Now let us define what is known as refined minimum (briefly, rmin) of two elements in D[0, 1].We also define the symbols "≥", "≤", and "=" in case of two elements in (2.4) and similarly we may have Definition 2.1.A fuzzy set µ in a BCI-algebra X is called a fuzzy BCI-subalgebra of X if µ(x * y) ≥ min µ(x), µ(y) for all x, y ∈ X.

Interval-valued fuzzy BCI-subalgebras.
In what follows, let X denote a BCIalgebra unless otherwise specified.We begin with the following two propositions.
Proof.For any x, y ∈ X, we have Proof.We first prove that Then we should consider the two cases: For the case (3.4), we have f [µ] y 1 = 0 or f [µ] y 2 = 0, and so Since µ is a fuzzy BCI-subalgebra of X, it follows from the definition of a fuzzy BCIsubalgebra that ) for all y 1 ,y 2 ∈ Y .This completes the proof.
∀x, y ∈ X. (3.9) Example 3.4.Let X = 0,a,b,c be a BCI-algebra with the following Cayley table: let an i-v fuzzy set A defined on X be given by μA It is easy to check that A is an i-v fuzzy BCI-subalgebra of X.
Proof.For every x ∈ X, we have (3.11) this completes the proof.
Theorem 3.6.Let A be an i-v fuzzy BCI-subalgebra of X.If there is a sequence {x n } in X such that ) Proof.Since μA (0) ≥ μA (x) for all x ∈ X, we have μA (0) ≥ μA (x n ) for every positive integer n.Note that

fuzzy BCI-subalgebra of X if and only if µ L
A and µ U A are fuzzy BCI-subalgebras of X.
Proof.Suppose that µ L A and µ U A are fuzzy BCI-subalgebras of X.Let x, y ∈ X.Then Hence A is an i-v fuzzy BCI-subalgebra of X.
Conversely, assume that A is an i-v fuzzy BCI-subalgebra of X.For any x, y ∈ X, we have

y) . Hence µ L
A and µ U A are fuzzy BCI-subalgebras of X.

Theorem 3.8. Let A be an i-v fuzzy set in X. Then A is an i-v fuzzy BCI-subalgebra of X if and only if the nonempty set
. Suppose there exist x 0 ,y 0 ∈ X such that μA x 0 * y 0 < rmin μA x 0 , μA y 0 . (3.18) Let μA . On the other hand, and so y)} for all x, y ∈ X.This completes the proof.
Theorem 3.9.Every BCI-subalgebra of X can be realized as an i-v level BCI-subalgebra of an i-v fuzzy BCI-subalgebra of X.
Proof.Let Y be a BCI-subalgebra of X and let A be an i-v fuzzy set on X defined by where Therefore A is an i-v fuzzy BCI-subalgebra of X, and the proof is complete.
Theorem 3.10.Let Y be a subset of X and let A be an i-v fuzzy set on X which is given in the proof of Theorem 3.9.If A is an i-v fuzzy BCI-subalgebra of X, then Y is a BCI-subalgebra of X.

Proof. Assume that
(3.28) This implies that x * y ∈ Y .Hence Y is a BCI-subalgebra of X.
Theorem 3.11.If A is an i-v fuzzy BCI-subalgebra of X, then the set is a BCI-subalgebra of X.
The following is a way to make a new i-v fuzzy BCI-subalgebra from old one.
Theorem 3.12.For an i-v fuzzy BCI-subalgebra A of X, the i-v fuzzy set A * in X defined by μA * (x) = μA (0 * x) for all x ∈ X is an i-v fuzzy BCI-subalgebra of X. for all x, y ∈ X. Therefore A * is an i-v fuzzy BCI-subalgebra of X.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: ∀x ∈ X and let D[0, 1] denotes the family of all closed subintervals of [0, 1].If µ L A (x) = µ U A (x) = c, say, where 0 ≤ c ≤ 1, then we have μA (x) = [c, c] which we also assume, for the sake of convenience, to belong to D[0, 1].Thus μA (x) ∈ D[0, 1], ∀x ∈ X, and therefore the i-v fuzzy set A is given by A = x, μA (x) , ∀x ∈ X, where μA : X → D[0, 1].