The Scientific World Journal

Volume 2015 (2015), Article ID 925040, 9 pages

http://dx.doi.org/10.1155/2015/925040

## (Fuzzy) Ideals of BN-Algebras

^{1}Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, Konstantynów 1H, 20-708 Lublin, Poland^{2}Institute of Mathematics and Physics, Siedlce University, 3 Maja 54, 08-110 Siedlce, Poland

Received 16 September 2014; Revised 17 December 2014; Accepted 28 December 2014

Academic Editor: Arsham B. Saeid

Copyright © 2015 Grzegorz Dymek and Andrzej Walendziak. 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 notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained.

#### 1. Introduction

In 1966, Imai and Iséki [1] introduced the notion of a BCK-algebra. There exist several generalizations of BCK-algebras, such as BCI-algebras [2], BCH-algebras [3], BCC-algebras [4], BH-algebras [5], and d-algebras [6]. Neggers et al. defined B/BM/BG-algebras [7–9] and showed that the class of all B-algebras is a proper subclass of the class of all BG-algebras. They also proved that an algebra is a BM-algebra if and only if it is a 0-commutative B-algebra (therefore, every BM-algebra is a B-algebra). In [10], it is shown that the class of 0-commutative B-algebras is the class of -semisimple BCI-algebras (and hence any BM-algebra is a BCI-algebra). The class of BM-algebras contains Coxeter algebras (see [8]). Some other connections between BM-algebras and its related topics are studied in [11]. Walendziak introduced in [12] the concept of BF-algebras, which is a generalization of B-algebras and BN-algebras defined by C. B. Kim and H. S. Kim [13]. An interesting result of [13] states that an algebra is a BN-algebra if and only if it is a 0-commutative BF-algebra.

We will denote by (resp., ) the class of all BCK-algebras (resp., BCI/BCH/BH/B/BM/BG/BF/BN-algebras). The interrelationships between some classes of algebras mentioned before are visualized in Figure 1 (an arrow indicates proper inclusion; that is, if and are classes of algebras, then means ).