Southeast Asian Bulletin of Mathematics (2000) 24: 243–253 Southeast Asian Bulletin of Mathematics  c Springer-Verlag 2000 q -Ideals and a -Ideals in BCI-Algebras ∗ Yong Lin Liu Department of Mathematics, Nanping Teachers College, Nanping 353000, Fujian, China E-mail: ylliun@163.net Jie Meng Department of Mathematics, Northwest University, Xian 710069, China E-mail: mengjie@pub.xa-online.sn.cn Xiao Hong Zhang and Zhen CaiYue Department of Mathematics, Hanzhong Teachers College, Hanzhong 723000, Shaanxi, China AMS Subject Classification (2000): 03G25, 06F35 Abstract. In this note we define the notions of q -ideals and a-ideals in BCI-algebras. We give several characterizations and the extensive theorems about q -ideals and a-ideals. We show that a non-empty subset of a BCI-algebra is a-ideal if and only if it is both q -ideal and p-ideal. Finally, we give four characterizations of associative BCI-algebras by a-ideals and eight characterizations of quasi-associative BCI-algebras by q -ideals. Keywords: q -ideal, a-ideal, p-ideal, associative BCI-algebra, quasi-associative BCI-algebra 1. Introduction In [3], Iséki introduced the notion of positive implicative ideals in BCK-algebras (i.e., Iséki’s implicative ideals). Iséki and Tanaka [4] applied this concept to characterize positive implicative BCK-algebras: A BCK-algebra X is positive implicative if and only if every ideal of X is positive implicative. Meng [7] introduced the notions of implicative ideals and commutative ideals in BCK-algebras and applied them to characterize implicative BCK-algebras and commutative BCK-algebras, respectively. Meng also showed that a non-empty subset of a BCK-algebra is an implicative ideal if and only if it is both a commutative ideal and a positive implicative ideal. All the above results motivate us to further investigate the relations between algebras and ideals ∗ Supported by Fujian Education Committee Foundation, China.