Abstract. In this paper we consider pseudo- BCK=BCI -algebras. Inparticular, we consider properties of minimal elements ( x  a implies x = a ) in terms of the binary relation  which is reexive and anti-symmetric along with several more complicated conditions. Some of theproperties of minimal elements obtained bear resemblance to properties of B -algebras in case the algebraic operations and ◦ are identical, includ-ing the property 0 ◦ (0 a ) = a . The condition 0 (0 ◦x ) = 0 ◦ (0 x ) = x forall x 2 X de nes the class of p -semisimple pseudo- BCK=BCI -algebras(0  x implies x = 0) as an interesting subclass whose further propertiesare also investigated below. 1. IntroductionY. Imai and K. Iseki introduced two classes of abstract algebras: BCK -algebras and BCI -algebras ([6, 7]). We refer useful textbooks for BCK=BCI -algebra to [5, 10, 11]. G. Georgescu and A. Iorgulescu ([3]) introduced thenotion of a pseudo BCK -algebra as an extension of BCK -algebra, and Y. B.Jun([8])characterizedpseudo