In this paper we consider pseudo-BCK/BCI-algebras. In particular, we consider properties of minimal elements ($x{leq}a$ implies x = a) in terms of the binary relation $leq$ which is reflexive and anti-symmetric along with several more complicated conditions. Some of the properties of minimal elements obtained bear resemblance to properties of B-algebras in case the algebraic operations $ast$ and $circ$ are identical, including the property $0{circ}(0{ast}a)$ = a. The condition $0{ast}(0{circ}x)=0{circ}(0{ast}x)=x$ all $x{in}X$ defines the class of p-semisimple pseudo-BCK/BCI-algebras($0{leq}x$ implies x = 0) as an interesting subclass whose further properties are also investigated below.