  In the sense of Palamodov, a preabelian category is semi-abelian if for every
morphism the natural morphism between the cokernel of its kernel and the kernel
of its cokernel is simultaneously a monomorphism and an epimorphism. In this
article we present several conditions which are all equivalent to
semi-abelianity. First we consider left and right semi-abelian categories in
the sense of Rump and establish characterizations of these notions via six
equivalent properties. Then we use these properties to deduce the
characterization of semi-abelianity. Finally, we investigate two examples
arising in functional analysis which illustrate that the notions of right and
left semi-abelian categories are distinct and in particular that such
categories occur in nature.
