Minimal Weak Drazin Inverses in Semigroups and Rings

In 1978, Campbell and Meyer proposed the notion of minimalrank weak Drazin inverses of complex matrices. In this paper, we defineminimal weak Drazin inverses of elements in semigroups using Green'spreorde r <= R,which generalize minimal rank weak Drazin inverses ofcomplex matrices. For two elementsa, yof a semigroup, it is provedthatyis a minimal weak Drazin inverse ofaif and only if ya(k+1)=a(k) for some nonnegative integer k and ay(2)=y