Minimal Weak Drazin Inverses in Semigroups and Rings

In 1978, Campbell and Meyer proposed the notion of minimal rank weak Drazin inverses of complex matrices. In this paper, we define minimal weak Drazin inverses of elements in semigroups using Green’s preorder \(\leqslant _{{\mathcal {R}}},\) ${⩽}_{\mathcal{R}},$ which generalize minimal rank weak Drazin inverses of complex matrices. For two elements a, y of a semigroup, it is proved that y is a minimal weak Drazin inverse of a if and only if \(ya^{k+1}=a^{k}\) $y{a}^{k+1}=a^k$ for some nonnegative integer k and \(ay^{2}=y.\) $a{y}^2=y.$