On the semiring of generalized inverses of a matrix

In this paper, we consider the set of all generalized inverses of an [Formula: see text] matrix [Formula: see text], denoted by [Formula: see text] and define two binary operations, ⊕ and ⋆ with some properties which offer the structure of a semi ring to [Formula: see text]. The compatibility of some partial orders like Sussman’s order, Conrad’s order, etc. is also checked which make it an ordered matrix semiring.