Some new properties on N semigroups

In this study we first show that  satisfies two important homological properties, namely Rees short exact sequence an d short five lemma. In addition, by defining inversive semigroup varieties of  we prove that strictly inverse semigroup  is isomorphic to the spined product of (C)-inversive semigroup and the idempotent semigroup of . Moreover, we give some consequences of the results to make a detailed classification over . It has been recently defined a new semigroup  based on Rees matrix and completely 0-simple semigroups. Further, it has been also proved finiteness conditions and the existence of some fundamental properties over .