In this paper, quasi-rigid $C_{0}$-semigroups are introduced and their properties are investigated.Also, some sufficient conditions for quasi-rigidity are stated.It is proved that quasi-rigid $C_{0}$-semigroups are recurrent.It is stated that if a $C_{0}$-semigroup contains a quasi-rigid operator, then it is quasi-rigid.The concept of topologically Cartridge Bags quasi-rigid $C_{0}$-semigroups is introduced.
It is proved that topologically quasi-rigidity and quasi-rigidity are equivalent.The direct sum of quasi-rigid $C_{0}$-semigroups is investigated.It is demonstrated that a $C_{0}$-semigroup is quasi-rigid if and only if the direct sum of the $C_{0}$-semigroup with itself is quasi-rigid.Moreover, the finite direct sum of a quasi-rigid operator with itself is recurrent.So, quasi-rigidity with respect to the recurrency RXOMEGA-3 FACTORS is similar to the weakly-mixing with respect to the hypercyclicity.