On intra-regular and left regular and left duo ordered $\mathrm{\Gamma }$-semigroups

For an intra-regular or a left regular and left duo ordered $\mathrm{\Gamma }$-semigroup $M$, we describe the principal filter of $M$ which plays an essential role in the structure of this type of $po$-$\mathrm{\Gamma }$-semigroups. We also prove that an ordered $\mathrm{\Gamma }$-semigroup $M$ is intra-regular if and only if the ideals of $M$ are semiprime and it is left (right) regular and left (right) duo if and only if the left (right) ideals of $M$ are semiprime.