Let $G$ be a group and $p$ a prime number. $G$ is said to be a $Y_p$-group if whenever $K$ is a $p$-subgroup of $G$ every subgroup of $K$ is an $S$-permutable subgroup in $N_G(K)$. The group $G$ is a soluble $PST$-group if and only if $G$ is a $Y_p$-group for all primes $p$. It is our purpose here to define a number of local properties related to $Y_p$ which lead to several new characterizations of soluble $PST$-groups.