On $\mathfrak{U}$-$\mathfrak{L}$-purity in Abelian groups

We give a complete description of the classes of $\mathfrak{U}$-$\mathfrak{L}$-divisible, $\mathfrak{U}$-$\mathfrak{L}$-flat, $\mathfrak{U}$-$\mathfrak{L}$-injective and $\mathfrak{U}$-$\mathfrak{L}$-projective groups.