Spaces Fréchet–Urysohn with respect to families of subsets

The study of the Fréchet–Urysohn property with respect to a family of subsets started by Reznichenko and this author in relation to the problem of Malykhin on the existence in ZFC of a separable nonmetrizable Fréchet–Urysohn topological group is continued. A simple characterization of spaces Fréchet–Urysohn with respect to finite subsets is given. It is proved that the Fréchet–Urysohn property in lattice-ordered spaces and groups and their subspaces often implies stronger Fréchet–Urysohn properties with respect to families of subsets in many situations.