Is it possible to save ps1 games on a ps2?also can u save ps2 games on the ps3 console?

[whats_my_name_again?]

so ps2 is backwards compatible apparrently with ps1 and yes it is true a ps1 game works in the ps2 ,but it doesn't have a save option when I play ps1 games on it. Is there a setting I need to change on the ps2 to get it to save? if not how can it be called backwards compatible ?is the ps3 equally as backward compatible?