  The fractions and dimension of bars, rings and lenses are studied in the
Near-IR S0 galaxy Survey (NIRS0S). We find evidence that multiple lenses in
some barred S0s are related to bar resonances in a similar manner as the inner
and outer rings, for which the outer/inner length ratio 2. Inner lenses in the
non-barred galaxies normalized to galaxy diameter are clearly smaller than
those in the barred systems. Interestingly, these small lenses in the
non-barred galaxies have similar sizes as barlenses (lens-like structures
embedded in a bar), and therefore might actually be barlenses in former barred
galaxies, in which the outer, more elongated bar component, has been destroyed.
We also find that fully developed inner lenses are on average a factor 1.3
larger than bars, whereas inner rings have similar sizes as bars. The fraction
of inner lenses is found to be constant in all family classes (A, AB, B).
Nuclear bars appear most frequently among the weakly barred (AB) galaxies,
which is consistent with the theoretical models by Maciejewski & Athanassoula
(2008). Similar sized bars as the nuclear bars were detected in seven
'non-barred' S0s. Galaxy luminosity does not uniquely define the sizes of bars
or bar-related structures, neither is there any upper limit in galaxy
luminosity for bar formation. Although all the family classes cover the same
range of galaxy luminosity, the non-barred (A) galaxies are on average 0.6 mag
brighter than the strongly barred (B) systems. Overall, our results are
consistent with the idea that bars play an important role in the formation of
the structure components of galaxies. The fact that multiple lenses are common
in S0s, and that at least the inner lenses can have very old stellar
populations, implies that the last destructive merger, or major gas accretion
event, must have taken place at a fairly high redshift.
