  The stabilization of one-dimensional solitons by a nonlinear lattice against
the critical collapse in the focusing quintic medium is a challenging issue. We
demonstrate that this purpose can be achieved by combining a
nonlinearlatticeandsaturationofthequinticnonlinearity.
Thesystemsupportsthreespeciesofsolitons, namely, fundamental (even-parity) ones
and dipole (odd-parity) modes of on- and off-site-centered types. Very narrow
fundamental solitons are found in an approximate analytical form, and
systematic results for very broad unstable and moderately broad partly stable
solitons, including their existence and stability areas, are produced by means
of numerical methods. Stability regions of the solitons are identified by means
of systematic simulations. The stability of all the soliton species obeys the
Vakhitov-Kolokolov criterion.
