  In this paper, we characterize the congruences of an arbitrary i--lattice,
investigate the structure of the lattice they form and how it relates to the
structure of the lattice of lattice congruences, then, for an arbitrary
non--zero natural number $n$, we determine the largest possible number of
congruences of an $n$--element i--lattice, along with the structures of the
$n$--element i--lattices with this number of congruences. Our characterizations
of the congruences of i--lattices have useful corollaries: determining the
congruences of i--chains, the congruence extension property of the variety of
distributive i--lattices, a description of the atoms of the congruence lattices
of i--lattices, characterizations for the subdirect irreducibility of
i--lattices. In terms of the relation between the above--mentioned problem on
numbers of congruences of finite i--lattices and its analogue for lattices,
while the $n$--element i--lattices with the largest number of congruences turn
out to be exactly the $n$--element lattices whose number of congruences is
either the largest or the second largest possible, we provide examples of pairs
of $n$--element i--lattices and even pseudo--Kleene algebras such that one of
them has strictly more congruences, but strictly less lattice congruences than
the other.
