Some Properties of Lattice Congruences Preserving Involutions and Their Largest Numbers in the Finite Case

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.