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.