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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 thestructure of the lattice of lattice congruences, then, for an arbitrarynon--zero natural number $n$, we determine the largest possible number ofcongruences of an $n$--element i--lattice, along with the structures of the$n$--element i--lattices with this number of congruences. Our characterizationsof the congruences of i--lattices have useful corollaries: determining thecongruences of i--chains, the congruence extension property of the variety ofdistributive i--lattices, a description of the atoms of the congruence latticesof i--lattices, characterizations for the subdirect irreducibility ofi--lattices. In terms of the relation between the above--mentioned problem onnumbers of congruences of finite i--lattices and its analogue for lattices,while the $n$--element i--lattices with the largest number of congruences turnout to be exactly the $n$--element lattices whose number of congruences iseither the largest or the second largest possible, we provide examples of pairsof $n$--element i--lattices and even pseudo--Kleene algebras such that one ofthem has strictly more congruences, but strictly less lattice congruences thanthe other.

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