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NATURAL EQUIVALENCES FROM LATTICE QUASIORDERS TO INVOLUTION LATTICES OF CONGRUENCES

The involution lattice Quord(A) of quasiorders of a lattice A is known to be isomorphic to the involution lattices Con(A) consisting of pairs of congruences of A. Moreover, the isomorphism described in [9] is supplied by a natural equivalence between the functors Quord and Con. The aim of the present paper is to describe and count the possible Quord → Con natural equivalences. The answer depends on the domain category L, always a prevariety of lattices with the surjective homomorphisms, of the functors Quord and Con; and the problem is solved only for very small prevarieties L. An overview on the most recent developments in the theory of involution lattices and quasiorders is also presented. To the memory of Milan Kolibiar