Congruence Lattice Problem - Semilattice Formulation of CLP

Semilattice Formulation of CLP

The congruence lattice Con A of an algebra A is an algebraic lattice. The (∨,0)-semilattice of compact elements of Con A is denoted by Con_c A, and it is sometimes called the congruence semilattice of A. Then Con A is isomorphic to the ideal lattice of Con_c A. By using the classical equivalence between the category of all (∨,0)-semilattices and the category of all algebraic lattices (with suitable definitions of morphisms), as it is outlined here, we obtain the following semilattice-theoretical formulation of CLP.