The Strength of the Grätzer-Schmidt Theorem

Paul Brodhead, Bjørn Kjos-Hanssen

Abstract: The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. A lattice is algebraic if it is complete and generated by its compact elements. We show that the set of indices of computable lattices that are complete is \(\Pi^1_1\)-complete; the set of indices of computable lattices that are algebraic is \(\Pi^1_1\)-complete; and that there is a computable lattice L such that the set of compact elements of L is \(\Pi^1_1\)-complete. As a corollary, there is a computable algebraic lattice that is not computably isomorphic to any computable congruence lattice.