The Strength of the Gr\"atzer-Schmidt Theorem

The Gr\"atzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also 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; the set of compact elements of a computable lattice is $\Pi^{1}_{1}$ and can be $\Pi^1_1$-complete; and the set of compact elements of a distributive computable lattice is $\Pi^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $\Pi^0_3$-complete.