  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.
