The Strength of the Gr\"atzer-Schmidt Theorem

TL;DR

This paper explores the reverse mathematics of the Grätzer-Schmidt theorem, establishing the complexity of various classes of computable lattices and their compact elements within descriptive set theory.

Contribution

It demonstrates the high logical complexity of the set of algebraic and complete computable lattices and their compact elements, revealing new insights into their classification.

Findings

The set of indices of complete computable lattices is $oldsymbol{ ext{Pi}}^1_1$-complete.

The set of indices of algebraic computable lattices is $oldsymbol{ ext{Pi}}^1_1$-complete.

The set of compact elements of a computable lattice can be $oldsymbol{ ext{Pi}}^1_1$-complete and $oldsymbol{ ext{Pi}}^0_3$-complete.

Abstract

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.