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.
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Brodhead, P., Kjos-Hanssen, B. (2009). The Strength of the Grätzer-Schmidt Theorem. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_7
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DOI: https://doi.org/10.1007/978-3-642-03073-4_7
Publisher Name: Springer, Berlin, Heidelberg
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