The Strength of the Grätzer-Schmidt Theorem
نویسندگان
چکیده
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 Π 1 -complete; the set of indices of computable lattices that are algebraic is Π 1 -complete; and that there is a computable lattice L such that the set of compact elements of L is Π 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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تاریخ انتشار 2009