Effective categoricity of equivalence structures

We investigate effective categoricity of computable equivalence structures A. We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively 0 3 categorical, we further investigate when they are 0 2 categorical. We also obtain results on the index sets of computable equivalence structures. The authors would like to thank the anonymous referee for his comments and suggestions. Calvert was partially supported by the NSF grants DMS-9970452, DMS-0139626, and DMS-0353748, Harizanov by the NSF grant DMS-0502499, and the last three authors by the NSF binational grant DMS-007589...

• Type: Journal
• Year: 2006
• Where: APAL
• Authors: Wesley Calvert, Douglas Cenzer, Valentina S. Harizanov, Andrei S. Morozov
• Keywords: APAL 2006, Computable Equivalence, Computable Equivalence Structures, NSF Grant