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Automatic presentations, also called FA-presentations, were introduced toextend finite model theory to infinite structures whilst retaining thesolubility of fundamental decision problems. A particular focus of research hasbeen the classification of those structures of some species that admitFA-presentations. Whilst some successes have been obtained, this appears to bea difficult problem in general. A restricted problem, also of significantinterest, is to ask this question for unary FA-presentations: that is,FA-presentations over a one-letter alphabet. This paper studies unaryFA-presentable binary relations.  It is proven that transitive closure of a unary FA-presentable binaryrelation is itself unary FA-presentable. Characterizations are then given ofunary FA-presentable binary relations, quasi-orders, partial orders,tournaments, directed trees and forests, undirected trees and forests, and theorbit structures of unary FA-presentable partial and complete mappings,injections, surjections, and bijections.

Unary FA-presentable binary relations: transitivity and classification results