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