An automatic presentation or FA-presentation is a description of a relational structure using regular languages. Informally, in an FA-presentation, elements of the structure are represented by a regular language of abstract representatives, and each relation (of arity $n$, say) can be recognized by a synchronous $n$-tape automaton. The concept arose from computer scientists' need to extend finite model theory to infinite structures. A particular research problem has been to try to characterize the structures of some species (say, groups, rings, or boolean algebras) that admit FA-presentations.

Since any FA-presentable structure admits an FA-presentation where the regular language of representatives is over a 2-letter alphabet, a related problem is to characterize the structures that admit FA-presentations over a 1-letter alphabet, so-called `unary'
FA-presentations.

This talk will describe a new diagrammatic method for reasoning about unary FA-presentations, and how it can be applied to characterize 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 mappings, injections, surjections, and bijections. This is joint work with Nik Ruskuc.