Unary FA-presentable binary relations: transitivity and classification results

TL;DR

This paper investigates unary FA-presentable binary relations, proving that their transitive closures are also unary FA-presentable and providing characterizations for various related structures.

Contribution

It introduces new characterizations of unary FA-presentable binary relations and proves that their transitive closures preserve FA-presentability.

Findings

Transitive closure of unary FA-presentable relations is unary FA-presentable.

Characterizations of unary FA-presentable quasi-orders, partial orders, and other structures.

Descriptions of orbit structures for unary FA-presentable mappings.

Abstract

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.