Semi-decidable equivalence relations obtained by composition and lattice join of decidable equivalence relations

TL;DR

This paper explores how semi-decidable equivalence relations can be constructed from decidable ones using composition and lattice join, revealing both possibilities and limitations.

Contribution

It demonstrates that all semi-decidable equivalence relations with infinite classes can be formed from decidable relations, and provides examples with finite classes showing both possibility and impossibility.

Findings

Semi-decidable relations with infinite classes are obtainable from decidable relations.

Some semi-decidable relations with finite classes cannot be obtained from decidable relations.

Examples illustrate the limits of constructing semi-decidable relations from decidable ones.

Abstract

Composition and lattice join (transitive closure of a union) of equivalence relations are operations taking pairs of decidable equivalence relations to relations that are semi-decidable, but not necessarily decidable. This article addresses the question, is every semi-decidable equivalence relation obtainable in those ways from a pair of decidable equivalence relations? It is shown that every semi-decidable equivalence relation, of which every equivalence class is infinite, is obtainable as both a composition and a lattice join of decidable equivalence relations having infinite equivalence classes. An example is constructed of a semi-decidable, but not decidable, equivalence relation having finite equivalence classes that can be obtained from decidable equivalence relations, both by composition and also by lattice join. Another example is constructed, in which such a relation cannot be obtained from decidable equivalence relations in either of the two ways.