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

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