Vol. 87, No. 1, 1980

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d-simple sets, small sets, and degree classes

Manuel Lerman and Robert Irving Soare

Vol. 87 (1980), No. 1, 135–155

A new notion of simplicity for recursively enumerable (r.e.) sets is introduced, that of d-simplicity or simplicity with respect to arrays of differences of r.e. sets (d.r.e. sets). This notion arose from the method used to generate automorphisms of , the lattice of r.e. sets modulo finite sets, and is a further step toward finding a complete set of invariants for the automorphism types of . The d-simple sets are closely related to the small sets defined by Lachlan as a key part of his decision procedure for the ∀∃-theory of . Finally, the degrees D of d-simple sets form a new invariant class of r.e. degrees, since H1 D but D splits L1 (where H1 and L1 are the high and low r.e. degrees respectively). This refutes conjectures of Martin and Shoenfield which imply that degrees C of any class of r.e. sets invariant under automorphisms of can be characterized by a finite set of equalities or inequalities involving the jump of degrees in C.

Mathematical Subject Classification 2000
Primary: 03D25
Secondary: 03D30
Received: 15 January 1978
Published: 1 March 1980
Manuel Lerman
Robert Irving Soare