Enumeration Order Equivalency

Definition 2.2 (Enumeration Order Reducibility on Listings and sets) 1. For listings , : we say is “Enumeration order reducible” to and write , if and only if, , ( . 2. For r.e. sets , , we say is “Enumeration order reducible” to and write , if and only if, for any listing of there exist some computable function (from the listings of to the listings of ) such that . 3. For two listings , (r.e. sets , ), we say is “Enumeration order equivalent” to and write ( ) ,if and only if, both and ( and )