Embedding the Diamond Lattice in the Recursively Enumerable Truth-table Degrees

It is shown that the four element Boolean algebra can be embedded in the recursively enumerable truth-table degrees with least and greatest elements preserved. Corresponding results for other lattices and other reducibilites are also discussed. For sets A, B ç co, we say that A is a truth-table (tt) reducible to B if there exists an effective procedure for reducing any question of the form "m e A?" to an equivalent finite Boolean combination of questions of the form "/c e BT' Then, A, B are said to have the same tt-degree if each is tt-reducible to the other, and tt-degrees have a natural ordering induced by tt-reducibility. (See [1, 6 and 8] for information on tt-degrees.) We show the existence of two incomparable recursively enumerable (r.e.) tt-degrees with supremum 0' (the highest r.e. tt-degree) and infimum 0 (the lowest). In other words, the four-element Boolean algebra (known also as the diamond lattice) can be embedded as a lattice in the r.e. tt-degrees with least and greatest elements preserved. We also obtain analogous results with the diamond lattice replaced by each of the two five-element nondistributive lattices (pentagon and 1-3-1) and with tt-reducibility replaced by many of its restricted forms, such as bounded truth-table and positive reducibility [2]. The history of this problem is as follows. A. H. Lachlan proved in his well-known "nondiamond theorem" [5, Theorem 5] that the diamond lattice cannot be embedded in the r.e. Turing degrees with 0 and 1 preserved. His proof simultaneously establishes the corresponding result for r.e. weak truth-table (wtt) degrees [6]. Lachlan also showed in [4] that no two incomparable r.e. many-one (m) degrees can have supremum 0', so the diamond lattice cannot be embedded in the r.e. w-degrees with 1 preserved. The trend of these results makes it reasonable to conjecture that the diamond lattice cannot be embedded in the r.e. tt-degrees with 0 and 1 preserved, although in the other direction D. Posner [7] proved that the Turing degrees below 0' are complemented. In [6, Theorem 6.6] P. G. Odifreddi announced that in fact the diamond lattice can be embedded in the r.e. tt-degrees with 0 and 1 preserved. His construction involved splitting a creative set K into two disjoint r.e. Received by the editors April 9, 1984. 1980 Mathematics Subject Classification. Primary 03D30; Secondary 03D25.