Intervals without Critical Triples

This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M5 (see Figure 1) cannot be embedded. Actually, we construct intervals I of computably enumerable degrees without any weak critical triples (this implies that M5 cannot be embedded in I, see Section 2). Our strongest result is that there is a low2 computably enumerable degree e such that there are no weak critical triples in either of the intervals [0, e] or [e, 0].

[1]  Manuel Lerman,et al.  Lattice embeddings into the recursively enumerable degrees , 1986, Journal of Symbolic Logic.

[2]  Peter Cholak,et al.  Lattice Nonembeddings and Intervals of the Recursively Enumerable Degrees , 1993, Ann. Pure Appl. Log..

[3]  Alistair H. Lachlan.  Embedding nondistributive lattices in the recursively enumerable degrees , 1972 .

[4]  Richard A. Shore,et al.  Working below a low2 recursively enumerably degree , 1990, Arch. Math. Log..

[5]  Rodney G. Downey,et al.  Degree theoretic definitions of the low2 recursively enumerable sets , 1995, Journal of Symbolic Logic.

[6]  V. Rich.  Personal communication , 1989, Nature.

[7]  Carl G. Jockusch,et al.  Pseudojump operators. I. The r.e. case , 1983 .

[8]  R. Soare,et al.  Not every finite lattice is embeddable in the recursively enumerable degrees , 1980 .

[9]  Manuel Lerman,et al.  A Finite Lattice without Critical Triple that cannot be Embedded into the Enumerable Turing Degrees , 1997, Ann. Pure Appl. Log..

[10]  Rodney G. Downey,et al.  Contiguity and distributivity in the enumerable Turing degrees , 1997, Journal of Symbolic Logic.

[11]  André Nies,et al.  Definability in the Recursively Enumerable Degrees , 1996, Bulletin of Symbolic Logic.

[12]  Richard A. Shore,et al.  The Recursively Enumerable Degrees , 1999, Handbook of Computability Theory.

[13]  Steven D. Leonhardi.  Generalized Nonsplitting in the Recursively Enumerable Degrees , 1997, J. Symb. Log..

[14]  Manuel Lerman,et al.  Lattice embeddings into the recursively enumerable degrees. II , 1989, Journal of Symbolic Logic.

[15]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[16]  Richard A. Shore.  RECURSIVELY ENUMERABLE SETS AND DEGREES A Study of Computably Generated Sets (Perspectives in Mathematical Logic) , 1988 .

[17]  Rodney G. Downey.  Lattice Nonembeddings and Initial Segments of the Recursively Enumerable Degrees , 1990, Ann. Pure Appl. Log..

[18]  Richard A. Shore,et al.  Lattice embeddings below a nonlow2 recursively enumerable degree , 1996 .

[19]  Juichi Shinoda,et al.  Preface - Papers presented at the International Symposium on Mathematical Logic and its Applications Nagoya, Japan, November 7-11, 1988 , 1991, Ann. Pure Appl. Log..