The algebraic structure of the isomorphic types of tally, polynomial time computable sets

Abstract

 We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P _T into six parts: D ⁻, D^ ⁻, C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets.

1. is a countable distributive lattice with the greatest element.

2. Infinitely many intervals in can be distinguished by first order formulas.

3. There exist infinitely many nontrivial automorphisms for .

4. is not distributive, but any interval in is a countable distributive lattice. RID=""ID="" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=""ID="" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID=""ID="" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.