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

We investigate the polynomial time isomorphic type structure of PT ⊂ {A : A ⊆ {0}∗} (the class of tally, polynomial time computable sets). We partition PT into six parts: D−, D̂−, C, S, F, F̂ , and study their p-isomorphic properties separately. The structures of 〈deg1(F ); ≤〉, 〈deg1(F̂ ); ≤〉, and 〈deg1(C); ≤〉 are obvious, where F , F̂ , 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 〈deg1(D̂); ≤〉 and 〈deg1(S); ≤〉 will be proved, where D̂ 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. 〈deg1(D̂); ≤〉 is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in 〈deg1(D̂); ≤〉 can be distinguished by first order formulas. 3. There exist infinitely many nontrivial automorphisms for 〈deg1(D̂); ≤〉. 4. 〈deg1(S); ≤〉 is not distributive, but any interval in 〈deg1(S); ≤〉 is a countable distributive lattice.