We study computable reducibility for computable isomorphic copies of the standard ordering of natural numbers. Following Andrews and Sorbi, we isolate the class of self-full degrees inside the induced degree structure Ω. We show that, over an arbitrary degree from Ω, there exists an infinite antichain of self-full degrees. This fact implies that the poset Ω has continuum many automorphisms. We prove that any non-self-full degree from Ω has no minimal covers, which implies that, inside Ω, the self-full degrees are precisely those elements that have a minimal cover. Bibliography: 18 titles.The paper studies computable reducibility for computable linear orders. Let R and S be binary relations on the set of natural numbers ω. The relation R is computably reducible to S (denoted by R c S) if there is a total computable function f (x) such thatIn this case, one also says that the function f is a computable reduction from R to S.The systematic study of computable reducibility for positive (computably enumerable) equivalence relations was initiated by Ershov [1] who constructed one of the first examples of a universal positive equivalence. Further developments led to sophisticated classifications for various * To whom the correspondence should be addressed.

The work of Bazhenov N.A., Zubkov M.V., Kalmurzayev B.S. started investigation of questions of the existence of joins and meets of positive linear preorders with respect to computable reducibility of binary relations. In the last section of this work, these questions were considered in the structure of computable linear orders isomorphic to the standard order of natural numbers. Then, the work of Askarbekkyzy A., Bazhenov N.A., Kalmurzayev B.S. continued investigation of this structure. In the last article, the notion of a self-full linear order played important role. A preorder R is called self-full, if for every computable function g(x), which reduces R to R, the image of this function intersects all supp(R)-classes. In this article, we measure exact algorithmic complexities of index sets of all self-full recursive linear orders isomorphic to the standard order of natural numbers and to the standard order of integers. Researching the index sets allows us to measure exact algorithmic complexities of different notions in constructive structures, that we are investigating. We prove that the index set of all self-full computable linear orders isomorphic to the standard order of that we are investigating. We prove that the index set of all self-full computable linear orders isomorphic to the standard order of integers is П3 0-complete.