Index Sets of Self-Full Linear Orders Isomorphic to Some Standard Orders

Abstract

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.