Autostability Spectra for Boolean Algebras

Abstract

Autostability spectra for Boolean algebras and their various enrichments are explored. The basic notions and definitions of computable structure theory used in the paper can be found in [1, 2]. As usual, we write “c.e.” as an abbreviation for “computably enumerable.” Recall that a computable structure M is said to be decidable if its full diagram FD(M) is computable. A structure M is strongly constructivizable if M has a decidable copy. Let d be a Turing degree. A constructivizable structure A is said to be d-autostable if, for any computable copies B and C of A, there exists a d-computable isomorphism f : B → C. A strongly constructivizable structure A is d-autostable relative to strong constructivizations if, for any decidable copies B and C for A, there exists a d-computable isomorphism f : B → C. If A is a constructivizable structure, then the autostability spectrum of A is the set of all Turing degrees d for which the structure A is d-autostable. The autostability spectrum relative to strong constructivizations for a strongly constructivizable structure A is the set of all degrees d such that A is d-autostable relative to strong constructivizations. Denote the autostability spectrum of a structure A by AutSpec (A), and the autostability spectrum relative to strong constructivizations for A by SCAutSpec (A). We say that a Turing degree d is the degree of autostability (degree of autostability relative to strong constructivizations) for A if d is least in AutSpec (A) (in SCAutSpec (A)). Note that in ∗Supported by RFBR (project No. 14-01-00376), by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1), and by the Russian Ministry of Education and Science (gov. contract No. 2014/139).