Groups of recursive automorphisms of constructive Boolean algebras
The author uses terminology and notations from \textit{Yu. L. Ershov}'s monograph ''Decision problems and constructivizable models'' (Russian) (1980; Zbl 0495.03009). An automorphism \(\phi\) of a constructive Boolean algebra (B.a.) (B,\(\nu)\) is said to be recursive if there exists a general recursive function f such that \(\phi\nu =\nu f\). Such automorphisms form the group \(Aut_ r(B,\nu)\). It is known that for any countable B.a. B the group Aut(B) implies rich information on B. In the present paper it is shown that some algebraic and algorithmic properties of a constructive B.a. (B,\(\nu)\) may be reconstructed by \(Aut_ r(B,\nu)\). The main results of this paper are the following: 1. If a constructive B.a. (B,\(\nu)\) has an atomless element then \(Aut_ r(B,\nu)\) is not constructivizable. 2. Let (B,\(\nu)\) be a constructive B.a. such that the set of its atoms and the set of its atomless elements are recursive. Then \(Aut_ r(B,\nu)\) is constructivizable iff B is finite. Let \((B_ 1,\nu_ 1)\) and \((B_ 2,\nu_ 2)\) be atomic strongly constructive B.a. Then \(Aut_ r(B_ 1,\nu_ 1)\cong Aut_ r(B_ 2,\nu_ 2)\) implies \((B_ 1,\nu_ 1)\cong (B_ 2,\nu_ 2).\) 4. There exist non-isomorphic strongly constructive B.a. with isomorphic groups of recursive authomorphisms. 5. There exist Boolean algebras \(B_ 1\) and \(B_ 2\) such that \(B_ 1\) is strongly constructivizable, \(B_ 2\) is non-constructivizable, and \(Aut(B_ 1)\cong Aut(B_ 2).\)
scientific article; zbMATH DE number 3875251