Abstract

A recursive algebra is a structure for which A is a recursive set of numbers and the Fi are uniformly recursive operations. We define an r.e. quotient algebra to be the quotient by an r.e. congruence .We say that is recursively stable among r.e. quotient algebras if, for each r.e. quotient algebra and each isomorphism from onto ′, the set {a,baA,bB and =[b]′} is r.e.We shall consider examples of recursive stability. Then, assuming that has a recursive existential diagram, we show that the task of determining its recursive stability among r.e. quotient algebras can be reduced to a more routine consideration of syntactical conditions. To this result, we provide a counter-example which demonstrates the necessity of having a recursive existential diagram. This result and counter-example are on similar lines to ones obtained by Goncharov , for the recursive stability of recursive structures