By `model of computability' I mean a precise characterization of which functions one counts as partially computable. So, for example, I say that Turing machines are a model of computability and Register machines are another model of computability. My question is the following. Suppose that we have two models of computability $A$ and $B$. Suppose that we know that the $A$-decidable sets are exactly the $B$-decidable sets. Can we conclude that the $A$-partially-computable functions are exactly the $B$-partially-computable functions?
Many thanks for your help.