Some people used the terms "computably enumerated in increasing order". But I can't find formal definition about it.
Computably enumerable set means that range of some total computable function. So I think that the set $S$ is computably enumerated in increasing order iff there exists total computable function $f:A \rightarrow S$ such that for $a,b\in S$, $a<b$ implies $f(a)\leq f(b)$. But I have two questions if my definition is right.
- Can we always think $A$ is subset of $N$?
- How to define the order in arbitrary computably enumerable set $S$?