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In , the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field
of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of
, i.e., a subfield of
that is an initial subtree of
. In this sequel to , analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of
that are themselves initial. It is further shown that an initial subdomain of
is discrete if and only if it is a subdomain of
’s canonical integer part
of omnific integers. Finally, making use of class models the results of  are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of