  The concepts of orthology, paralogy, and xenology play a key role in
molecular evolution. Orthology and paralogy distinguish whether a pair of genes
originated by speciation or duplication. The corresponding binary relations on
a set of genes form complementary cographs. Allowing more than two types of
ancestral event types leads to symmetric symbolic ultrametrics. Horizontal gene
transfer, which leads to xenologous gene pairs, however, is inherent asymmetric
since one offspring copy "jumps" into another genome, while the other continues
to be inherited vertically. We therefore explore here the mathematical
structure of the non-symmetric generalization of symbolic ultrametrics. Our
main results tie non-symmetric ultrametrics together with di-cographs (the
directed generalization of cographs), so-called uniformly non-prime
2-structures, and hierarchical structures on the set of strong modules. This
yields a characterization of relation structures that can be explained in terms
of trees and types of ancestral events. This framework accommodates a
horizontal-transfer relation in terms of an ancestral event and thus, is
slightly different from the the most commonly used definition of xenology.
