  Three-way dissimilarities are a generalization of (two-way) dissimilarities
which can be used to indicate the lack of homogeneity or resemblance between
any three objects. Such maps have applications in cluster analysis, and have
been used in areas such as psychology and phylogenetics, where three-way data
tables can arise. Special examples of such dissimilarities are three-way
tree-metrics and ultrametrics, which arise from leaf-labelled trees with edges
labelled by positive real numbers. Here we consider three-way maps which arise
from leaf-labelled trees where instead the interior vertices are labelled by an
arbitrary set of values. For unrooted trees we call such maps three-way
symbolic tree-maps; for rooted trees we call them three-way symbolic
ultrametrics since they can be considered as a generalization of the (two-way)
symbolic ultrametrics of B\"ocker and Dress. We show that, as with two- and
three-way tree-metrics and ultrametrics, three-way symbolic tree-maps and
ultrametrics can be characterized via certain $k$-point conditions. In the
unrooted case, our characterization is mathematically equivalent to one
presented by Gurvich for a certain class of edge-labelled hypergraphs. We also
show that it can be decided whether or not an arbitrary three-way symbolic map
is a tree-map or a symbolic ultrametric using a triplet-based approach that
relies on the so-called BUILD algorithm for deciding when a set of 3-leaved
trees or triplets can be displayed by a single tree. We envisage that our
results will be useful in developing new approaches and algorithms for
understanding 3-way data, especially within the area of phylogenetics.
