The DAG has a distinguished top node all.
As an example, in the dimension of Figure 4, we have p5 < b3 < c2 < d1 < all. The
descendant/ancestor relationship, denoted <, is the transitive closure of the relation
<. As an example, in the dimension of Figure 4, from the existence of a child/parent
path from 5 to all, it follows p5 < all. The hierarchy domain may have many bottom
elements, but it is not allowed to have shortcuts. The reason for this is that a shortcut
from an element e1to an element e2 is redundant information. As examples, the
graphs in the right-hand sides of Figures 1, 2, and 4 are hierarchy domains.
6 Hurtado & Gutierrez
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Dimension
A dimension comprises a hierarchy schema (C, ??°), a hierarchy domain (M, <), and
a function d : M ?†’ C that defines the category to which each element belongs. In
particular, the element all is mapped to the category All, that is, d(all) = All . The
fundamental property of the function d is that whenever two elements satisfy e1 <
e2 in the hierarchy domain, the corresponding categories to which they belong satisfy
d(e1) ??° d(e2). In other words, an edge in the hierarchy domain implies an edge
between the corresponding categories in the hierarchy schema.
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