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Section 19.3 Graph for a partial order
Hasse diagram
a diagram for the graph for a partial order on a
finite set
\(A \text{,}\) omitting reflexive loops and transitive “composite” edges, and placing “smaller” elements
lower on the diagram instead of using arrows
Example 19.3.1 . Hasse diagram for division of integers.
Let
\(A = \{2,4,6,8,10,12\} \text{.}\) The Hasse diagram of the “divides” partial order
\(\mathord{\mid} \) on
\(A \) appears in
Figure 19.3.2 . Notice that
\(2 \) is not joined directly to either
\(8 \) or
\(12 \text{,}\) since we can use transitivity and the facts that
\(2 \mid 4 \) and
\(2 \mid 6 \) to infer
\(2 \mid 8 \) and
\(2 \mid 12 \text{,}\) respectively, from the diagram.
Figure 19.3.2. The Hasse diagram for the “divides” partial order on a finite set of integers.
Example 19.3.4 . Hasse diagram for subset order.
The graph from
Example 14.4.1 has been reproduced in
Figure 19.3.5 as a Hasse diagram, and represents the partial order
\(\mathord{\subseteq} \) on
\(\powset{\{a,b,c\}} \text{.}\)
Figure 19.3.5. The Hasse diagram for the subset partial order on the power set of a finite set.
