Hasse diagram
In
mathematics, a
Hasse diagram (pronounced
HAHS uh) is a simple picture of a finite
partially ordered set. One says of two members
x and
y of a partially ordered set
S that "
y covers
x" if
x ≤
y and no element of
S is between
x and
y. The partial ordering is then just the
transitive closure of the cover relation. The Hasse diagram of
S may then be defined abstractly as the set of all ordered pairs (
x,
y) such that
y covers
x, i.e., the Hasse diagram may be identified with the cover relation. Concretely, one represents each member of
S as a black dot on the page and draws a line that goes upward from
x to
y if
y covers
x.
[An illustration here might be useful.]
For example, if a Hasse diagram was drawn of the poset of all the divisors of a number, partially ordered by divisibility, then the number itself would be at the top of the diagram, the number 1 would be at the bottom, and the smallest (prime) divisors would cover the bottom element.