Examples
Standard examples of posets arising in mathematics include:
- The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).
- The set of natural numbers equipped with the relation of divisibility.
- The vertex set of a directed acyclic graph ordered by reachability.
- The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right).
- The set of subspaces of a vector space ordered by inclusion.
- For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn)n∈ℕ if and only if an ≤ bn for all n in ℕ.
- For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all x in X.
- A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...
Read more about this topic: Partially Ordered Set
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)