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
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