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Definition & Example
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- A (finite) **poset** (or **partially ordered set**) is a finite set $P$ together with a partial order $\leq$ satisfying
- $\leq$ is *reflexive*: $x \leq x$ for all $x \in P$,
- $\leq$ is *transitive*, $x \leq y \leq z$ implies $x \leq z$ for all $x,y,z \in P$,
- $\leq$ is *antisymmetric*, $x \leq y \Rightarrow y \not\leq x$ for $x,y \in P$ such that $x \neq y$,
see <> and <>.
- One often writes $a < b$ for $a \leq b$ and $a \neq b$.
- A **cover relation** $a \prec b$ is a pair of elements $a < b$ such that there exists no $c \in P$ for which $a < c < b$ <>.
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- Posets are graphically represented by their **Hasse diagram** which is the directed graph of cover relations.
- Two posets $(P,\leq_P)$ and $(P',\leq_{P'})$ are **isomorphic** if there exists a bijection $\pi: P\ \tilde\longrightarrow\ P'$ such that $x \leq_P y$ if and only if $\pi(x) \leq_{P'} \pi(y)$ for all $x,y \in P$.
- This project considers **unlabelled posets**. This is, two posets are considered to be equal if they are isomorphic.
- For the number of unlabelled posets see [OEIS:A000112](https://oeis.org/A000112).
Additional information
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Notations
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- $x,y \in P$ are called **comparable** if $x \leq y$ or $y \leq x$. A poset is called **linear**, **linearly ordered**, or **totally ordered** if any two elements are comparable.
- $x \in P$ is **minimal** if there is no $y \in X$ such that $y \leq x$,
- $x \in P$ is **maximal** if there is no $y \in X$ such that $x \leq y$.
- A subset $X$ of $P$ is called **chain** if it is pairwise comparable, and **antichain** if it is pairwise not comparable. A **maximal chain** is containment-maximal chain.
- An **order ideal** or **down-closed set** is a subset $X$ of $P$ such that $x \leq y \in X$ implies $x \in X$. There is a one-to-one correspondence between order ideals and antichains.
Extensions of Posets
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- Let $P$ be a set with two partial orders $\leq_1$ and $\leq_2$. Then $\leq_2$ is an **extension** of $\leq_1$ if $x \leq_1 y$ implies $x \leq_2 y$.
- Linear extensions of posets play a particularly important role, see <>.
References
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Sage examples
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{{{#!sagecell
for n in [1,2,3,4]:
Ps = Posets(n)
print n, Ps.cardinality()
for P in Posets(3):
print (P.cover_relations(),P.cardinality())
}}}
Technical information for database usage
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- A poset is uniquely represented as a tuple `(E,n)` where `E` is the sorted list of cover relations and `n` is the number of elements. For this representation, we consider a **canonical labelling** of a poset. This is, a labelling of the elements by $\{0,1,\ldots,n-1\}$ such that any two posets are isomorphic if and only if their canonical labellings coincide.
- Posets are graded by the number of elements.
- The database contains all posets of size at most 7.