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Definition & Example
 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 $x \neq y$,

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 $a < c < b$ [Wei14].
the 5 Posets of size 3  
([],3)  ([(1,2)],3)  ([(0,1),(0,2)],3)  ([(0,2),(2,1)],3)  ([(0,2),(1,2)],3) 

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.
Additional information
Notations
 $x,y \in P$ 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 containmentmaximal chain.
 An order ideal or downclosed set is a subset $X$ of $P$ such that $x \leq y \in X$ implies $x \in X$. There is a onetoone correspondence between order ideals and antichains.
Extensions of Posets
 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 [Sta10].
References
 [Bru10] R. Brualdi, Partial orders and equivalence relations, Combinatorics Fifth Edition (2010)
 [Sta10] R. Stanley, Enumerative Combinatorics Vol 1, Second edition, Cambridge Studies in Advanced Mathematics (2011)
 [Wei14] E. Weisstein, Cover relation, Math World (2014)
Sage examples
Technical information for database usage
 A poset is uniquely represented as a tuple
(E,n)
whereE
is the sorted list of cover relations andn
is the number of elements. For this representation, we consider a canonical labelling of a poset. This is, a labelling of the elements by $\{0,1,\ldots,n1\}$ 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.