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# Definition & Example

• A (finite) lattice 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 $x \neq y$,

such that any two elements $x$ and $y$ have a least upper bound $x \vee y$, called join, and a greatest lower bound $x \wedge y$, called meet.

• 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 Lattices of size 5     ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) ([(0,4),(2,3),(3,1),(4,2)],5) ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)

• Lattices are graphically represented by their Hasse diagram which is the directed graph of cover relations.

• Two lattices $(P,\leq_P)$ 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 lattices. This is, two lattices are considered to be equal if they are isomorphic.

• For the number of unlabelled lattices see OEIS:A006966.

## 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.

• [Wei14] E. Weisstein, Cover relation, Math World (2014)

# Technical information for database usage

• A lattice 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 lattices are isomorphic if and only if their canonical labellings coincide.
• Lattices are graded by the number of elements.
• The database contains all posets of size at most 9.

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