The number of doubly irreducible elements of a lattice.
An element $d$ of a lattice $L$ is doubly irreducible if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$.
In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.

The lattice of maximal antichains in a poset.
An antichain $A$ in a poset is maximal if there is no antichain of larger cardinality which contains all elements of $A$.
The set of maximal antichains can be ordered by setting $A \leq B \Leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow}B$, where $\mathop{\downarrow}A$ is the order ideal generated by $A$.

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.

The lattice of congruences of a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.