The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ordering: if $m<n \in \mathbb{N}\setminus\{0\}$ every point of layer $m$ is smaller than every point in layer $n$.


This poset is far from being a lattice. Which leads to the


**Question.** Is there a lattice $(L,\leq)$ such that for every antichain $A\subseteq L$ there is an antichain $A' \subseteq L$ with $|A'|>|A|$?



Antichains of maximum cardinality: posets vs lattices
Yes. For $n \in \mathbb{N}$ even let "layer" $n$ consist of a single point, and for $n \in \mathbb{N}$ odd let "layer" $n$ consist of $n$ points. The ordering is the same as yours, one point is less than another if and only if it lies in a lower layer.


