Let $(P,\le)$ be a poset, and $C$ be its set of (saturated, if needed) chains, i.e., set of totally ordered subsets of $(P,\le).$ Is there any canonical ordering and meet, join operations on $C$ that makes it into a lattice? Can we make this into a distributive lattice?
I was thinking about this question for few hours, but couldn't construct anything satisfactory. Therefore thought to ask the opinion of people familiar with order theory. Thank you in advance for your help.
Intersection will make C a
meet semilattice.
If C is lower complete, then intersection
will make C is a complete meet semilattice.
There is, in general, no join.
The chains of 0 < a,b are
empty set, {0}, {a}, {b}, {0,a}, {0,b}.
As {{0,a}, {0,b}} has no upper bound, it has no join.
Trivally, linear orders are lattices and
complete linear orders are complete lattices.