(a) Define partially ordered set and order isomorphism. Prove that two finite partially
ordered sets can be represented by the same diagam if and only if they are order
isomorphic.
Answers
Answer:
The theory of partially ordered sets (or posets) plays an important unifying role in enumerative combinatorics. In particular, the theory of Möbius inversion on a partially ordered set is a far-reaching generalization of the Principle of Inclusion-Exclusion, and the theory of binomial posets provides a unified setting for various classes of generating functions. These two topics will be among the highlights of this chapter, though many other interesting uses of partially ordered sets will also be given.
Isomorphism order:
order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into equivalence classes, families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called order types.