Thursday, September 19, 2013

Orders derived from group action

Each permutation group produces a partition of the power set of the underlying set of the group. By the definition of partitioned partial orderings this produces a partial ordering relation from the group action. The partial ordering of a disjoint union of permutation groups is the product order of the partial orderings of each of those permutation groups.

The symmetric group and the alternating group are both completely transitive so they produce a total ordering relation $T_n$. Permutation groups that are defined as the disjoint union of such completely transitive groups effectively produce a multiset inclusion lattice. The cyclic group $C_4$ produces the free distributive lattice of size 2:

The cyclic group $C_2$ is the smallest group that doesn't produce a complete lattice as its underlying partial order and it is the smallest group that isn't the product of transitive groups:

The cyclic group $C_5$ is an example of a transitive group whose partial ordering is also not a complete lattice:

As I already mentioned in my post on ranking elements of partial orders the size of an element of one of these partial orders is its height and the cardinality of one of these partial orders is the total height minus one.

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