  Partially ordered sets (posets) have a universal appearance as an abstract
structure in many areas of mathematics. Though, even their explicit enumeration
remains unknown in general, and only the counts of all partial orders on sets
of up to 16 unlabelled elements have been calculated to date, see sequence
A000112 in the OEIS.
  In this work, we study automorphisms of posets in order to formulate a
classification by local symmetries. These symmetries give rise to a division
operation on the set of all posets and lead us to the construction of symmetry
classes that are easier to characterise and enumerate. Additionally to the
enumeration of symmetry classes, I derive polynomial expressions that count
certain subsets of posets with a large number of layers (a large height). As an
application in physics, I investigate local symmetries (or rather their lack
of) in causal sets, which are discrete spacetime models used as a candidate
framework for quantum gravity.
