The Shape and Symmetry of Posets

What's the best way to rank the players on a baseball team? You could compare batting averages, but a star pitcher might not have a good batting average. It doesn't make sense to compare a pitcher's strikeouts to another player's batting average. It's hard (or even impossible) to properly rank the team. This is an example of a set where some elements can be compared, but not every element. We call sets like this partially-ordered sets, or "posets" for short. Posets appear in many areas of mathematics.

Posets can be visualized through a type of graph called a Hasse diagram. By studying the shape of these Hasse diagrams, we can better understand the structure of a poset. Students will be looking for patterns in these diagrams by studying symmetries and loops within the graph. Once we obtain a better understanding of the shape of some Hasse diagrams, we can attempt to develop an algorithm to identify this information for any diagram.

The only prerequisite for this project is a familiarity with the language of sets, functions and relations. Math 11 or an equivalent course will be sufficient. Students can expect to draw lots of graphs and examine even more. Anyone with knowledge of topology or abstract algebra may explore a surprising connection between these subjects and posets.