Robert's Mistakes

Diagram 11: Hasse diagram / poset

Nov 03, 2013

The Hasse diagram, named after Helmut Hasse (1898–1979), shows a finite poset (partial ordered set). That's any structure on which a comparison relation "≤" is defined that behaves like this (from wikipedia on "poset"):

a ≤ a (reflexivity); if a ≤ b and b ≤ a then a = b (antisymmetry); if a ≤ b and b ≤ c then a ≤ c (transitivity).

Finite posets can be represented as a directed graph without loops. The computer scientist's acronym is DAG (directed acyclic graph). A Hasse diagram is a picture of a DAG, representing a poset.

The rational numbers are a simple example for a poset. But they're furthermore totally ordered , because for any two rationals we can decide which is smaller. Their Hasse diagram is a line.

Integers are even well ordered . That means that every subset has a least element, something not true about the real numbers. But today, we're only looking at finite examples...

In general, a poset may not have a minimal or maximal element, as illustrated by the first diagram below.

Let me use this post as an excuse to tell you about prime ideals. An ideal is kind of an attractive subset. If you multiply anything with a member of an ideal you get another member of that ideal. The even numbers are an ideal. 2 is a factor of every even number and that's why 2 is called a prime ideal in Z .

Most boringly, every integer spans up an ideal, and the prime ideals all have size 1: Each prime number is a prime ideal and there is none other. The picture below illustrates this. The colors refer to the definition of the prime ideal as an ideal that is both semisimple (has only powers of 1 in its factorization, red) and also primary (has only one factor to some power, blue).

So prime ideals are a generalization of the concept of a prime number. A more interesting example is given by the ring of polynomials with integer coefficients. As the fundamental theorem of algebra states: Every polynomial can be written as a product of linear factors of the form (x-x_i)^m_i. The x_i are the solutions (zeros, roots) of the polynomial. m_i is a natural number, the multiplicity of such a zero.

Now, (x-1)(x+1) is a more interesting prime ideal.

The set of all prime ideals for a given ring is called it's spectrum . And here's the connection to the image in my first post about algebraic geometry (to be continued):

I think Mumford's treasure map is going to make a terriffic diagram post of its own.