Sunday, April 25, 2010

Modular Fun

After Einstein and Grossmann developed relativity, physicists grew to love Riemannian geometry. After Heisenberg rediscovered matrices, physicists grew to love linear algebra, and with the 20th century development of particle physics they came to absolutely love groups, especially continuous ones.

When they use matrices to describe groups, it is usually in the context of representation theory, where the matrices are linear maps between vector spaces over a familiar continuum, such as the real or complex numbers. For the field with one element, however, matrices are no such thing. But we still like to multiply matrices, and to study interesting finitely generated sets of matrices.

For example, consider allowing a zero element to be attached to a group like set. Groups themselves don't have zeroes, because a zero has no inverse. Let all other members have inverses, but note that members may be zero divisors. Although this is not a ring, because there are only products, we love the zero divisors. So taking two familiar matrix generators: we have an infinite example analogous to the $12$ element group $A_4$ (which has generators $S$ and $T$ such that $S^2 = 1$ and $T^3 = 1$). Here the relations are $S^2 = S$, $T^3 = 1$ and $(ST)^3 = (STT)^3 = (TS)^3 = 0$. The list consists of all words in $S$ and $T$ such that $S$ occurs singly and $T$ singly or doubly. Most strings collapse or terminate eventually, but $(ST^2 ST)^n$ (which is circulant) just keeps on going. Everything has an inverse and everything is a zero divisor. Idempotents like $S$ are useful for creating zero divisors.

In summary, this object has a zero, an identity and multiplication, just like the field characters that we like to put into matrices. It is a bit like a noncommutative semiring, except that semirings have addition, and it is certainly a monoid and a semigroup, but these don't usually involve zeroes. Unromantic, it is called a semigroup with zero.

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