Thursday, June 13, 2013

Implementing the Tamari lattice

The catalan numbers enumerate several different structures in combinatorics including monotone paths, dyck words, and semiorders. A catalan lattice is a class of disjoint lattices each having a catalan number of elements. The tamari lattice is a catalan lattice that can be implemented by applying min/max to monotone paths.

Sunday, June 9, 2013

Ordinal numbers

Like all surreal numbers, ordinal numbers can be described by a transfinite sequence of terms that are ordered by order of magnitude. The predecessors of the ordinal number $\omega^\omega$ can all be described by a sequence of ordinal monomials $m \omega^n$ totally ordered by the value of the exponent $n$.

The predecessors of the ordinal number $\epsilon_0$ contain all the ordinal numbers that can be represented by exponential polynomials. Such ordinal numbers can be represented by sequences of ordinal numbers raised to the power of some other ordinal sequence such as $w^{w^2+2} + w^2 + 2$.