Monday, September 3, 2012

Iteration properties

Every monoid element has two sets associated with it that define the properties of its iteration:
  • Iteration set: this define the set of numbers with which this function can be iterated over. Injective functions include -1 as in their iteration set.
  • Iteration equivalence relation: this defines equivalent iteration numbers, for example, all iterations other then zero are equal to one with idempotent elements.
These sets can be described as a single partition of the real numbers that includes a specially flagged set of invalid values. The iteration properties of an element form a distributive lattice, and the meet operation in this lattice describes the iteration properties of the composition of several elements.

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