Operator Symmetry on Functions

Abstract

All functions possess symmetries over their input with certain operators. So-called Symmetry-Sets over a given function and operator are the sets of objects that can be operated with the input to the function under that operator without effecting the output of the function. This work shows that when the domain of a function forms algebraic structure – e.g. a Monoid, Group, Ring, etc. – with a given operator or pair of operators, the Symmetry-Sets over the same operator(s) have many nice properties. The work develops and enumerates many interesting results on so-called Tessellations – functions from the integers to some at-least cancellative-algebra – using the structure of Symmetry-Sets on them, i.e. Period-Sets when speaking of Tessellations. The behavior of the principal period of any given Tesselation is detailed, as well as how they interact with each-other when Tesselations are operated together using generalized function-operators. Briefly, a venue is developed for studying these Symmetry-Sets more thoroughly, by introducing the notion of Allgebras, an element set paired with the set of all definable operators on the element set. In this context, algebraic structures are relations between subsets of elements and operators.