Morphisms, Symbolic sequences, and their Standard Forms

TL;DR

This paper introduces methods to select unique standard forms for morphisms and symbolic sequences, aiding classification and storage of these objects in databases, with applications demonstrated on k-symbol Fibonacci sequences.

Contribution

It proposes a novel approach to identify unique representatives for morphisms and symbolic sequences, facilitating their classification and database storage.

Findings

Unique representatives for morphisms are identified.

Standard forms for symbolic sequences are established.

Applications demonstrated on k-symbol Fibonacci sequences.

Abstract

Morphisms are homomorphisms under the concatenation operation of the set of words over a finite set. Changing the elements of the finite set does not essentially change the morphism. We propose a way to select a unique representing member out of all these morphisms. This has applications to the classification of the shift dynamical systems generated by morphisms. In a similar way, we propose the selection of a representing sequence out of the class of symbolic sequences over an alphabet of fixed cardinality. Both methods are useful for the storing of symbolic sequences in databases, like The On-Line Encyclopedia of Integer Sequences. We illustrate our proposals with the $k$-symbol Fibonacci sequences.