Morphisms, Symbolic Sequences, and Their Standard Forms

Abstract

Morphisms are homomorphisms under the concatenation operation of the set of words over a finite alphabet. Changing the elements of the finite alphabet does not change the morphism in an essential way. We propose a method to select a unique representative from 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, such as The On-Line Encyclopedia of Integer Sequences. We illustrate our proposals with the k-symbol Fibonacci sequences.