Factorization of homotopies of nanophrases

TL;DR

This paper introduces a factorization method for homotopy data triples in nanophrases, enabling simplified computation of homotopy equivalences through prime factorization, and proves the uniqueness of this factorization.

Contribution

It defines a product on homotopy data triples and proves that any such triple can be uniquely factorized into prime triples, simplifying homotopy analysis.

Findings

Prime factorization of homotopy data triples is unique.

Homotopy equivalence can be computed using prime factors.

Factorization simplifies the analysis of nanophrase homotopies.

Abstract

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a homotopy data triple is composite, we show that equivalence of nanophrases under the corresponding homotopy can be calculated just by using the homotopies given by its prime factors.