Bi-resolving graph homomorphisms and extensions of bi-closing codes

TL;DR

This paper explores the extension of bi-resolving graph homomorphisms to bi-covering ones and applies these findings to extend bi-closing codes between subshifts to n-to-1 codes between irreducible shifts of finite type.

Contribution

It provides new sufficient conditions for extending bi-resolving homomorphisms to bi-covering extensions with irreducible domains and applies this to extend bi-closing codes.

Findings

Bi-resolving homomorphisms can be extended to bi-covering homomorphisms under certain conditions.

Bi-closing codes between subshifts can be extended to n-to-1 codes for large n.

Extension results hold for irreducible shifts of finite type.

Abstract

Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving homomorphisms and give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, we prove that a bi-closing code between subshifts can be extended to an n-to-1 code between irreducible shifts of finite type for all large n.