Quantum partial automorphisms of finite graphs

TL;DR

This paper introduces the quantum analogue of the semigroup of partial automorphisms of finite graphs, expanding the framework of quantum symmetries to include partial automorphisms and exploring the infinite vertex case.

Contribution

It defines and develops the basic theory of quantum partial automorphism semigroups for finite graphs, extending quantum symmetry concepts beyond automorphism groups.

Findings

Quantum partial automorphism semigroup $ ilde{G}^+(X)$ is well-defined for finite graphs.

Contains both classical automorphism group and quantum automorphism group.

Extension to infinite graphs shows $ ilde{G}^+(X)$ remains well-defined, unlike $G^+(X)$.

Abstract

The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$, which contains the automorphism group $G(X)$. We discuss here the quantum analogue of this construction, with a definition and basic theory for the quantum semigroup of quantum partial automorphisms $\widetilde{G}^+(X)$, which contains both $G(X)$, and the quantum automorphism group $G^+(X)$. We comment as well on the case $N=\infty$, which is of particular interest, due to the fact that $\widetilde{G}^+(X)$ is well-defined, while its subgroup $G^+(X)$, not necessarily, at least with the currently known methods.