On Computational Aspects of Finding Inverse Monoids of Partial Automorphisms (Logic, Algebraic system, Language and Related Areas in Computer Science)

Abstract

A partial automorphism of a combinatorial structure is an isomorphism between its two induced substructures. It is a natural generalization of a total automorphism which is a classical tool to study symmetries. As all total automorphisms of a combinatorial structure form a group, all partial automorphisms form an inverse monoid, called inverse monoid of partial automorphisms of a combinatorial structure. This monoid is a richer and more complex object that contains more information about the structure than its automorphism group used in the classical Group Theory. In our paper, we review the results we obtained for the inverse monoids of partial automorphisms of graphs in the study of questions analogous to those concerning automorphism groups of graphs. We also address some computational aspects of finding these inverse monoids.