On codes defined by bio-operations

We consider the classes of ⊕-codes and ⊗-codes, which are superclasses of outfix and hyper-codes, respectively. These restrictions are based on the synchronized insertion operation, which serves as a model for the gene rearrangement function in certain unicellular organisms. We investigate the classes of ⊕-codes and ⊗-codes from a theoretical perspective, examine their relationships with traditional code classes and consider related decidability problems.