This book constitutes the proceedings of the 9th International Conference on Algebraic Informatics, CAI 2022, held as virtual event, in October 27-29, 2022. The 2 abstracts, 3 full papers of invited speakers, and 12 contributed papers presented in this volume were carefully reviewed and selected from 17 submissions. The papers contain original and unpublished research; the topics of them lie in automata theory, cryptography, coding theory, DNA computation, computer algebra, and theory of software architectures.