Multidirectional associative memory (MAM) enables associations among many items. Its architecture is very simple and provides high parallelism. However, recall results of MAMs are inconsistent for contradictory inputs. The recall results depend on the order of update. If a given input includes an incorrect pattern, we expect that the incorrect pattern will be corrected, and the recalled pattern is determined by the majority. In this work, we propose MAMs with self-connections and a new learning algorithm. These MAMs provide recall results independent of the order of update. Furthermore, they maintain the advantages of MAMs, such as their simple architecture, high parallelism and stability.