A Study on Groupoids, Ideals and Congruences via Cubic Sets

The inclusion, the intersection and the union between cubic sets are each defined in two ways. From this point of view, we introduce the concepts of cubic subgroupoids, cubic ideals, cubic subgroups, and cubic congruences as two types, respectively, and discuss their various properties. In particular, we give a relationship between the set of all cubic normal subgroups of a group and all cubic congruences on the group.