Recently, many combinatorial notions have been found to have infinite partition and almost disjoint properties, which state that in a given semigroup of cardinality \(\kappa \) , these notions can be partitioned into \(\kappa \) subsets with the same property; moreover, if \(\kappa \) contains \(\delta \) almost disjoint subsets, then the notion contains \(\delta \) almost disjoint subsets of that type. In this paper, we investigate two notions: C-sets and J-sets in non-commutative semigroups. We obtain that in some non-commutative structures, such as free semigroups, free groups, general linear groups, and the Heisenberg group, C-sets and J-sets still have infinite partition and almost disjoint properties.
