  The most powerful formulation of the Central Sets Theorem in an arbitrary
semigroup was proved in the work of De, Hindman, and Strauss. The sets which
satisfy the conclusion of the above Central Sets Theorem are called $C$-sets.
The original Central Sets Theorem was extended by J. McLeod for adequate
commutative partial semigroups. In this work, we will extend the Central Sets
Theorem obtained by taking all possible adequate sequences in a commutative
adequate partial semigroup. We shall also discuss a sufficient condition for
being a set $C$-set in our context.
