  This paper develops the theory of enriched toric $[\vec{D}]$-partitions.
Whereas Stembridge's enriched $P$-partitions give rises to the peak algebra
which is a subring of the ring of quasi-symmetric functions $\text{QSym}$, our
enriched toric $[\vec{D}]$-partitions will generate the cyclic peak algebra
which is a subring of cyclic quasi-symmetric functions $\text{cQSym}$. In the
same manner as the peak set of linear permutations appears when considering
enriched $P$-partitions, the cyclic peak set of cyclic permutations plays an
important role in our theory. The associated order polynomial is discussed
based on this framework.
