  Gessel's fundamental and Stembridge's peak functions are the generating
functions for (enriched) $P$-partitions on labelled chains. They are also the
bases of two significant subalgebras of formal power series, respectively the
ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao
introduced the monomial peak functions, a basis of the algebra of peaks indexed
by odd integer compositions whose relation to peak functions mimics the one
between the monomial and fundamental bases of QSym. We show that the extension
of monomial peaks to any composition is a new basis of QSym and generalise
Hsiao's results including the product rule. To this end we introduce a weighted
variant of posets and study their generating functions.
