  Skew monoidal categories are monoidal categories with non-invertible
`coherence' morphisms. As shown in a previous paper bialgebroids over a ring R
can be characterized as the closed skew monoidal structures on the category Mod
R in which the unit object is R. This offers a new approach to bialgebroids and
Hopf algebroids. Little is known about skew monoidal structures on general
categories. In the present paper we study the one-object case: skew monoidal
monoids (SMM). We show that they possess a dual pair of bialgebroids describing
the symmetries of the (co)module categories of the SMM. These bialgebroids are
submonoids of their own base and are rank 1 free over the base on the source
side. We give various equivalent definitions of SMM, study the structure of
their (co)module categories and discuss the possible closed and Hopf structures
on a SMM.
