  We prove fixed point theorems in a space with a distance function that takes
values in a partially ordered monoid. On the one hand, such an approach allows
one to generalize some fixed point theorems in a broad class of spaces,
including metric and uniform spaces. On the other hand, compared to the
so-called cone metric spaces and $K$-metric spaces, we do not require that the
distance function range has a linear structure. We also consider several
applications of the obtained fixed point theorems. In particular, we consider
the questions of the existence of solutions of the Fredholm integral equation
in $L$-spaces.
