  This paper is devoted to give all the technical constructions and definitions
that will lead to the construction of an algorithm of resolution of
singularities for binomial ideals.
  We construct a resolution function that will provide a resolution of
singularities for binomial ideals, over a field of arbitrary characteristic.
For us, a binomial ideal means an ideal generated by binomial equations without
any restriction, including monomials and $p$-th powers, where $p$ is the
characteristic of the base field.
  This resolution function is based in a modified order function, called
$E$-order. The $E$-order of a binomial ideal is the order of the ideal along a
normal crossing divisor $E$.
  The resolution function allows us to construct an algorithm of
$E$-\emph{resolution of binomial basic objects}, that will be a subroutine of
the main resolution algorithm.
