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This paper is devoted to give all the technical constructions and definitionsthat will lead to the construction of an algorithm of resolution ofsingularities for binomial ideals.  We construct a resolution function that will provide a resolution ofsingularities for binomial ideals, over a field of arbitrary characteristic.For us, a binomial ideal means an ideal generated by binomial equations withoutany restriction, including monomials and $p$-th powers, where $p$ is thecharacteristic 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 anormal 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 ofthe main resolution algorithm.

Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties