The paper presents results of the algebraic dimension theory for exponential analytic sets, which, by definition are the set of common zeros of a finite number of entire functions each of which being of the form \[ \sum c_\lambda\exp\langle z,\lambda\rangle, \] where \(c_\lambda\in\mathbb{C}\) and \(\lambda\) runs over a finite subset \(\Lambda\) of the exponent space (i.e., the conjugate space \(\mathbb{C}^{n*})\).