Summary: A new logic model is presented in this paper for subsets of \(R^n \times R^m\) known as \(n\)-input \(m\)-output \(r\)-valued multiple-valued logic (MVL) relations, where \(n, m >0\) and \(r >1\) are integers, and \(R = \{0,1,\dots, r-1\}\) is an enumeration of the finite ordered set \(\text E = \{e_0, e_1,\dots, e_{r-1}\}\). The model, called E2 systems (or shortly E2), represents an extension of an existing generalized cube representation for MVL relations called set functions. E2 systems consist of two components: logic implication (LI) systems, and logic equivalence (LE) systems. Some properties of the E2 systems are presented, and applying the model in cell-based combinatorial MVL circuit synthesis is discussed.