A necessary and sufficient condition for Lukasiewicz logic functions

The literal, TSUM, min and max operations employed in multiple-valued logic design can be expressed in terms of the implication and the negation of Lukasiewicz logic. We can easily show that the set of multiple-valued functions composed of the above four operations and the negation is equivalent to the set of all multiple-valued functions composed of the Lukasiewicz implication and the negation. This implies that from the viewpoint of the multiple-valued logic design, Lukasiewicz multiple-valued logic is a fundamental system. In this paper, we clarify a necessary and sufficient condition for a multiple-valued function to be a Lukasiewicz logic function, which is defined as a function in terms of the Lukasiewicz implication and the negation