On some transformation theorems in many-valued logical systems

Since the well-known DeMorgan theorem in two-valued logical systems may be considered as a special transformation theorem, we are interested in finding its counterpart in many-valued logical systems. Some basic concepts and theorems regarding the existence of duality and N-ality relationships due to transformations in many-valued logical systems are introduced in terms of isomorphism. On the basis of these concepts and theorems, a number of special transformation theorems involving some practical logical operations are developed. Several aspects of the generalization of the DeMorgan theorem to transformation theorems are presented. It is shown that biuniqueness, monotonicity and simplicity are three significant properties of transformations in many-valued logical systems. In terms of the concept of isomorphism, transformations even with composite ones including those of two or more variables such as from two variables to two variables may be studied rather conveniently. Functional relationships involving transformations may exist not only in duality, but also in N-ality, in many-valued logical systems. It can be seen that transformations are important in the study of equivalence, interrelations and conversions between logical functions or operations and are useful in both analysis and synthesis of logical systems.