A New Set of Unary and Binary Operators 
With A New Algebraic System
For Multiple-Valued Logic Systems:
The Algebra Of Priority 
(AOP)

By
Abu-Msameh, R. K.

            The aim of this paper is to introduce a new set of unary and binary operators with their algebraic system that will allow the design of MVL digital circuits in a way that is more simpler and much more efficient than would be by the traditional operators of MVL systems. The algebra associated with these operators is called the algebra of priority (AOP).  It is a new multi-valued multi-operational switching algebra. This newly introduced algebra was developed based on the priority concept. This paper (1) presents the priority concept and principle; (2) presents the development of AOP based on the priority principle; (3) presents the new binary operators of AOP which are called "prioritors" for binary, ternary and quaternary systems;  (4) proves that the number of prioritors in a z-radix digital system is z!; (5) presents the basic intrinsic and extrinsic theorems of AOP; (6) presents the orthogonal theorem-I and II, which extends the Post representations of MVL functions from two representations (sum-of-products and product-of-sums) to z! representations; (7) presents the expansion theorem I and II, which extends the Post expansions of MVL functions from two expansions (sum-of-products and product-of-sums) to z! expansions (8) presents the  uniform image-scaling theorem which replaces DeMorgan's laws (9) presents the absorption theorem-III which replaces Kleene's laws. (10) presents the uniform degeneracy theory which replaces the duality theory; (11) shows how to derive Boolean, Post and Kleenean algebras from AOP; (12)  presents design examples using AOP.

Index Terms: multiple-valued logic, algebra of priority, prioritors, priority principle, quaternary system, STAS systems, ternary system, switching algebras, Post algebras, Kleenean algebra, function representations, degeneracy, representations.