CONCLUSION

AOP is characterized by its insights and the simplicity of its concepts, notations, and mathematical operations. It is a multi-valued multi-operational switching algebra and it is a generalization to the formal generalizations of binary and multi-valued switching algebras. We have shown that AOP in a z-radix system has z! binary operations called prioritors, has z! conservative unary operators and has z2(z-1) unary orthogonal operators.  Further, we have shown: (1) the development of AOP from the priority concept and principle; (2) the TAS systems of AOP; (3) the intrinsic and extrinsic theorems AOP; (4)  the advanced theorems of AOP: the image-scaling theorem, uniform degeneracy theorem, orthogonal theorem I, orthogonal theorem II, expansion theorem I, expansion theorem II; (5) (6) the proofs of the basic and advanced theorems of AOP; (4) the prioritors of the binary, ternary, and quaternary systems; (5) that Boolean and Post algebras are special cases of AOP. 

Furthermore, we have shown: (1) how the uniform degeneracy theory of AOP extended the duality theory used by Boolean and Post algebras; (2) how a MVL equation can be degenerated into z! equations; (3) how the uniform degeneracy operation replaced the "dual" operation used by Boolean and Post algebras; (4) how the orthogonal theorems I & II of AOP extended the representations of MVL functions from two representations (sum-of-products and product-of-sums) to z! representations; (5) how the expansions theorems I&II of AOP extended the expansion of MVL functions from two expansions to z! expansions; (6) how the image-scaling theorem of AOP replaced DeMorgan's laws; (7) how the absorption theorem-III‎ of AOP replaced Kleene's laws; (8) how Boolean, Post and Kleenean algebras are special cases of AOP;  (9) how AOP reduces MVL circuits complexity.  

Expectations

Multiple-Operational Logic (MOL) is a new area that uses multiple-operators from unary and binary operators to design digital circuits. It is aimed on introducing, into logical systems, a variety of new operators that will make design more flexible than would be using just the MVL traditional operators.  AOP is just a starting point in this field. AOP opens a new wide area for research.  The large number of prioritors in various radii needs to be investigated more in terms of their use in digital circuits. If researchers get interested in this field, then they can work toward the means that will develop the concepts of the MOL field as they did for the field of MVL.

There are many directions that researchers can seek depending on their interests and skills. Here are some:

1. Concentrate on "prioritors" and "conservative operators". Prioritors as operators are not important but what is important about them is how do we combine them to get digital circuits that can be applied in ourlives.  For example, look at the AND, OR and NOT operators.  The power they gain is from the various efforts done by researchers who combined these circuits to design gates, flip-flops, registers, counters, comparators, arithmetic units, microprocessors, controllers, computers, ...etc.  

   1. Under this direction there are few sub-directions and here are some:

      1. Researchers may try to develop a representation theory that does not use unary operators at all. 

      2. Researchers may try to develop a scheme to represent 2-variable functions by prioritors and conservative operators and without using orthogonal operators.  If they have to be used they should be few. 

2. Concentrates on the classifications of the binary and unary operators of a z-radix system.  There are sets of operators that have common algebraic behavior.  The study of one operator or a few in the set can be generalized for the entire set. This will simplify the study of binary operators
   

3. AOP can have a great impact on artificial intelligence.  Humans have the ability to change their priorities on the same events that they have to execute in their daily lives. "Intelligence" under the priority concept can be viewed as a well organized priority-schemes taken  at different times to set the execution order of events in a productive manner. The solution of a problem requires some events that has to be executed.  However, in the course of execution, new information may surfaces and requires a change to a different priority scheme. Try to find applications in this area for AOP by using programmable prioritors that change their priorities during the course of execution. 

There are too many problems in this field that need to be investigated.  Researchers should concentrates on all the binary operators of a z-radix digital system. For example, the ternary system has 19,683 binary operators. The quaternary system has 4,294,967,296 binary operators. Prioritors in the ternary system are 6 and in quaternary are 24.  Compare 6 to 19,683 and 24 to 4,294,967,296. This shows that our knowledge is so limited on both systems not to mention other high radix systems.