1-AOP Versus Boolean Algebra
Boolean algebra is a perfect algebra.  It completely describes the binary system.  AOP did not reveal new information on Binary system that Boolean algebra did not.  All results obtained for the binary system by AOP  are identical to that of Boolean algebra.  There is no difference between these two algebras at the binary system level.  However, they are different in none binary systems where Boolean algebras does not work for none binary systems but AOP does.  Also, we can derive Boolean algebra from AOP but we can not derive AOP from Boolean algebra.

 2-AOP Versus Post Algebra

There are similarities and differences between  AOP and Post algebras.  The differences between AOP and Post prove that AOP is not same as Post and because of these differences, we can not derive AOP from Post. On the other hand, we can derive Post algebra from AOP.  The differences between AOP and Post exist on the followings levels: Operators Theorems Concepts Notations.

Operators Differences

The development of AOP was not sought to enrich the literature of mathematical science.  In my analysis to the complexity of MVL circuits and on how we can reduce such complexity, I  proposed a solution which states that " to reduce circuit complexity of MVL circuits we should increase the basic operators of design and not rely on the classical operators (MIN & MAX)".  Based on this solution, I developed the set of unary and binary operators (prioritors, conservative operators, orthogonal operators) to serve as the BASIC SET OF DESIGN. The BINARY OPERATORS SET, which is prioritors, has a huge number of operators equal to z!.  For example, this number is 40,320 operators in the octal system.  Since, the design of any MVL digital circuit is going to be achieved by composing these prioritors into various configurations, there has to be a mathematical tool which describes these operators and show how to compose such circuits and how to minimize them.  It is clear that Boolean algebra can not be selected as a tool for this purpose.  Also, Post algebra can not be selected, except for limited cases, for this purpose for two reasons: firstly functional reason and secondly notational reason. The first reason is the most important reason because there are too many operators that Post algebra is incapable to handle.  Here is a simple example: when we run in the expression  (A+B)¯ where "¯" means the image of  Max(A,B) under MV-NOT operator, we simply say, by the inherited DeMorgan's law in Post, this expression can be broken as A¯ *B¯, but how Post will handle this operation when the image operation is not taken based on "MV-NOT operator but based on different operators obtained from one-to-one functions?.  Here is a practical example: the following design of the ternary multiplication by AOP is composed from three prioritors labeled a, b, and m with two conservative unary operators labeled “f” and “y” and it written as  A*B= (AaB^¾f)m (A^{¾ y} b B). There are various prioritors and conservative operators that can compose this circuit.  For example,  a=T1, b=T4, m=T2, f=T4 and Y=T3. Assume, we are going to simplify this equation by Post.  How are we going to go about this with the followings facts: T1 and T4 do not satisfy Post's axioms; T1 & T2 do not satisfy Post's axioms, the conservative operators T4 and T3 are not the same as MV-NOT, and thus DeMorgan's laws cannot be used. Here is a simplification by AOP to this circuit where we reduce the number of conservative operators to one instead of two: A*B= (AmB)a(AaB)^¾f   where a=T1, m=T2, and f=T4.  It is obvious that Post cannot handle this equation and get this reduction.

In summary, the operators domain of AOP is different from that of Post and the operators domain of Post is always a subset of the operators domain of AOP.  So, Post is a special case of AOP.  This is like saying, the variables domain of Post is different from that of Boolean algebra and the variables domain of Boolean algebra is always a subset of the variables domain of Post. So, Boolean algebra cannot work for the entire variable domain of Post.  In a similar way, we say Post cannot work for the entire operators domain of AOP.

Note: prioritors at the electronic implementation level are equivalent in hardware structure.  Thus there is no cost differences between different prioritors.  They all have the same number of transistors, resistors, diodes ...etc.  They only differ in internal connections which result in different functions.  View this as RAM.  No matter what data you store in the RAM,  it still has the same structure but with different data for various RAM of similar types.

Theorems Differences

Due to the large domain of AOP operators, it is natural to have theorems for those in AOP domain that do not lie in Post Domain.  Three are too many theorems that exist in AOP and do not exist in Post. Here are the major theorems of AOP that do not exist in Post and cannot be derived from Post.

• AOP Uniform Image-Scaling Theorem
• AOP Orthogonal Theorem-II
• Local theorems: ex. AOP Virtual theorems
• AOP Uniform Degeneracy theorems
• AOP none-uniform Image-Scaling Theorem
• AOP none-uniform Degeneracy Theorems 

AOP Uniform Image-Scaling (UIS) Theorem is one of the most powerful theorems in AOP and it replaces DeMorgan's Laws.  The UIS theorem simply breaks the image of binary operations under any conservative unary operator into three components as    (AaB)¯^f=A¯^f a¯^f B¯^f where "f" is a conservative operator, 'a' is a prioritor and the image operation acting on 'a' is applied to it priority-assignment . This break up is useful when this binary operation exist in equations that need to be simplified or minimized.  Post algebra does not have a similar theorem that operates for any image operation done by a one-to-one function.  If we use functional notation in mathematics, which is not a good practice, we would write this theorem as f(AaB)=f(A) f(a) f(B).

AOP Orthogonal Theorem-II does not exist in Post and cannot be derived from Post algebra because its orthogonal operators do not satisfy Post conditions of "complementation functions".  Also, this theorem has  better representations to MVL functions than the Post representations.  This theorem also offers z! representations for any function.

AOP Uniform Degeneracy theorem is also one of the most powerful theorems in AOP and it replaces the concepts of duality in Post algebra. Duality means that a MVL equation can have two different forms with different operators and constants but with the same variables. AOP uniform degeneracy theorem states that every MVL equation can have Z! different forms with different operators and constants but with the same variables.  Unlike AOP, Post algebra follows the same steps of Boolean and says that every MVL equations has two different forms. 

AOP none-uniform Image-Scaling Theorem and AOP none-uniform Degeneracy Theorem are completely beyond the scope of this comparison.  But we just listed them without the need to present what are these theorems.  They follow exactly the theorems of uniform degeneracy.

Concepts Differences

Due to the limitations imposed by logic concept on our world, limited to 'true' and 'false', people tried to find middle states between these two states and developed the term "Multiple-Valued Logic" meaning a logic with many logic-values.  What are these logical values in the world of logic?.  Are they '0', '1', '2'... etc.?  Are they 'true', 'false', 'half-true', 'half-false' ..etc.? The term "Multiple-Valued Logic" has no existence in the logical world but the term is used in a mathematical sense rather than in a logical sense.  The term "Multiple-Valued Logic" sounds as a "gray-scaled" logic. It views our world as a black and white but with different degrees. Post algebra picks on this term and provides a mathematical tool to work with multiple-valued logic systems. The algebra does not have a natural concept, as the case in Boolean algebra, to derive its operators and theorems from. The algebra is designed based on a limited set of axioms.  I would characterize Post algebra as man-made flowers.  They have the look of flowers but they do not have the natural smell and feeling of real flowers.  On the other hand, AOP is an algebra that has a solid concept from which AOP derived its operators and theorems. There are no axioms that exist in AOP. Only its priority concept represents the keystone of AOP.  The operators of AOP are naturally derived from this single concept.  Also, all the theorems for AOP are proved by this concept.  There is no axioms restrictions on AOP.  It applies the priority concept to MVL systems and reveals what is found about these systems.  Based on this concept, AOP discovered many facts about MVL system that Post could not do.  For example, 

• AOP discovered that every z-radix system has a basic set of operators called prioritors whose number depends on system radix and is given by z!.  
• AOP discovered that any MVL equation can have z! distinct forms.  
• AOP discovered that there exist z! representations for any MVL functions.  
• AOP discovered that there exist z! expansions for any MVL functions.  
• AOP discovered its Image-Scaling theorem to replace DeMorgan's Laws
• AOP discovered its absorptions theorem-III to replace Kleene's laws 

In AOP, all operations and theorems rely on the priority concept which is a common sense to man, animals, birds, insects, ...etc everywhere on this planet.  It is a concept that all livings work by.  When you feel hungry, the first priority is to go and eat.  When your home is burning the first priority is to rescue your family not go to the movies or read the newspapers. When an animal is attacked the priority is to defend itself. So, the priority concept is a universal and global concept that AOP did not found, designed or create but used as a natural resource to build and design digital systems based on this concept. If God implemented this concept in all of his creations, so why not use it in man-made machines and hope one day to achieve real smart-thinking machines.  Also, AOP is a very useful tool in artificial intelligence because its prioritors can be programmed to have different priority schemes.  These priorities can be triggered (selected) by environmental factors.