General Degeneracy Theory

The Degeneracy theory is completely independent of AOP.  The degeneracy theory is much like "Object Programming theory". The concepts of "objects", "inheritance", "ancestors", ..etc can be seen indirectly in the degeneracy theory.  Object programming theory deals with data and code.  The degeneracy theory deals with operators (like data) and equations (like code).

The degeneracy theory classifies operators  into two parts: "elements" (like ancestor-objects) and degenerate operators (like descendants).  Elements are the ancestors of degenerate operators.  Therefore, degenerate operators are derived from their elements and inherit all the properties of their elements (ancestors).  The part of degeneracy which deals with elements is called "GTODE" from (General Theory of Digital Elements).  For example, the ternary system has 19683 binary operators.  Based on GTODE, the number of elements for the ternary system is "139".  This means that the remaining "19533" operators of the ternary system are the descendants of their "139" ancestors and they inherit all of their properties.  This simplifies the study of operators and the developments of new algebras that describe their operations. See ternary system elements.

Also, degeneracy deals with equations in the same manner as it does with operators. Degeneracy organizes equations into families.  All the members of one family are degenerate from one equation that exist in the same family called the "parent equation".  The rest of the equations are called the "child-equations".  All child-equations are degenerate from their parent-equation and inherit the same properties.  The degeneracy generates the child-equations by applying its various degeneracy operators on the parent-equation.  These degeneracy operators are categorized into two parts: uniform-degeneracy operators and none-uniform degeneracy operators. Uniform degeneracy operators result in equations that are called "uniformly degenerate equations".  A one-parent equation can uniformly generate a maximum of z! child-equations and a minimum of zero child equation.  The same one-parent equation can generate none-uniformly degenerate equations.  Their numbers is very huge and it is directly proportional to powers of system z!, number for variables and number of binary operations in the equation.  

Duality is a very limited special case of degeneracy theory.  Duality states that for every parent-equation their is one child-equation and both the parent and child-equation form a family of "2" equations, which led researchers to call it duality (having two forms for one action).  Duality is also limited to certain operators.  For example, duality is restricted to AND and OR for Boolean algebra and to MIN and MAX for Post algebra.  However, degeneracy is for all binary operators.  For example, the "XOR", "ADD" and "SUB" operators have  degenerate operators and under the binary system we say, by the degeneracy theory, that they have dual operators too.

I still did not post any file on this web-site that discuss the degeneracy theory.  You can see part of the uniform degeneracy theory applied to AOP (Uniform Degeneracy In AOP). Also see Comments On Degeneracy

Degeneracy has very important benefits.  We can design many different circuits that do the same action because of degeneracy. For example, we subtract numbers by an adder because of degeneracy.  Degeneracy is what allows us to use the complement method to subtract numbers by addition.  This is because the "+" and "-" operators are degenerate operators.