Binary Operations

Contents	Notations, Infimum-Operation, Supremum Operation, Star Operation, Costar Operation
See also	TOC, unary operations
Abstract	Introduction	Priority	Unary	Prioritors	Operations
TAS	Theorems	Orthogonal	Expansion	Image-Scaling	Degeneracy
Design	Derive	AOP Versus	Tables	Figures	Proofs
Computer	Bibliography

AOP Binary operations are operations that operate on the prioritors of AOP. Some operate on the priority assignment and some operate on the prioritor function-table.

In Boolean and Post algebras we use the ‘+, Ú’ or ‘·, Ù’ symbols to stand for the OR (MAX) and AND (MIN) binary operators. AOP uses the Greek alphabets as symbols to stand for prioritors in its algebraic equations. In this paper, we use the a symbol to stand for prioritors in general.

Each prioritor has an infimum digit and a supremum-digit which are called the prioritor switches. The infimum digit (a switch to open) allows the data flow to pass through the output of its prioritor. The supremum digit (a switch to close) blocks the data flow out the prioritor. This physical process at the hardware level is expressed mathematically by AOP in the following definitions.

Definition 5.1.1 Infimum digit: The infimum digit of a prioritor operator, denoted by a^─Ú, is a digit in the logic-set such that a^─Ú a A =Aa a^─Ú =A.

The infimum operation of the a operator is read as “the infimum digit of a” where “^─Ú” is called the inferiority operator. Table-1

lists the infimum digit of all prioritors in the quaternary, ternary and binary digital systems under the 'a^─Ú' column. Using Table-1

, the infimum of a=QA is a^─Ú=0, of a=Q5 is a^─Ú=2, of a=QE is a^─Ú=1, of a=Q1 is a^─Ú=3, of a=QO is a^─Ú=0. In Boolean algebra, the “0” digit is the infimum digit of the OR operator (since 0+A=A). The “1” digit is the infimum digit of the AND operator (since 1·A=A).

Definition 5.1.2 Supremum digit: The supremum digit of a prioritor operator, denoted by a^─^L, is a digit in the logic-set such that a^─^LaA=Aaa^─^L =a^─^L

The supremum operation of the a operator is read as “the supremum digit of a” where “^─^L” is called the superiority operator. Table-1

lists the supremum digit of all prioritors in the quaternary, ternary and binary digital systems under the 'a^─^L' column. Using Table-1

, the supremum of a=QA is a^─^L=1; of a=Q5 is a^─^L=0; of a=QE is a^─^L=2; of a=Q1 is a^─^L=0; of a=QO is a^─^L=3. In Boolean algebra, the “1” digit is the supremum digit of the OR operator (since 1+A=1). The “0” digit is the supremum digit of the AND operator (since 0·A=0).

Star Operation on Prioritors

The star operation and the costar operation are the major unary operations in AOP that operate on the priority-assignments of prioritors. Table-1

lists the star under the a* column and the costar under the a# column of each priority-assignment expressed by the priority s-code and by Q's, T's and B's codes for the quaternary, ternary, and binary systems. These two operations are very important operations in digital applications of AOP. They are used by STAS systems of AOP.

When we apply the star operation on a priority assignment of a prioritor, we reverse (transpose) the order of its priorities. Thus, the star of a prioritor is a prioritor.

Example-4.1 On The Star Operation: In the binary system, for a=2S01 a*=2S10; a=2S10 a*=2S01. In the ternary system, for a=3S012 a*=3S210; a=3S021 a*=3S120; In the quaternary system, for a=4S0123, a*=4S3210; a=4S3210, a*=4S0123.

Costar Operation on Prioritors

The costar of a STAS system always corresponds to the NOT operator in Boolean, Post and Kleenean algebras. It is a very important operation in using prioritors to design MVL flip-flop circuits.

Example-4.2 On The Costar Operation: Let a=4S3021=QK. The a# is obtained by taking the image of its inverse by its star. Using Table-1

, a-=4S3102 and a*=4S1203=Q9 . Thus, the costar of a is a#=a-^─^a*=4S3102^─^4S1203=4S1032=Q8.