An abelian group (G, *) is a group where the binary operation
is also commutative.
A binary operation
[??]: [0, 1] x [0, 1] [right arrow] [0, 1] is said to be continuous t-conorm (s-norm) if [??] satisfies the following conditions:
A complete lattice [Q.sub.t] having associative binary operation
* is called a quantale if it satisfies
The improvement comes in part from the use of parallel and binary operations
, as well as our fast BCD additions.
If we define the binary operation
"*" on U as [a.sub.i] * [a.sub.j] = [alpha][a.sub.ti + [[alpha].sup.2][a.sub.j]([alpha] + [[alpha].sup.2] = 1), then this algebra satisfies (10) but [a.sub.i] * [a.sub.i] = [a.sub.i].
Let Y be a nonempty set and * be a binary operation
on Y such that (Y, *) is a semigroup, and suppose that X [subset] Y is a set of plaintexts.
If we denote [A.sub.[alpha]] = [[a.sub.1[alpha]]; [a.sub.2[alpha]]] and [B.sub.[alpha]] = [[b.sub.1[alpha]]; [b.sub.2[alpha]]] then the extended binary operation
for increasing binary operation
For a binary operation
 on labeled or colored graphs, and a graph parameter f, we define the Hankel matrix H(f, ) with
Let * 6 (+, -, ., /) be a binary operation
on the set of real numbers.
In the following example, we give some normed binary operations
[??] on [0, [infinity]) x [0, [infinity]) with properties (PI) and P(II).
Because any Boolean function can be implemented in terms of binary operations
, in principle there is no reason why this would be impossible.
The mapping [phi]: Q [right arrow] Q is called an endomorphism of Q if [phi] preserves the binary operation
x, that is [phi](x x y) = [phi](x) x [phi](y) for all x, y [member of] Q.