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 * is

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.