For a binary operation [] on labeled or colored graphs, and a graph parameter f, we define the Hankel matrix H(f, []) with

We extend any binary operation [] to quantum graphs by

In the following example, we give some normed binary operations [?

The follwing lemma defines a normed binary operation exploting some properties of a self map on [0, [infinity]).

Immediate computations show that if [phi][member of] End Q, then [phi] preserves also the binary operations \ and /.

From Definition 7 it follows that the binary operation x is commutative and there exists the [mapping.

A quantale is a complete lattice Q with an associative binary operation "&" satisfying:

Let Q be a Girard quantale with cyclic dualizing element d and a, b [member of] Q, define the binary operation "[parallel]" by a[parallel]b = ([a.

We can still define a pairing as in the case of a

binary operation but now it is a map

In her book [16] and first paper [17] on Smarandache concept in loops, she defined a Smarandache loop as a loop with at least a subloop which forms a subgroup under the

binary operation of the loop.

For any order of these double binary operations in O([?

As we known, a set R with two binary operation "+" and "[?

Not loss of generality, assume the order of

binary operations in [?

We have defined Smarandache quasigroup rings which are again non-associative structures having two

binary operations.

defines a

binary operation on the set of closed intervals.