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  and first paper  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
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.