Intereting Posts

Does the nontrivial commutants of the Volterra operator admit a strictly continuous spectrum?
Is my solution for proving $3^n > n^2$ using induction correct?
Row vector vs. Column vector
Power sums over additive subgroups of finite fields
Uniqueness of meets and joins in posets
Topological Property of the Set of Integers
Why is $n$ divided by $n$ equal to $1$?
Definite integral using the method of residues
Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?
What's the geometrical interpretation of the magnitude of gradient generally?
Convergence and uniform convergence of a sequence of functions
Every section of a measurable set is measurable? (in the product sigma-algebra)
How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$
Homomorphsim from a finite group into a divisible abelian group.
Solving $\sin(5x) = \sin(x)$

**Foreword:**

I have read R.H. Bruck’s *A Survey of binary systems*, where the notion of **halfoperation** is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined for all ordered pairs $(a,b)$. For example, from the Cayley table

$$\begin{array}{c|cc}\ast & 0 & 1\\ \hline 0 & 0 & 1\\1 & & 0\end{array}$$

- Can the cubic be solved this way?
- Is duality an exact functor on Banach spaces or Hilbert spaces?
- Homomorphisms of graded modules
- Non-associative, non-commutative binary operation with a identity
- Modules over commutative rings
- A question about the proof that $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic

we deduce that $0\ast 1=1$ while $1\ast 0$ is not defined ($(0,1)$ belongs to the *range* of the halfoperation, $(1,0)$ does not).

Let us now consider two well known operations over $\mathbb{R}^3$:

*vector product*: ${\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3}\qquad{(a,b)\mapsto a\wedge b}$,*scalar product*: ${\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}}\qquad{(a,b)\mapsto a\cdot b}$.

In italian (my native language) we call both of them binary operation:

*operazione binaria interna*one like vector product (*interna*: “inner” or “internal”),*operazione binaria esterna*one like scalar product (*esterna*: “outer” or “external”).

**Question:**

Does such distinction exist in english? I found no traces of it in literature and, reading papers, forums and SE, I have experienced that this matter may sometimes give raise to ambiguities or misunderstanding.

How should we properly call an operation like subtraction $-$ over $\mathbb{N}$, since $a-b$ only belongs to $\mathbb{N}$ if $a\ge b$? Halfoperation over $\mathbb{N}$?

How should we properly call an operation like scalar product $\cdot$ over $\mathbb{R}^3$, since $a\cdot b$ never belongs to $\mathbb{R}^3$? Is this not a binary operation while vector product is?

**And, above all:**

In the end the word *binary*, alone, means that the operation is a function of two variables (its arity is $2$), but does not deal with closure…

…Right?

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