4
$\begingroup$

Let $$f : X_0 \rightarrow Y_0, \;\;\; g:X_1 \rightarrow Y_1$$ and define that the "direct product" of $f$ and $g$ is a map $$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times Y_1), \mbox{ such that } (f \otimes g)(x_0,x_1)=(f(x_0),g(x_1)).$$

Question 1. What is the standard terminology/notation for this concept?

Now given a function $h$, it's possible that $h$ has the property that there exist $f$ and $g$ such that $$h = f \otimes g$$ Question 2. What is the name of this property?

$\endgroup$
4
  • 1
    $\begingroup$ What context are you working in? In additive categories I would denote this $\begin{pmatrix}f&0\\0&g\end{pmatrix}$ $\endgroup$ Sep 6, 2012 at 11:31
  • 4
    $\begingroup$ I know this as the "product of two functions", since it is just the result of applying the product functor $- \times -: Set \times Set \to Set$ to a pair of functions/morphisms. $\endgroup$
    – Todd Trimble
    Sep 6, 2012 at 11:49
  • 2
    $\begingroup$ This condition comes up often enough for it really to deserve a name. I never found a standard one and so have used the phrase "$h$ splits multiplicatively" in publications. Despite its apparent simplicity it is useful to have criteria which one can use to determine if $h$ has this property. Two very straightforward ones which are valid under suitable assumptions are: a) for all pairs $x_0$, $x_1$ and $y_0$, $y_1$ we have: $h(x_0,x_0)h(x_1,y_1)=h(x_0,y_1)h(x_1,y_0)$. b) $h h_{xy} = h_x h_y$ (this for smooth functions). $\endgroup$
    – jbc
    Sep 6, 2012 at 13:22
  • $\begingroup$ It is Cartesian product of two functions: en.wikipedia.org/wiki/… $\endgroup$
    – porton
    Feb 15, 2020 at 1:48

2 Answers 2

2
$\begingroup$

I'd say that a map $F:\prod_{i=1}^n X_i \to \prod_{i=1}^n Y_i$ is diagonal if there are functions $f_i:X_i\to Y_i$ such that $$ F(x_1,\ldots,x_n) = \bigl( f_1(x_1),\ldots,f_n(x_n) \bigr). $$ I've also seen the map $F$ called triangular if there are maps $f_i:\prod_{j=1}^i X_j\to Y_i$ such that $$ F(x_1,\ldots,x_n) = \bigl( f_1(x_1),f_2(x_1,x_2),f_3(x_1,x_2,x_3)\ldots,f_n(x_1,\ldots,x_n) \bigr). $$ The terminology comes from the fact that if $F$ is a linear map $F:K^n\to K^n$, then $F$ is given by a diagonal, respectively (upper) triangular, matrix if the map $F$ is diagonal, respectively triangular. In the case that $Y_i=X_i$, both diagonal maps and triangular maps have been studied in dynamical systems.

$\endgroup$
0
$\begingroup$

Question 1. It is Cartesian product of two functions, see Wikipedia.

Question 2. I think the second property may be called "two variables don't interfere", however I am unsure whether this phrase is well defined in mathematics.

$\endgroup$
1
  • $\begingroup$ I think it is more common to refer to separation of variables than to non-interfering variables. $\endgroup$
    – LSpice
    Feb 15, 2020 at 2:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.