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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?

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    $\begingroup$ What context are you working in? In additive categories I would denote this $\begin{pmatrix}f&0\\0&g\end{pmatrix}$ $\endgroup$ Commented Sep 6, 2012 at 11:31
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    $\begingroup$ 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
    Commented Sep 6, 2012 at 11:49
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    $\begingroup$ 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
    Commented Sep 6, 2012 at 13:22
  • $\begingroup$ It is Cartesian product of two functions: en.wikipedia.org/wiki/… $\endgroup$
    – porton
    Commented Feb 15, 2020 at 1:48

2 Answers 2

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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.

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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.

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

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