# Functions that can be written as direct products of other functions; question about terminology and notation

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?

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

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.