What is this property to be called?

Question

Two functions, $f$ and $g$, that satisfy the following identity: $$f(g(a_1,...,a_n), g(b_1,...,b_n),...) = g(f(a_1,b_1,....), f(a_2,b_2,...)...)$$ (notice the "transposition" of the arguments), do they possess a named and/or known property? Or as an alternative, where can I find out more about it?

(Apologies if I asked a duplicate question, but I didn't know how to search for it)

Answer 1

Wikipedia calls it "$f$ and $g$ commute" for multivariate functions:

The notion of commutation also finds an interesting generalization in the multivariate case; a function $f$ of arity $n$ is said to commute with a function $g$ of arity $m$ if $f$ is a homomorphism preserving $g$, and vice versa i.e.: $${f\big(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm})\big)=g\big(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})\big)} $$

The given reference is Universal Algebra: Fundamentals and Selected Topics by Clifford Bergman.

Answer 2

In section 5 of https://www.google.at/url?sa=t&source=web&rct=j&url=https://www.imbs.uci.edu/files/personnel/luce/2000/AczelFalmagneLuce_Mathematica%2520Japonica_2000.pdf&ved=2ahUKEwjS_cWRuqXcAhWNZ1AKHZ7BDGwQFjAEegQIAhAB&usg=AOvVaw3WLjcxovgzVUfeuIABNdBJ

you my find this equation of generalized bisymmetry.