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Abstract

We show that every function $f: A \times B \to A \times B$, where $|A|\le 3$ and $|B|< \omega $, can be represented as a composition $f_1 \circ f_2 \circ f_3 \circ f_4 $ of four axial functions, where $f_1$ is a vertical function. We also prove that for every finite set $A$ of cardinality at least 3, there exist a finite set $B$ and a function $f: A \times B \to A \times B$ such that $f\not =f_1 \circ f_2 \circ f_3 \circ f_4$ for any axial functions $f_1, f_2, f_3, f_4$, whenever $f_1$ is a horizontal function.