We show that every function f: A × B → A × B, where |A| ≤ 3 and |B| < ω, can be represented as a composition f₁ ∘ f₂ ∘ f₃ ∘ f₄ of four axial functions, where f₁ 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 × B → A × B such that f ≠ f₁ ∘ f₂ ∘ f₃ ∘ f₄ for any axial functions f₁, f₂, f₃, f₄, whenever f₁ is a horizontal function.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-3,
author = {Krzysztof P\l otka},
title = {Composition of axial functions of products of finite sets},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {15-20},
zbl = {1121.03061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-3}
}
Krzysztof Płotka. Composition of axial functions of products of finite sets. Colloquium Mathematicae, Tome 107 (2007) pp. 15-20. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-3/