A nontrivial pairing of finite \(T_0\) spaces
A map of pointed spaces \(\mu:X\times Y\to Z\) is called a pairing with axes \(f:X\to Z\) and \(g:Y\to Z\) if \(\mu\) restricted to \(X\vee Y\) is homotopic to \(f\vee g\). This paper exhibits examples of nontrivial connected finite \(T_0\)-spaces \(X\) and \(Z\) that admit a pairing \(\mu:X\times X\to Z\) with axes \(f\) and \(f\) where \(f:X\to Z\) is a weak homotopy equivalence. The authors propose this concept as a good notion of \(H\)-space for finite spaces. They use it to exhibit finite space realizations of the product on \(S^1\), the Hopf map \(S^3\to S^2\), and the spaces \(\mathbb{R} P^2\) and \(\mathbb{C} P^2\).
scientific article; zbMATH DE number 1879144