Green's relations and regularity for semigroups of transformations that preserve double direction equivalence.
For a set \(X\) and an equivalence relation \(E\) on \(X\) the authors study the subsemigroup \(T_E(X)\) of all full transformations \(f\) on \(X\) satisfying the condition that for any two elements \(x,y\in X\) we have \((x,y)\in E\) if and only if \((f(x),f(y))\in E\). The authors describe Green's relations for \(T_E(X)\) in combinatorial terms and discuss regularity of elements in \(T_E(X)\). In particular, it is shown that \(T_E(X)\) is regular if and only if \(|X/E|\) is finite.
scientific article; zbMATH DE number 5730952