On semigroups in which each left or right translation is idempotent (Q2739049)

Let \(T\) be a semigroup. For \(x\in T\), take a set \(Z_x\) such that \(Z_x\cap T=\{x\}\) and \(Z_x\cap Z_y=\emptyset\), \(\forall x,y\in T\), \(x\not=y\). Let \(S=\bigcup\{Z_x\mid x\in T\}\). Then \((S,*)\) is a semigroup, called the inflation of \(T\), with respect to the multiplication ``\(*\)'' defined by: \(a*b=xy\) for \(a\in Z_x\), \(b\in Z_y\). The main result of this paper is: Let \(S\) be a semigroup. Every left or right translation is idempotent if and only if \(S\) satisfies (1) \(E(S)\) is a band; (2) \(S=\{Z_e\mid e\in E(S)\}\) is a translation of \(E(S)\); (3) Every left or right translation of \(E(S)\) is idempotent and \(|Z_e=\{ x\in S\mid x^2=e\}|\leq 2\), \(\forall e\in E(S)\); (4) Let \(|Z_e|=2\), \(\forall e\in E(S)\). If for any \(f\in E(S)\setminus\{e\}\), \(\lambda_f=\lambda_{fe}\) or \(\lambda_e=\lambda_{ef}\) or \(\rho_f=\rho_{ef}\) or \(\rho_e=\rho_{fe}\), then \(|Z_f|=1\). The authors also obtain four corollaries for some special cases.