Inequalities connected with averaging operators (Q1946800)

Let \(\mathcal{R}\) be a partially-ordered ring and, as usual, denote \(\mathcal{R}^{\mathcal{R}}\) the set of all mappings from \(\mathcal{R}\) inti itself. Put \[ A=\left\{ T\in\mathcal{R}^{\mathcal{R}}:T\left( f+Tg\right) \geq Tf+Tg\text{ for all }f,g\in\mathcal{R}\right\} \] and \[ M=\left\{ T\in\mathcal{R}^{\mathcal{R}}:T\left( f.Tg\right) \geq Tf.Tg\text{ for all }f,g\in\mathcal{R}\right\} . \] The paper under review instigates sets \(A\) and \(M\). A typical result states that if \(T\in A\) is onto, then \[ Tf=f+T\left( 0\right) \text{ for all }f\in\mathcal{R}. \] For further facts, the reader can take a look at this interesting article.