On diagonal acts of monoids (Q2716972)

For a monoid \(M\), the set \(M\times M\) (with natural multiplication) is called a diagonal \(M\)-act. The power monoid \({\mathcal P}_f(M)\) of \(M\) is defined as the set of all finite subsets of \(M\), under the multiplication \(AB=\{ab:a\in A,\;b\in B\}\). Let \(R_N\) be the monoid of all partial recursive functions in one variable. It is proved that \(R_N\) is finitely generated but not finitely presented. The paper gives an example of an infinite finitely presented monoid \(P\) such that \(P\times P\) is both a cyclic right and cyclic left \(P\)-act as well as an example of a monoid \(C\) for which \(C\times C\) is finitely generated as a right \(C\)-act but not finitely generated as a left \(C\)-act. Some links between diagonal and power acts are established. For example: if \({\mathcal P}_f(M)\) is finitely generated then \(M\times M\) is a finitely generated bi \(M\)-act.