Absolutely annihilator-flat monoids (Q1849639)

For elements \(s,t\) of a monoid \(S\), a left annihilator ideal \(L_{s,t}\) is defined as \(\{u\in S\mid us=ut\}\). A right \(S\)-act \(A_S\) is called `annihilator-flat', if the natural mapping \(A\otimes L_{s,t}\to A\) is injective for each \(s,t\in S\). A monoid is called `right absolutely annihilator-flat' (r.a.a-f.) if all its right acts are annihilator-flat. It is proved that a monoid \(S\) is r.a.a-f. if and only if, for all \(u,v,s,t\in S\), if \(u,v\in L_{s,t}\), then there exist \(e_1,\dots,e_{2m+1}\in L_{s,t}\) such that \(u=ue_1\), \(ve_1=ve_2\), \(ue_2=ue_3\),\dots, \(ue_{2m}=ue_{2m+1}\), \(ve_{2m+1}=v\). A structural characterization is given of the r.a.a-f. completely (0-)simple semigroups and of the r.a.a-f. normal bands (with 1 adjoined). It is also proved that all full transformation monoids are both left and right absolutely annihilator-flat.