A note on amalgams of inverse semigroups (Q2726650)

Let \(I(X)\) be the inverse monoid of all partial bijections of a set \(X\). An action \(\varphi\colon I\to I(X)\) of an inverse monoid \(I\) on \(X\) is called graded if there exists a function \(p\colon X\to E(I)\) (\(E(I)\) the set of idempotents of \(I\)) such that \(\text{dom}(\varphi(e))=p^{-1}([e])\) and if \(t^{-1}t=p(x)\), then \(tt^{-1}=p(tx)\), where \([e]=\{f\in E(I),\;f\leq e\}\) and \(\varphi(t)x\) is denoted as \(tx\), \(t\in I\), \(x\in X\). Graded actions are used to find a condition for an inverse monoid to be a full amalgam of inverse semigroups; essentially all full amalgams are obtained in this way.