Transformation semigroup diagram and the \({\mathcal J}\)-relation (Q1318957)

A semigroup \(S\) of one-to-one partial transformations on a nonempty set \(X\) is said to be \({\mathcal G}_ X\)-normal if \(h^{-1}Sh \subseteq S\) for each bijection \(h\) on \(X\). The domain and range of a transformation \(f \in S\) are denoted by \(D(f)\) and \(R(f)\) respectively. Let \(\nabla f = | X\setminus D(f)|\) and \(\Delta f = | X \setminus R(f)|\). The semigroups under investigation here are those \({\mathcal G}_ X\)-normal semigroups where \(\nabla f\) and \(\Delta f\) are both infinite and \(| R(f)| = | X|\) for all \(f \in S\). The diagram referred to in the title is the set \(G(S) = \{(\nabla f,\Delta f): f \in S\}\). An equivalence relation \({\mathcal J}_ G\) on \(G(S)\) is defined by \((\alpha_ 1,\beta_ 1){\mathcal J}_ G(\alpha_ 2,\beta_ 2)\) if for all \(f,g \in S\) with \(\nabla f = \alpha_ 1\), \(\Delta f = \beta_ 1\), \(\nabla g = \alpha_ 2\) and \(\Delta g = \beta_ 2\), one has \(f{\mathcal J} g\) where, as usual, \(f {\mathcal J}g\) means that \(f\) and \(g\) generate the same principal ideal. The problem of describing the \(\mathcal J\)-relation on \(S\) is reduced to that of describing the \({\mathcal J}_ G\)-relation on \(G(S)\) and the author does this.