Remarks on isomorphisms of regressive transformation semigroups (Q1916048)

Let \(X\) be a partially ordered set with the property that each element is comparable to at least one other element. Denote by \(T(X)\) the full transformation semigroup on \(X\) and let \(T_{RE}(X)=\{f\in T(X):f(x)\leq x\) for all \(x\in X\}\). Let \(U(A)=\{x\in X:x\geq a\}\) and \(SU(a)=\{x\in X:x>a\}\) and denote by \(\text{Min}(X)\) the collection of all minimal elements of \(X\). Let \(\leq^e\) be the equivalence relation generated by \(\leq\). A subset \(Y\) of \(X\) is said to be connected if \(x\leq^e y\) for all \(x,y\in Y\). A partially ordered set \(X\) is defined to be adjusted if either \(\text{Min}(X)=\emptyset\) or \(\text{Min}(X)\) is connected. The authors show that for each partially ordered set \(X\), there exists an adjusted partially ordered set \(Y\) such that the semigroups \(T_{RE}(X)\) and \(T_{RE}(Y)\) are isomorphic. They go on to show that if \(X\) and \(Y\) are adjusted partially ordered sets, then the semigroups \(T_{RE}(X)\) and \(T_{RE}(Y)\) are isomorphic if and only if \(X\) and \(Y\) are order isomorphic. In fact, they show that for each isomorphism \(\varphi\) from \(T_{RE}(X)\) onto \(T_{RE}(Y)\), there exists an order isomorphism \(h\) from \(X\) onto \(Y\) such that \(\varphi(f)=h\circ f\circ h^{-1}\) for all \(f\in T_{RE}(X)\). In particular, if \(X\) is adjusted and \(\varphi\) is an automorphism of the semigroup \(T_{RE}(X)\), then there exists an order automorphism \(h\) of \(X\) such that \(\varphi(f)=h\circ f\circ h^{-1}\) for all \(f\in T_{RE}(X)\). Since the order automorphism \(h\) is unique, it readily follows that the map \(\Phi\), which is defined by \(\Phi(\varphi)=h\), is an isomorphism from the automorphism group of the semigroup \(T_{RE}(X)\) onto the automorphism group of the partially ordered set \(X\).