Semigroups of order-decreasing transformations: The isomorphism theorem (Q1924520)

Let \(X\) be a totally ordered set. Let \(T(X)\) denote the full transformation semigroup on \(X\) and let \(S^-(X)=\{\alpha\in T(X):x\alpha\leq x\) for all \(x\in X\}\) and \(S^+(X)=\{\alpha\in T(X):x\alpha\geq x\) for all \(x\in X\}\). Then \(S^-(X)\) is a semigroup which consists of all order decreasing selfmaps of \(X\) and, similarly, \(S^+(X)\) is a semigroup which consists of all order increasing selfmaps of \(X\). In the main result of the paper, the author proves that the semigroups \(S^-(X)\) and \(S^-(Y)\) are isomorphic if and only if the totally ordered sets \(X\) and \(Y\) are order isomorphic. An immediate corollary is the fact that \(S^-(X)\) and \(S^+(Y)\) are isomorphic if and only if \(X\) and \(Y\) are order anti-isomorphic.