An embedding theorem for semigroups of continuous selfmaps (Q791672)

It is proved: If S(X) is the semigroup of all continuous selfmaps of the topological space X and X and Y are local dendrites with finite branch numbers then the following four statements are equivalent: (1) There exists a monomorphism from S(X) into S(Y) and also a monomorphism from S(Y) into S(X). (2) S(X) and S(Y) are isomorphic. (3) There exists a homeomorphism from X into Y and also a homeomorphism from Y into X. (4) X and Y are homeomorphic.