Ordering the local subsemigroups of a semigroup (Q1820875)

For a space X, S(X) denotes the semigroup of continuous functions from X into X. The authors discuss conditions under which there are idempotents e and f in S(X) such that the local semigroup eS(X)e is not contained in fS(X)f but is isomorphic to a subsemigroup of the local semigroup fS(X)f. An extensive discussion of L-admissibility is presented by the authors with regard to dendrites and Peano continua. The paper is concluded with a pair of enlightening examples.