Generators and factorisations of transformation semigroups

If E is the set of idempotents and G the group of units within a full transformation semigroup F-X. then EG = GE = F-X, if X is finite. The question of identifying the subsemigroup EG = GE = < G boolean OR E > in the case where X is infinite leads to an investigation of interrelations among various naturally occurring subsemigroups of F-X. In the final section it is shown that precisely two additional elements mu, nu are needed in order that G boolean OR E boolean OR {mu, nu} should generate F-X.