The existence and structure of Irr(X)

It has been shown that, for a compact totally ordered set X, the category of irreducible semigroups with idempotents X and idempotent separating homomorphisms has a generator, Irr(X) 7]. In this article we outline the proof of the existence of Irr(X), along with the results obtained previously about the existence of Irr(X). We also survey the known results about the structure of an Irr(X) and about its uniqueness. These latter results were obtained by K. H. Hofmann and the author in a seminar on compact semigroups at Tulane University during the academic year 1970–1971. Supported by NSF Grant GP28655.