Semigroup Forum Vol. 66 (2003) 236–272 c  2002 Springer-Verlag New York Inc. DOI: 10.1007/s002330010133 RESEARCH ARTICLE Perfect Semigroups and Aliens Brigitte E. Breckner ∗ and Wolfgang A. F. Ruppert Communicated by Jimmie D. Lawson Abstract A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided, that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2, R). 2000 Mathematics Subject Classification: 22E15, 22E67, 22E46, 22A15, 22A25. Key words and phrases: Alien elements, divisors of an element, perfect semigroups, proper semigroups, compact order intervals, Bohr compactification, subsemigroups of Sl(2, R), Lie semigroups, Lie semialgebras, umbrella sets. 1. Introduction An element s of a topological semigroup S is said to be compactlydivided if the set {(x,y) ∈ S × S | xy = s} is compact. If s is not compactly divided (that is, if it can be factored by elements living arbitrarily far away) then we say that it is purely alien. Finally, if for every compact subset K of S the element s is contained in the closure of (S\K)S ∪ S(S\K) then s is called an alien. The above three concepts appear implicitly or under other names in the literature (cf., e.g., [12], [11], [19]), they show up in the following contexts: • In the harmonic analysis of semisimple Lie groups we need information about the compactness of intervals with respect to an order induced by a Lie subsemigroup (compactness is needed for integration). Compactly divided elements correspond to compact order intervals. • In the general theory of locally compact topological semigroups a natural question asks for conditions under which the product AB of two closed sets A and B is closed as well. If A , B are closed and Al(S) denotes the set of all aliens in S then AB ∪ Al(S) is always closed. * This author was supported by the Deutsche Forschungsgemeinschaft.