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A new notion of a canonical extension A σ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) A. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice A, A σ has the same meaning as before.A novel feature is the introduction of several topologies on the universe of the canonical extension of a DL. One of these topologies is used to define the canonical extension f σ : A σ → B σ of an arbitrary map f : A → B between DLs, and hence to define the canonical extension A σ of an arbitrary DLE A. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.

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Introduction. An operator group with a principal series can obviously be written as a direct product of finitely many directly indecomposable admissible subgroups, and the classical WedderburnRemak-Krull-Schmidt Theorem asserts that this representation i& unique up to isomorphism. Numerous generalizations of this theorem are known in the literature.1 Thus it follows from results in Baer [1, 2] that if the admissible center of an operator group G satisfiesthe minimal and the local maximal conditions, then any two direct decompositions of G (with arbitrarily many factors) have isomorphic refinements. In a different direction, it is shown in Crawley [4] that if an operator group G has a direct decomposition each factor of which has a principal series, then any two direct decompositions of G haveisomorphic refinements.The results of this paper yield sufficient conditions for a group-(with or without operators) to have the isomorphic refinement property* For operator groups a common generalization of the theorems mentioned above is obtained:

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