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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. In his pioneering work, Matheron (1975) has shown that the set of translationinvariant (t.i.) set mappings, whose domain is a collection S (closed under translation) of subsets of a Euclidean space, is isomorphic to the set of all subcollections of Sd. In order to clarify this point, Matheron introduced the concept of a kernel of a t.i. set mapping qi, which is defined as the subcollection of subsets in i, whose transforms by qi contain the null element of the Euclidean space. Hence, for example, the kernel of an erosion by a structuring element A is the subcollection of all subsets in d containing A. Society for Industrial and Applied MathematicsBy using the fact that the kernel of an increasing t.i. set mapping is a dual ideal (Matheron says that the kernel is U-hereditary), Matheron (1975) has shown that any increasing t.i. set mapping qi can be represented as the supremum of erosions by structuring elements in the kernel of qi.

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