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This paper discusses the admissibility of agglomerative hierarchical clustering algorithms with respect to space distortion and monotonicity which were defined by Yadohisa et al. and Batagelj, respectively. Several admissibilities and their properties are given for selecting a clustering algorithm. Necessary and sufficient conditions for an updating formula, as introduced by Lance and Williams, are provided for the proposed admissibility criteria. A detailed explanation of the admissibility of eight popular algorithms is also given.