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The algebraic structure of the attractors in a dynamical system determines much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods., Author(s): William D. Kalies[sup.1] , Konstantin Mischaikow[sup.2] , Robert C. A. M. Vandervorst[sup.3] Author Affiliations: (1) Florida Atlantic University, 777 Glades Road, 33431, Boca Raton, FLUSA (2) Rutgers University, 110 [...]