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Deciding whether the union of two convex polyhedra is itself a convexpolyhedron is a basic problem in polyhedral computations; having importantapplications in the field of constrained control and in the synthesis,analysis, verification and optimization of hardware and software systems. Insuch application fields though, general convex polyhedra are just one amongmany, so-called, numerical abstractions, which range from restricted familiesof (not necessarily closed) convex polyhedra to non-convex geometrical objects.We thus tackle the problem from an abstract point of view: for a wide range ofnumerical abstractions that can be modeled as bounded join-semilattices --thatis, partial orders where any finite set of elements has a least upper bound--,we show necessary and sufficient conditions for the equivalence between thelattice-theoretic join and the set-theoretic union. For the case of closedconvex polyhedra --which, as far as we know, is the only one already studied inthe literature-- we improve upon the state-of-the-art by providing a newalgorithm with a better worst-case complexity. The results and algorithmspresented for the other numerical abstractions are new to this paper. All thealgorithms have been implemented, experimentally validated, and made availablein the Parma Polyhedra Library.

Exact Join Detection for Convex Polyhedra and Other Numerical Abstractions