A convex algebra on the unit interval [0,1] assigns to each pair of points and each weight a convex combination. The operation is monotone (larger inputs give larger outputs) and semicontinuous (the operation is well-behaved at limits). These structures arise in probability theory, where convex combinations of distributions are weighted mixtures, and in optimization, where convex combinations of feasible points stay feasible.
The full classification of convex algebras on [0,1] with semicontinuous monotone operations reveals unexpected richness. The algebras are not all equivalent — different semicontinuity conditions (upper vs. lower) produce genuinely different algebraic structures with different equational theories.
The classification connects to Mio's quantitative equational theories, a framework for extending classical equational logic to settings where operations produce quantitative rather than qualitative outputs. The convex algebras on [0,1] are precisely the models of specific quantitative equational theories, and the classification determines which theories have which models.
The structural insight: the interaction between convexity and semicontinuity creates a richer algebraic landscape than either property alone. Convex algebras without semicontinuity are well-understood (they're convex sets). Semicontinuous operations without convexity are well-understood (they're lattice operations). Together, they generate structures that neither framework predicts.
The classification is complete: every convex algebra on [0,1] with semicontinuous monotone operations falls into one of the identified classes, and the classes are distinguished by their quantitative equational theories. The interval remembers which equations it satisfies.