An algebraic setting for near-unanimity consensus

Some sets of taxonomic models can be structured as meet-semilattices having the properties that (i) every principal ideal is a distributive lattice and (ii) each finite subset has an upper bound whenever each of its (n–1)-subsets is bounded above, where n3 is a fixed number. Every such semilattice is endowed with an n-ary near-unanimity operation. We show that for n4 one can define these semilattices solely in terms of this n-ary operation. The resulting algebras are subdirect powers of two-element algebras, and of course, generalize median algebras.