Median algebra

Abstract:

A study of algebras with a ternary operation $(x, y, z)$ satisfying some identities, equivalent to embeddability in a lattice with $(x, y, z)$ realized as, simultaneously, $(x \wedge (y \vee z)) \vee (y \wedge z)$ and $(x \vee (y \wedge z)) \wedge (y \vee z)$. This is weaker than embeddability in a modular lattice, where those expressions coincide for all x, y, and z, but much of the theory survives the extension. For actual embedding in a modular lattice, some necessary conditions are found, and the investigation is carried much further in a special, geometrically described class of examples ("2-cells"). In distributive lattices $(x, y, z)$ reduces to the median $(x \wedge y) \vee (x \wedge z) \vee (y \wedge z)$, previously studied by G. Birkhoff and S. Kiss. It is shown that Birkhoff and Kiss found a basis for the laws; indeed, their algebras are embeddable in distributive lattices, i.e. in powers of the 2-element lattice. Their theory is much further developed and is connected into an explicit Pontrjagin-type duality.