Resumen de Square means and geometric means in lattice-ordered groups and vector lattices

Gerard Buskes, R. H. Redfield

• The geometric mean and the square mean of two positive real numbers are determined by the order in the sense that they can be written as (respectively) an infinite meet and an infinite join. We use this observation to define the geometric mean and the square mean on the positive elements of an Abelian lattice-ordered group. For certain totally ordered fields and vector lattices, the square mean can be used to define a new compatible addition. The resulting structure lives inside the complexification of the original structure and is constructed by using a general method for extending commutative lattice-ordered monoids to Abelian lattice-ordered groups. We describe this method and use it as the starting point for similar constructions of lattice-ordered rings and vector lattices.