Uniform lattices and modular functions. (English) Zbl 0989.28007

The paper contains many interesting results for \(G\)-valued functions (\((G,+)\) being a commutative group) defined on uniform lattices, i.e., lattices with uniformity making the lattice operations \(\vee\) and \(\wedge\) uniformly continuous. These lattices include topological Boolean algebras and locally solid Riesz spaces. A natural generalization of finitely additive measures on Boolean algebras and linear operators on Riesz spaces are the modular functions \(\mu: L\to (G,+)\) satisfying the identity \(\mu(x)+ \mu(y)= \mu(x\vee y)+ \mu(x\wedge y)\).


28B10 Group- or semigroup-valued set functions, measures and integrals
06B30 Topological lattices
46A40 Ordered topological linear spaces, vector lattices