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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 $$\mu(x)+ \mu(y)= \mu(x\vee y)+ \mu(x\wedge y)$$.

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