Convergence structures induced by scales

By a (fine) scale on a lattice L, we mean a (strictly) isotone real-valued function μ on L. The coarsest topology on L such that μ composed with the unary lattice operations becomes continuous is denoted by τμ. The following convergence structures on L are compared with each other: 1. der convergence, onvergence in the order topology, τμ-convergence, μ-convergence, where ρμ(x, y) = μ(x ∨ y) − μ(x ∧ y). ow that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρμ-convergence and with τμ-convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρμ-convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τμ-convergence. From these and related results, we derive various applications to the theory of measures and valuations on orthomodular lattices. For example, if μ is a fine scale on a complete orthomodular lattice then order convergence agrees with τμ-convergence iff μ is continuous and L is algebraic (or atomic and meet-continuous).