Convergence structures induced by scales

Abstract

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 τ μ . The following convergence structures on L are compared with each other: 1. (i) order convergence, 2. (ii) convergence in the order topology, 3. (iii) τ μ -convergence, 4. (iv) ρ μ -convergence, where ρ μ ( x, y) = μ( x ∨ y) − μ( x ∧ y). We show that for any scale μ on an arbitrary complete lattice, order convergence agrees with ρ μ -convergence and with τ μ -convergence iff μ is a fine continuous scale such that join and meet operations of arbitrary arity are continuous with respect to ρ μ -convergence. Furthermore, for any fine continuous scale μ on a bi-algebraic lattice, order convergence agrees with τ μ -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 τ μ -convergence iff μ is continuous and L is algebraic (or atomic and meet-continuous).