Interval convexity of scale effect algebras.

In this paper, we first propose the concept of DC-nets, where its domain is a DCPO, in convexity spaces and discuss some related properties. And then, we obtain that f : (X , C 1) → (Y , C 2) is convexity-preserving if and only if (f (a α)) α ∈ Λ converges to f(a) with respect to C 2 when (a α) α ∈ Λ converges to a with respect to C 1. Hence, we could discuss convexity-preserving properties of partial binary operations + and – of effect algebras by convergence of DC-nets in convexity spaces. Concretely, we prove that if (E , + , 0 , 1) is a scale effect algebra and C is an interval convexity on E, then + and – are separately convexity-preserving with respect to C. Finally, we provide an example to show that + and – are not jointly convexity-preserving with respect to C when (E , + , 0 , 1) is a lattice effect algebra. Communicated by Ángel del Río Mateos [ABSTRACT FROM AUTHOR]