Interval convexity of scale effect algebras

Abstract

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 $f:(X,{\mathcal{C}}_1)\to (Y,{\mathcal{C}}_2)$ is convexity-preserving if and only if ${(f(a_{\alpha }))}_{\alpha \in \Lambda }$ converges to f(a) with respect to ${\mathcal{C}}_2$ when ${(a_{\alpha })}_{\alpha \in \Lambda }$ converges to a with respect to ${\mathcal{C}}_1.$ 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 $\mathcal{C}$ is an interval convexity on E, then + and – are separately convexity-preserving with respect to $\mathcal{C}$. Finally, we provide an example to show that + and – are not jointly convexity-preserving with respect to $\mathcal{C}$ when $(E,+,0,1)$ is a lattice effect algebra.

Communicated by Ángel del Río Mateos

KEYWORDS:

• Convex filter
• DC-net
• effect algebra
• interval convexity
• interval operator

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 08A05
• 52A01

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Natural Science Foundation of China (11871097, 12271036, 12071033, 11971448) and Beijing Institute of Technology Science and Technology Innovation Plan Cultivation Project (2021CX01030).