Order Topology and Frink Ideal Topology of Effect Algebras

TL;DR

This paper investigates the relationships between order topology, Frink ideal topology, and algebraic properties of effect algebras, establishing conditions for their equivalence, continuity, and topological characteristics.

Contribution

It provides new characterizations of effect algebras' topological properties and continuity conditions, especially relating order and Frink ideal topologies in complete atomic lattice effect algebras.

Findings

Equivalence of (o)-continuity, order-topological, and algebraic properties in complete atomic lattice effect algebras.

Finer relationship between Frink ideal topology and order topology in complete atomic distributive lattice effect algebras.

Conditions under which the order topology is Hausdorff and when the operation is continuous in effect algebras.

Abstract

In this paper, the following results are proved: (1) $ $ If $E$ is a complete atomic lattice effect algebra, then $E$ is (o)-continuous iff $E$ is order-topological iff $E$ is totally order-disconnected iff $E$ is algebraic. (2) $ $ If $E$ is a complete atomic distributive lattice effect algebra, then its Frink ideal topology $\tau_{id}$ is Hausdorff topology and $\tau_{id}$ is finer than its order topology $\tau_{o}$, and $\tau_{id}=\tau_o$ iff 1 is finite iff every element of $E$ is finite iff $\tau_{id}$ and $\tau_o$ are both discrete topologies. (3) $ $ If $E$ is a complete (o)-continuous lattice effect algebra and the operation $\oplus$ is order topology $\tau_o$ continuous, then its order topology $\tau_{o}$ is Hausdorff topology. (4) $ $ If $E$ is a (o)-continuous complete atomic lattice effect algebra, then $\oplus$ is order topology continuous.