# Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras

## TL;DR

This paper investigates the topological properties of atomic lattice effect algebras, establishing conditions under which their interval topology is Hausdorff and exploring implications for order convergence and state smearing.

## Contribution

It proves that the interval topology of certain atomic lattice effect algebras is Hausdorff under almost orthogonality, and links this to order continuity and compactness, extending the understanding of their topological structure.

## Key findings

- Interval topology is Hausdorff under almost orthogonality.
- Order convergence coincides with topological convergence in complete cases.
- State smearing theorem applies to these effect algebras.

## Abstract

We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements. For block-finite Archimedean atomic lattice effect algebras the equivalence of almost orthogonality and s-compact generation is shown. As the main application we obtain the state smearing theorem for these effect algebras, as well as the continuity of $\oplus$-operation in the order and interval topologies on them.