Order Topology and Frink Ideal Topology of 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.