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 $τ_{id}$ is Hausdorff topology and $τ_{id}$ is finer than its order topology $τ_{o}$, and $τ_{id}=τ_o$ iff 1 is finite iff every element of $E$ is finite iff $τ_{id}$ and $τ_o$ are both discrete topologies. (3) $ $ If $E$ is a complete (o)-continuous lattice effect algebra and the operation $\oplus$ is order topology $τ_o$ continuous, then its order topology $τ_{o}$ is Hausdorff topology. (4) $ $ If $E$ is a (o)-continuous complete atomic lattice effect algebra, then $\oplus$ is order topology continuous.