效应代数的序拓扑与Frink理想拓扑
Order Topology and Frink Ideal Topology of Effect Algebras
摘要：
本文证明了：（1）$ $若$E$是完备原子格效应代数，则$E$是（o）-连续的当且仅当$E$是序拓扑的当且仅当$E$是全序不连通的当且仅当$E$是代数的. (2)如果$ $是一个完全原子分配格效应代数，则它的Frink理想拓扑$τ_{id}$是Hausdorff拓扑，$τ_{id}$比它的序拓扑$τ_{o}$更精细，并且$τ_{id}=τ_o$当且仅当1是有限的当且仅当$E$的每个元素都是有限的当且仅当$τ_{id}$和$τ_o$都是离散拓扑. (3)若$ $是完备（o）-连续格效应代数，且运算$\oplus$是序拓扑$τ_o$连续的，则它的序拓扑$τ_{o}$是Hausdorff拓扑. (4)若$ $是（o）-连续的完备原子格效应代数，则$\oplus$是序拓扑连续的.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
