Order convergence,order and interval topologies on posets and lattice effect algebras
| Authors | |
|---|---|
| Year of publication | 2008 |
| Type | Article in Proceedings |
| Conference | UNCERTAINTY 2008 |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | order convergence; topological convergence; lattice effect algebra; de Morgan lattice |
| Description | We can say that topology is practically equivalent with the concept of convergence. In the modern topology the concept of convergence of filters is discussed rather than convergence of nets. That is because, while convergence of nets is more intuitive, filters are easier to handle. Nevertheless, from the probability (on Boolean algebras or quantum structures) point of view the convergence of nets is the main tool. It is not only because of properties of states, probabilities or observables, but for algebraic properties of sets of events as well. In section 2 we review some of the standard facts about order convergence and topological convergence of nets and posets. Sections 3 and 4 include new results. In every complete atomic MV-effect algebra (MV-algebra) the order convergence of nets is a topological convergence, at which this order topology is a compact Hausdorff topology. As a consequence we obtain that on every complete atomic and block-finite lattice effect algebra the order topology is compact and Hausdorff. In Section 4 a uniform topology on every compactly generated de Morgan lattice is constructed and its connection with the order topology and order convergence is shown. |
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