Definition:Limit Inferior of Moore-Smith Sequence

It has been suggested that this page or section be merged into Definition:Limit Inferior. In particular: There is already a page for limit inferior which this could be merged with. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code.

Definition

This article, or a section of it, needs explaining. In particular: Why is the preorder $\preceq$ and the order $\precsim$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Let $\left({S, \preceq}\right)$ be a directed set.

Let $L = \left({T, \precsim}\right)$ be a complete lattice.

Let $N: S \to T$ be a Moore-Smith sequence in $T$.

Then limit inferior of $N$ is defined as follows:

$\liminf N := \sup_L \left\{ {\inf_L \left({N \left[{\preceq \left({j}\right)}\right]}\right): j \in S}\right\}$

where

$\preceq \left({j}\right)$ denotes the image of $j$ by $\preceq$,

$N \left[{\preceq \left({j}\right)}\right]$ denotes the image of $\preceq \left({j}\right)$ under $N$.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL11:def 6