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Etymologie

Was (wer) ist lower set - definition

A SUBSET OF A PARTIAL ORDER THAT INCLUDES ALL SUCCESSORS OF ITS ELEMENTS

Lower set; Down set; Down-set; Down-set lattice; Down set lattice; Downset lattice; Downward closed; Initial segment; Principal down set; Principal lower set; Principal down-set; Up-set; Upward closed set; Final segment; Upward closure; Downward closure; Downward closed set; Upper set and lower set; Upper and lower sets

lower set

<mathematics> A finite non-empty downward closed subset of a partial order. (1999-03-17)

downward closed

closure

upward closure

closure

Wikipedia

Upper set

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set $(X,\leq )$ is a subset $S\subseteq X$ with the following property: if s is in S and if x in X is larger than s (that is, if $s<x$ ), then x is in S. In other words, this means that any x element of X that is $\,\geq \,$ to some element of S is necessarily also an element of S. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is $\,\leq \,$ to some element of S is necessarily also an element of S.

https://en.wikipedia.org/wiki/Upper set