lower set

In mathematics, an upper set, or upward set, is a subset Y of a given partially ordered set (X,≤) such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y. More formally, The dual notion is lower set (alternatively, down set, decreasing set, initial segment; the set is downward closed), which is any subset Y of a given partially ordered set (X,≤) such that, for all elements x and y, if x is less than or equal to y and y is an element of Y, then x is also in Y. More formally,
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<mathematics> A finite non-empty downward closed subset of a partial order.
(1999-03-17)