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Given an element of a preordered set the upper closure or upward closure of is defined by
:while the lower closure or downward closure of by
:

Upper and lower sets of the form and are called principal. The upper closure of an element is the same thing as the principal filter generated by that element, since it is also directed downward.

More generally, given a subset the upper closure and lower closure of are defined as and ; they are,

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