Extreme values in abstract spaces with order

In the case of a general partial order one should not confuse a least element (smaller than all other) and a minimal element (nothing is smaller). Likewise, a greatest element of a poset is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if mb (for any b in A) then m = b.

Any least element or greatest element of a poset will be unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element.

If a chain is finite then it will always have a maximum (maximal element, greatest element) and a minimum (minimal element, least element). If a chain is infinite then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum.

If an infinite chain S is bounded, then the closure Cl(S) of the set will have a minimum and a maximum, which are the greatest lower bound and the least upper bound of the set S, and which either belong to S or are accumulation points of S.

(source: Wikipedia)

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