Definition·D053
Least and greatest element
A member of a subset below (or above) every member of that subset.
Let
be a poset and
. An element
is the least element (minimum) of
when
and
is the greatest element (maximum) of
when
In words
The least element of a subset is a member that relates below every member of the subset; the greatest element is a member that every member relates below. A subset need not have either.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
A least (or greatest) element, when it exists, is unique: two least elements
satisfy
and
, so
by antisymmetry, justifying the article 'the'. It differs from a minimal element, one with nothing strictly below it: a minimal element need not be comparable to everything and need not be unique, whereas a least element is below everything. In a total order the two notions coincide. A well order is precisely a total order in which every nonempty subset has a least element.