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Inmathematics, especially inorder theory, thegreatest element of a subset${\displaystyle S}$ of apartially ordered set (poset) is an element of${\displaystyle S}$ that is greater than every other element of${\displaystyle S}$. The termleast element is defineddually, that is, it is an element of${\displaystyle S}$ that is smaller than every other element of${\displaystyle S.}$

Let${\displaystyle (P,\le )}$ be apreordered set and let${\displaystyle S\subseteq P.}$ An element${\displaystyle g\in P}$ is said to beagreatest element of${\displaystyle S}$ if${\displaystyle g\in S}$ and if it also satisfies:

${\displaystyle s\le g}$ for all${\displaystyle s\in S.}$

By switching the side of the relation that${\displaystyle s}$ is on in the above definition, the definition of a least element of${\displaystyle S}$ is obtained. Explicitly, an element${\displaystyle l\in P}$ is said to bealeast element of${\displaystyle S}$ if${\displaystyle l\in S}$ and if it also satisfies:

${\displaystyle l\le s}$ for all${\displaystyle s\in S.}$

If${\displaystyle (P,\le )}$ is also apartially ordered set then${\displaystyle S}$ can have at most one greatest element and it can have at most one least element. Whenever a greatest element of${\displaystyle S}$ exists and is unique then this element is calledthe greatest element of${\displaystyle S}$. The terminologythe least element of${\displaystyle S}$ is defined similarly.

If${\displaystyle (P,\le )}$ has a greatest element (resp. a least element) then this element is also calledatop (resp.abottom) of${\displaystyle (P,\le ).}$

Greatest elements are closely related toupper bounds.

Let${\displaystyle (P,\le )}$ be apreordered set and let${\displaystyle S\subseteq P.}$ Anupper bound of${\displaystyle S}$ in${\displaystyle (P,\le )}$ is an element${\displaystyle u}$ such that${\displaystyle u\in P}$ and${\displaystyle s\le u}$ for all${\displaystyle s\in S.}$ Importantly, an upper bound of${\displaystyle S}$ in${\displaystyle P}$ isnot required to be an element of${\displaystyle S.}$

If${\displaystyle g\in P}$ then${\displaystyle g}$ is a greatest element of${\displaystyle S}$ if and only if${\displaystyle g}$ is an upper bound of${\displaystyle S}$ in${\displaystyle (P,\le )}$and${\displaystyle g\in S.}$ In particular, any greatest element of${\displaystyle S}$ is also an upper bound of${\displaystyle S}$ (in${\displaystyle P}$) but an upper bound of${\displaystyle S}$ in${\displaystyle P}$ is a greatest element of${\displaystyle S}$ if and only if itbelongs to${\displaystyle S.}$ In the particular case where${\displaystyle P=S,}$ the definition of "${\displaystyle u}$ is an upper bound of${\displaystyle S}$in${\displaystyle S}$" becomes:${\displaystyle u}$ is an element such that${\displaystyle u\in S}$ and${\displaystyle s\le u}$ for all${\displaystyle s\in S,}$ which iscompletely identical to the definition of a greatest element given before. Thus${\displaystyle g}$ is a greatest element of${\displaystyle S}$ if and only if${\displaystyle g}$ is an upper bound of${\displaystyle S}$in${\displaystyle S}$.

If${\displaystyle u}$ is an upper bound of${\displaystyle S}$in${\displaystyle P}$ that is not an upper bound of${\displaystyle S}$in${\displaystyle S}$ (which can happen if and only if${\displaystyle u\notin S}$) then${\displaystyle u}$ cannot be a greatest element of${\displaystyle S}$ (however, it may be possible that some other elementis a greatest element of${\displaystyle S}$). In particular, it is possible for${\displaystyle S}$ to simultaneouslynot have a greatest elementand for there to exist some upper bound of${\displaystyle S}$in${\displaystyle P}$.

Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negativereal numbers. This example also demonstrates that the existence of aleast upper bound (the number 0 in this case) does not imply the existence of a greatest element either.

A greatest element of a subset of a preordered set should not be confused with amaximal element of the set, which are elements that are not strictly smaller than any other element in the set.

Let${\displaystyle (P,\le )}$ be apreordered set and let${\displaystyle S\subseteq P.}$ An element${\displaystyle m\in S}$ is said to be amaximal element of${\displaystyle S}$ if the following condition is satisfied:

whenever${\displaystyle s\in S}$ satisfies${\displaystyle m\le s,}$ then necessarily${\displaystyle s\le m.}$

If${\displaystyle (P,\le )}$ is apartially ordered set then${\displaystyle m\in S}$ is a maximal element of${\displaystyle S}$ if and only if there doesnot exist any${\displaystyle s\in S}$ such that${\displaystyle m\le s}$ and${\displaystyle s\ne m.}$ Amaximal element of${\displaystyle (P,\le )}$ is defined to mean a maximal element of the subset${\displaystyle S:=P.}$

A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.

In atotally ordered set the maximal element and the greatest element coincide; and it is also calledmaximum; in the case of function values it is also called theabsolute maximum, to avoid confusion with alocal maximum.^[1] The dual terms areminimum andabsolute minimum. Together they are called theabsolute extrema. Similar conclusions hold for least elements.

Role of (in)comparability in distinguishing greatest vs. maximal elements

One of the most important differences between a greatest element${\displaystyle g}$ and a maximal element${\displaystyle m}$ of a preordered set${\displaystyle (P,\le )}$ has to do with what elements they are comparable to. Two elements${\displaystyle x,y\in P}$ are said to becomparable if${\displaystyle x\le y}$ or${\displaystyle y\le x}$; they are calledincomparable if they are not comparable. Because preorders arereflexive (which means that${\displaystyle x\le x}$ is true for all elements${\displaystyle x}$), every element${\displaystyle x}$ is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable aredistinct pairs.In general, however, preordered sets (and evendirected partially ordered sets) may have elements that are incomparable.

By definition, an element${\displaystyle g\in P}$ is a greatest element of${\displaystyle (P,\le )}$ if${\displaystyle s\le g,}$ for every${\displaystyle s\in P}$; so by its very definition, a greatest element of${\displaystyle (P,\le )}$ must, in particular, be comparable toevery element in${\displaystyle P.}$ This is not required of maximal elements. Maximal elements of${\displaystyle (P,\le )}$ arenot required to be comparable to every element in${\displaystyle P.}$ This is because unlike the definition of "greatest element", the definition of "maximal element" includes an importantif statement. The defining condition for${\displaystyle m\in P}$ to be a maximal element of${\displaystyle (P,\le )}$ can be reworded as:

For all${\displaystyle s\in P,}$IF${\displaystyle m\le s}$ (so elements that are incomparable to${\displaystyle m}$ are ignored) then${\displaystyle s\le m.}$

Example where all elements are maximal but none are greatest

Suppose that${\displaystyle S}$ is a set containingat least two (distinct) elements and define a partial order${\displaystyle \le }$ on${\displaystyle S}$ by declaring that${\displaystyle i\le j}$ if and only if${\displaystyle i=j.}$ If${\displaystyle i\ne j}$ belong to${\displaystyle S}$ then neither${\displaystyle i\le j}$ nor${\displaystyle j\le i}$ holds, which shows that all pairs of distinct (i.e. non-equal) elements in${\displaystyle S}$ areincomparable. Consequently,${\displaystyle (S,\le )}$ can not possibly have a greatest element (because a greatest element of${\displaystyle S}$ would, in particular, have to be comparable toevery element of${\displaystyle S}$ but${\displaystyle S}$ has no such element). However,every element${\displaystyle m\in S}$ is a maximal element of${\displaystyle (S,\le )}$ because there is exactly one element in${\displaystyle S}$ that is both comparable to${\displaystyle m}$ and${\displaystyle \ge m,}$ that element being${\displaystyle m}$ itself (which of course, is${\displaystyle \le m}$).^{[note 1]}

In contrast, if apreordered set${\displaystyle (P,\le )}$ does happen to have a greatest element${\displaystyle g}$ then${\displaystyle g}$ will necessarily be a maximal element of${\displaystyle (P,\le )}$ and moreover, as a consequence of the greatest element${\displaystyle g}$ being comparable toevery element of${\displaystyle P,}$ if${\displaystyle (P,\le )}$ is also partially ordered then it is possible to conclude that${\displaystyle g}$ is theonly maximal element of${\displaystyle (P,\le ).}$ However, the uniqueness conclusion is no longer guaranteed if the preordered set${\displaystyle (P,\le )}$ isnot also partially ordered. For example, suppose that${\displaystyle R}$ is a non-empty set and define a preorder${\displaystyle \le }$ on${\displaystyle R}$ by declaring that${\displaystyle i\le j}$always holds for all${\displaystyle i,j\in R.}$ Thedirected preordered set${\displaystyle (R,\le )}$ is partially ordered if and only if${\displaystyle R}$ has exactly one element. All pairs of elements from${\displaystyle R}$ are comparable andevery element of${\displaystyle R}$ is a greatest element (and thus also a maximal element) of${\displaystyle (R,\le ).}$ So in particular, if${\displaystyle R}$ has at least two elements then${\displaystyle (R,\le )}$ has multipledistinct greatest elements.

Throughout, let${\displaystyle (P,\le )}$ be apartially ordered set and let${\displaystyle S\subseteq P.}$

• A set${\displaystyle S}$ can have at mostone greatest element.^{[note 2]} Thus if a set has a greatest element then it is necessarily unique.
• If it exists, then the greatest element of${\displaystyle S}$ is anupper bound of${\displaystyle S}$ that is also contained in${\displaystyle S.}$
• If${\displaystyle g}$ is the greatest element of${\displaystyle S}$ then${\displaystyle g}$ is also a maximal element of${\displaystyle S}$^{[note 3]} and moreover, any other maximal element of${\displaystyle S}$ will necessarily be equal to${\displaystyle g.}$^{[note 4]}
  • Thus if a set${\displaystyle S}$ has several maximal elements then it cannot have a greatest element.
• If${\displaystyle P}$ satisfies theascending chain condition, a subset${\displaystyle S}$ of${\displaystyle P}$ has a greatest elementif, and only if, it has one maximal element.^{[note 5]}
• When the restriction of${\displaystyle \le }$ to${\displaystyle S}$ is atotal order (${\displaystyle S=\{1,2,4\}}$ in the topmost picture is an example), then the notions of maximal element and greatest element coincide.^{[note 6]}
  • However, this is not a necessary condition for whenever${\displaystyle S}$ has a greatest element, the notions coincide, too, as stated above.
• If the notions of maximal element and greatest element coincide on every two-element subset${\displaystyle S}$ of${\displaystyle P,}$ then${\displaystyle \le }$ is a total order on${\displaystyle P.}$^{[note 7]}

• A finitechain always has a greatest and a least element.

The least and greatest element of the whole partially ordered set play a special role and are also calledbottom (⊥) andtop (⊤), orzero (0) andunit (1), respectively.If both exist, the poset is called abounded poset.The notation of 0 and 1 is used preferably when the poset is acomplemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top.The existence of least and greatest elements is a specialcompleteness property of a partial order.

Further introductory information is found in the article onorder theory.

• The subset ofintegers has no upper bound in the set${\displaystyle \mathbb{R}}$ ofreal numbers.
• Let the relation${\displaystyle \le }$ on${\displaystyle \{a,b,c,d\}}$ be given by${\displaystyle a\le c,}$${\displaystyle a\le d,}$${\displaystyle b\le c,}$${\displaystyle b\le d.}$ The set${\displaystyle \{a,b\}}$ has upper bounds${\displaystyle c}$ and${\displaystyle d,}$ but no least upper bound, and no greatest element (cf. picture).
• In therational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
• In${\displaystyle \mathbb{R},}$ the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
• In${\displaystyle \mathbb{R},}$ the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
• In${\displaystyle {\mathbb{R}}^2}$ with theproduct order, the set of pairs${\displaystyle (x,y)}$ with has no upper bound.
• In${\displaystyle {\mathbb{R}}^2}$ with thelexicographical order, this set has upper bounds, e.g.${\displaystyle (1,0).}$ It has no least upper bound.

• Essential supremum and essential infimum
• Initial and terminal objects
• Maximal and minimal elements
• Limit superior and limit inferior (infimum limit)
• Upper and lower bounds
• Well-order — a non-strict order such that every non-empty set has a least element

• Davey, B. A.; Priestley, H. A. (2002).Introduction to Lattices and Order (2nd ed.).Cambridge University Press.ISBN 978-0-521-78451-1.

• Order theory
• Superlatives

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