In

preordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$(P,\; \backslash leq)$ does happen to have a greatest element $g$ then $g$ will necessarily be a maximal element of $(P,\; \backslash leq)$ and moreover, as a consequence of the greatest element $g$ being comparable to element of $P,$ if $(P,\; \backslash leq)$ is also partially ordered then it is possible to conclude that $g$ is the maximal element of $(P,\; \backslash leq).$
However, the uniqueness conclusion is no longer guaranteed if the preordered set $(P,\; \backslash leq)$ is also partially ordered.
For example, suppose that $R$ is a non-empty set and define a preorder $\backslash ,\backslash leq\backslash ,$ on $R$ by declaring that $i\; \backslash leq\; j$ holds for all $i,\; j\; \backslash in\; R.$ The

partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

and let $S\; \backslash subseteq\; P.$
* A set $S$ can have at most greatest element.If $g\_1$ and $g\_2$ are both greatest, then $g\_1\; \backslash leq\; g\_2$ and $g\_2\; \backslash leq\; g\_1,$ and hence $g\_1\; =\; g\_2$ by

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, especially in order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

, the greatest element of a subset $S$ of a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

(poset) is an element of $S$ that is greater than every other element of $S$. The term least element is defined dually, that is, it is an element of $S$ that is smaller than every other element of $S.$
Definitions

Let $(P,\; \backslash leq)$ be apreordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and let $S\; \backslash subseteq\; P.$
An element $g\; \backslash in\; P$ is said to be if $g\; \backslash in\; S$ and if it also satisfies:
:$s\; \backslash leq\; g$ for all $s\; \backslash in\; S.$
By using $\backslash ,\backslash geq\backslash ,$ instead of $\backslash ,\backslash leq\backslash ,$ in the above definition, the definition of a least element of $S$ is obtained. Explicitly, an element $l\; \backslash in\; P$ is said to be if $l\; \backslash in\; S$ and if it also satisfies:
:$l\; \backslash leq\; s$ for all $s\; \backslash in\; S.$
If $(P,\; \backslash leq)$ is a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

then $S$ can have at most one greatest element and it can have at most one least element. Whenever a greatest element of $S$ exists and is unique then this element is called greatest element of $S$. The terminology least element of $S$ is defined similarly.
If $(P,\; \backslash leq)$ has a greatest element (resp. a least element) then this element is also called (resp. ) of $(P,\; \backslash leq).$
Relationship to upper/lower bounds

Greatest elements are closely related toupper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

s.
Let $(P,\; \backslash leq)$ be a preordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and let $S\; \backslash subseteq\; P.$
An is an element $u$ such that $u\; \backslash in\; P$ and $s\; \backslash leq\; u$ for all $s\; \backslash in\; S.$ Importantly, an upper bound of $S$ in $P$ is required to be an element of $S.$
If $g\; \backslash in\; P$ then $g$ is a greatest element of $S$ if and only if $g$ is an upper bound of $S$ in $(P,\; \backslash leq)$ $g\; \backslash in\; S.$ In particular, any greatest element of $S$ is also an upper bound of $S$ (in $P$) but an upper bound of $S$ in $P$ is a greatest element of $S$ if and only if it to $S.$
In the particular case where $P\; =\; S,$ the definition of "$u$ is an upper bound of $S$ " becomes: $u$ is an element such that $u\; \backslash in\; S$ and $s\; \backslash leq\; u$ for all $s\; \backslash in\; S,$ which is to the definition of a greatest element given before.
Thus $g$ is a greatest element of $S$ if and only if $g$ is an upper bound of $S$ .
If $u$ is an upper bound of $S$ that is not an upper bound of $S$ (which can happen if and only if $u\; \backslash not\backslash in\; S$) then $u$ can be a greatest element of $S$ (however, it may be possible that some other element a greatest element of $S$).
In particular, it is possible for $S$ to simultaneously have a greatest element for there to exist some upper bound of $S$ .
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
This example also demonstrates that the existence of a least upper bound
are equal.
Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond).
In mathematic ...

(the number 0 in this case) does not imply the existence of a greatest element either.
Contrast to maximal elements and local/absolute maximums

A greatest element of a subset of a preordered set should not be confused with amaximal element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the set, which are elements that are not strictly smaller than any other element in the set.
Let $(P,\; \backslash leq)$ be a preordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and let $S\; \backslash subseteq\; P.$
An element $m\; \backslash in\; S$ is said to be a if the following condition is satisfied:
:whenever $s\; \backslash in\; S$ satisfies $m\; \backslash leq\; s,$ then necessarily $s\; \backslash leq\; m.$
If $(P,\; \backslash leq)$ is a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

then $m\; \backslash in\; S$ is a maximal element of $S$ if and only if there does exist any $s\; \backslash in\; S$ such that $m\; \backslash leq\; s$ and $s\; \backslash neq\; m.$
A is defined to mean a maximal element of the subset $S\; :=\; P.$
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 a totally ordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...

.The notion of locality requires the function's domain to be at least a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...

.
The dual terms are minimum and absolute minimum.
Together they are called the absolute 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 $g$ and a maximal element $m$ of a preordered set $(P,\; \backslash leq)$ has to do with what elements they are comparable to.
Two elements $x,\; y\; \backslash in\; P$ are said to be if $x\; \backslash leq\; y$ or $y\; \backslash leq\; x$; they are called if they are not comparable.
Because preorders are reflexive (which means that $x\; \backslash leq\; x$ is true for all elements $x$), every element $x$ is always comparable to itself.
Consequently, the only pairs of elements that could possibly be incomparable are pairs.
In general, however, preordered sets (and even directed
Director may refer to:
Literature
* Director (magazine), ''Director'' (magazine), a British magazine
* The Director (novel), ''The Director'' (novel), a 1971 novel by Henry Denker
* The Director (play), ''The Director'' (play), a 2000 play by Nanc ...

partially ordered sets) may have elements that are incomparable.
By definition, an element $g\; \backslash in\; P$ is a greatest element of $(P,\; \backslash leq)$ if $s\; \backslash leq\; g,$ for every $s\; \backslash in\; P$; so by its very definition, a greatest element of $(P,\; \backslash leq)$ must, in particular, be comparable to element in $P.$
This is not required of maximal elements.
Maximal elements of $(P,\; \backslash leq)$ are required to be comparable to every element in $P.$
This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important statement.
The defining condition for $m\; \backslash in\; P$ to be a maximal element of $(P,\; \backslash leq)$ can be reworded as:
:For all $s\; \backslash in\; P,$ $m\; \backslash leq\; s$ (so elements that are incomparable to $m$ are ignored) then $s\; \backslash leq\; m.$
;Example where all elements are maximal but none are greatest
Suppose that $S$ is a set containing (distinct) elements and define a partial order $\backslash ,\backslash leq\backslash ,$ on $S$ by declaring that $i\; \backslash leq\; j$ if and only if $i\; =\; j.$
If $i\; \backslash neq\; j$ belong to $S$ then neither $i\; \backslash leq\; j$ nor $j\; \backslash leq\; i$ holds, which shows that all pairs of distinct (i.e. non-equal) elements in $S$ are comparable.
Consequently, $(S,\; \backslash leq)$ can not possibly have a greatest element (because a greatest element of $S$ would, in particular, have to be comparable to element of $S$ but $S$ has no such element).
However, element $m\; \backslash in\; S$ is a maximal element of $(S,\; \backslash leq)$ because there is exactly one element in $S$ that is both comparable to $m$ and $\backslash geq\; m,$ that element being $m$ itself (which of course, is $\backslash leq\; m$).Of course, in this particular example, there exists only one element in $S$ that is comparable to $m,$ which is necessarily $m$ itself, so the second condition "and $\backslash geq\; m,$" was redundant.
In contrast, if a directed
Director may refer to:
Literature
* Director (magazine), ''Director'' (magazine), a British magazine
* The Director (novel), ''The Director'' (novel), a 1971 novel by Henry Denker
* The Director (play), ''The Director'' (play), a 2000 play by Nanc ...

preordered set $(R,\; \backslash leq)$ is partially ordered if and only if $R$ has exactly one element. All pairs of elements from $R$ are comparable and element of $R$ is a greatest element (and thus also a maximal element) of $(R,\; \backslash leq).$ So in particular, if $R$ has at least two elements then $(R,\; \backslash leq)$ has multiple greatest elements.
Properties

Throughout, let $(P,\; \backslash leq)$ be aantisymmetry
In linguistics, antisymmetry is a theory of syntax, syntactic linearization presented in Richard Kayne's 1994 monograph ''The Antisymmetry of Syntax''. The crux of this theory is that hierarchical structure in natural language maps universally onto ...

. Thus if a set has a greatest element then it is necessarily unique.
* If it exists, then the greatest element of $S$ is an upper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

of $S$ that is also contained in $S.$
* If $g$ is the greatest element of $S$ then $g$ is also a maximal element of $S$If $g$ is the greatest element of $S$ and $s\; \backslash in\; S,$ then $s\; \backslash leq\; g.$ By antisymmetry
In linguistics, antisymmetry is a theory of syntax, syntactic linearization presented in Richard Kayne's 1994 monograph ''The Antisymmetry of Syntax''. The crux of this theory is that hierarchical structure in natural language maps universally onto ...

, this renders ($g\; \backslash leq\; s$ and $g\; \backslash neq\; s$) impossible. and moreover, any other maximal element of $S$ will necessarily be equal to $g.$If $M$ is a maximal element, then $M\; \backslash leq\; g$ since $g$ is greatest, hence $M\; =\; g$ since $M$ is maximal.
** Thus if a set $S$ has several maximal elements then it cannot have a greatest element.
* If $P$ satisfies the ascending chain conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...

, a subset $S$ of $P$ has a greatest element if, and only if
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argumen ...

, it has one maximal element.''Only if:'' see above. — ''If:'' Assume for contradiction that $S$ has just one maximal element, $m,$ but no greatest element. Since $m$ is not greatest, some $s\_1\; \backslash in\; S$ must exist that is incomparable to $m.$ Hence $s\_1\; \backslash in\; S$ cannot be maximal, that is, $s\_1\; <\; s\_2$ must hold for some $s\_2\; \backslash in\; S.$ The latter must be incomparable to $m,$ too, since $m\; <\; s\_2$ contradicts $m$'s maximality while $s\_2\; \backslash leq\; m$ contradicts the incomparability of $m$ and $s\_1.$ Repeating this argument, an infinite ascending chain $s\_1\; <\; s\_2\; <\; \backslash cdots\; <\; s\_n\; <\; \backslash cdots$ can be found (such that each $s\_i$ is incomparable to $m$ and not maximal). This contradicts the ascending chain condition.
* When the restriction of $\backslash ,\backslash leq\backslash ,$ to $S$ is a total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

($S\; =\; \backslash $ in the topmost picture is an example), then the notions of maximal element and greatest element coincide.Let $m\; \backslash in\; S$ be a maximal element, for any $s\; \backslash in\; S$ either $s\; \backslash leq\; m$ or $m\; \backslash leq\; s.$ In the second case, the definition of maximal element requires that $m\; =\; s,$ so it follows that $s\; \backslash leq\; m.$ In other words, $m$ is a greatest element.
** However, this is not a necessary condition for whenever $S$ 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 $S$ of $P,$ then $\backslash ,\backslash leq\backslash ,$ is a total order on $P.$If $a,\; b\; \backslash in\; P$ were incomparable, then $S\; =\; \backslash $ would have two maximal, but no greatest element, contradicting the coincidence.
Sufficient conditions

* A finitechain
Image:Kettenvergleich.jpg, Roller chains
A chain is a wikt:series#Noun, serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compres ...

always has a greatest and a least element.
Top and bottom

The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is acomplemented lattice
In the mathematical discipline of order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...

, 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 special completeness property of a partial order.
Further introductory information is found in the article on order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

.
Examples

* The subset ofinteger
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

s has no upper bound in the set $\backslash mathbb$ of real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s.
* Let the relation $\backslash ,\backslash leq\backslash ,$ on $\backslash $ be given by $a\; \backslash leq\; c,$ $a\; \backslash leq\; d,$ $b\; \backslash leq\; c,$ $b\; \backslash leq\; d.$ The set $\backslash $ has upper bounds $c$ and $d,$ but no least upper bound, and no greatest element (cf. picture).
* In the rational number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
* In $\backslash mathbb,$ the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
* In $\backslash mathbb,$ the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
* In $\backslash mathbb^2$ with the product order
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, the set of pairs $(x,\; y)$ with $0\; <\; x\; <\; 1$ has no upper bound.
* In $\backslash mathbb^2$ with the lexicographical order
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, this set has upper bounds, e.g. $(1,\; 0).$ It has no least upper bound.
See also

* Essential supremum and essential infimum * Initial and terminal objects * Maximal and minimal elements * Limit superior and limit inferior (infimum limit) * Upper and lower boundsNotes

References

* {{cite book , last1=Davey , first1=B. A. , last2=Priestley , first2=H. A. , year = 2002 , title = Introduction to Lattices and Order , title-link= Introduction to Lattices and Order , edition = 2nd , publisher = Cambridge University Press , isbn = 978-0-521-78451-1 Order theory Superlatives