Definition:Greatest Element
Definition
Let $\struct {S, \preceq}$ be an ordered set.
An element $x \in S$ is the greatest element (of $S$) if and only if:
- $\forall y \in S: y \preceq x$
That is, every element of $S$ precedes, or is equal to, $x$.
The Greatest Element is Unique, so calling it the greatest element is justified.
Thus for an element $x$ to be the greatest element, all $y \in S$ must be comparable to $x$.
Greatest Element of Subset
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be a subset of $S$.
An element $x \in T$ is the greatest element of $T$ if and only if:
- $\forall y \in T: y \preceq \restriction_T x$
where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
Class Theory
In the context of class theory, the definition follows the same lines:
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be an ordering.
Let $A$ be a subclass of the field of $\RR$.
An element $x \in A$ is the greatest element of $A$ if and only if:
- $\forall y \in A: y \mathrel \RR x$
Comparison with Maximal Element
Compare the definition of a maximal element with that of a greatest element.
Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.
An element $x \in T$ is the greatest element of $T$ if and only if:
- $\forall y \in T: y \preceq x$
That is, $x$ is comparable with, and succeeds, or is equal to, every $y \in S$.
An element $x \in S$ is a maximal element of $T$ if and only if:
- $x \preceq y \implies x = y$
That is, $x$ succeeds, or is equal to, every $y \in S$ which is comparable with $x$.
If all elements are comparable wth $x$, then such a maximal element is indeed the greatest element.
Note that when an ordered set is in fact a totally ordered set, the terms maximal element and greatest element are equivalent.
Also known as
The greatest element of a collection is also called:
- The largest element (or biggest element, etc.)
- The last element
- The maximum element (but beware confusing with maximal - see above)
- The unit element (in the context of boolean algebras and boolean rings)
Examples
Finite Subsets of Natural Numbers
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.
Consider the ordered set $\struct {\FF, \subseteq}$.
$\struct {\FF, \subseteq}$ has no greatest element.
Finite Subsets of Natural Numbers less Empty Set
Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.
Let $\GG$ denote the set $\FF \setminus \O$, that is, $\FF$ with the empty set excluded.
Consider the ordered set $\struct {\GG, \subseteq}$.
$\struct {\FF, \subseteq}$ has no greatest element.
Also see
- Results about greatest elements can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.11 \ \text{(a)}$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.7$: Maximum and Minimum
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): greatest
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): greatest