Definition:Greatest Element/Class Theory
Definition
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:
- $\forall y \in A: y \mathrel \RR x$
Thus for an element $x$ to be the greatest element, all $y \in S$ must be comparable to $x$.
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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering