Definition:Maximal Element/Class Theory
Definition
Let $A$ be a class.
An element $x \in A$ is a maximal element of $A$ if and only if:
- $\forall y \in A: x \not \subset y$
That is, $x$ a proper subset of no element of $A$.
Comparison with Greatest Element
Compare the definition of 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 see
- Results about maximal 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 $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles: Definition $5.1$