Definition:Maximal Element/Comparison with Greatest Element
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Definition
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.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets