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Maximum or maximal set with property $P$

When I was reading some textbooks, I noticed that I do not get the meaning of the following two phrases.

($P1$) $\quad$ maximum set with property $P$

($P2$) $\quad$ maximal set with property $P$

In this regard, I have the following two questions.

($Q1$) $\quad$ Are the phrases equivalent?

($Q2$) $\quad$ What are their meanings? (generally accepted meanings, meanings specific to particular theories)

As for ($Q2$), I suspect the following three meanings.

$\quad$ A set is a $\textit{maxim... set with property}$ $P$ if and only if there is $\dots$

($M1$) $\quad \dots$ no proper superset with property $P$.

($M2$) $\quad \dots$ no set with property $P$ that has greater cardinality.

($M3$) $\quad \dots$ no other set with property $P$ that has greater or equal cardinality.

Thanks in advance!

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1 Answer 1

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If $(A,\leq)$ is a partial order, then we define these two definitions for $a\in A$:

  • $a$ is maximal if whenever $a\leq b$, then $a=b$.
  • $a$ is maximum if for every $b\in A$, $b\leq a$.

You can prove that every maximum is maximal, but a maximal element need not be a maximum. In particular there can be many maximal elements. So being maximal and maximum are not the same thing in general.

Now you can consider the collection $A=\{X\mid X\text{ has property }P\}$, and $\leq$ as set inclusion. Then a maximal set with property $P$ is just a maximal element of this partial order; and a maximum is a maximum in this partial order. If those even exist, of course.

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  • $\begingroup$ With the subtelty that, sometimes, if $(A, \leq)$ already has a greatest element $\top$, maximality may mean maximality in $A \setminus \{ \top \}$ (see e.g. the notion of maximal ideal of a ring). $\endgroup$
    – polmath
    Sep 17, 2014 at 15:20
  • $\begingroup$ @Paul: No, with the subtlety that $P$ may exclude $\top$ in the first place. $\endgroup$
    – Asaf Karagila ♦
    Sep 17, 2014 at 15:21
  • $\begingroup$ Asaf: What do you mean? If $A$ is a ring then $A$ is an ideal of $A$, and the definition given e.g. by Wikipedia of a maximal ideal is "an ideal that is maximal (with respect to set inclusion) amongst all proper ideals". $\endgroup$
    – polmath
    Sep 17, 2014 at 15:24
  • $\begingroup$ @Paul: Yes. And therefore a maximal ideal is a maximal element of the partial order of proper ideals, thus excluding $A$ itself from having the very property required to enter the partial order in the first place. $\endgroup$
    – Asaf Karagila ♦
    Sep 17, 2014 at 15:26
  • $\begingroup$ Asaf: Yes, I think we agree. In a nutshell my point was just to recall that we don't say "maximal proper ideal" but only "maximal ideal", which could literally be understood as a maximal element in the set of all ideals. $\endgroup$
    – polmath
    Sep 17, 2014 at 15:33

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