# What are the usual definitions of minimality and maximality with respect to an arbitrary relation?

Let $\mathcal R$ be a relation on $S$ and let $T \subseteq S$.

It seems there are two notions floating around of an $\mathcal R$-minimal element of $T$:

1. $x$ is an $\mathcal R$-minimal element of $T$ iff $\forall y \in T: (y \mathrel{\mathcal R} x \implies y = x)$
2. $x$ is an $\mathcal R$-minimal element of $T$ iff $\forall y \in T: y \not\mathrel{\mathcal R} x$

Which of these is more common? Does the other one go by some other name?

The first is compatible with the usual notion of minimality in ordered sets, but the few pages I've found that mention the topic mostly seem to prefer the second.

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They’re the same thing, one expressed for $\le$, the other for $<$. – Brian M. Scott Feb 22 '13 at 20:24
In the second do you mean $\forall y\neq x$? Otherwise it is not a very good definition for reflexive relations... – rschwieb Feb 22 '13 at 20:25
Do texts really define minimality for arbitrary relations? Equality is a relation on $\Bbb N$, and in the sense of the first definition, every element is minimal. This isn't meant to sound like a criticism... I am frankly interested to know if this happens somewhere. – rschwieb Feb 22 '13 at 20:28
@rschwieb Yes, for example for well-founded relations (such as $\in$) in set theory. – Trevor Wilson Feb 22 '13 at 20:37
@TrevorWilson Ah OK! I see how it's natural for well-founded relations. Thanks! – rschwieb Feb 22 '13 at 20:46

In my experience (2) is more common when dealing with arbitrary relations. Under this definition a reflexive relation will have no minimal elements, so if you want to use this definition for posets you will want to talk about $<$-minimal elements instead of $\le$-minimal elements. (Here we use the convention that $<$ denotes a strict partial order and $\le$ denotes the corresponding weak partial order.)
People who only deal with posets and prefer to talk about about $\le$ rather than $<$ would probably prefer definition (1). I would call an $x$ as in (1) "weakly minimal" or "minimal" and call an $x$ as in (2) "minimal", "strictly minimal", or "strongly minimal", but I don't know if this agrees with others' usage.